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Spherical image
In differential geometry, the spherical image of a unit-speed curve is given by taking the curve's tangent vectors as points, all of which must lie on the unit sphere. The movement of the spherical image describes the changes in the original curve's direction[1] If $\alpha $ is a unit-speed curve, that is $\|\alpha ^{\prime }\|=1$, and $T$ is the unit tangent vector field along $\alpha $, then the curve $\sigma =T$ is the spherical image of $\alpha $. All points of $\sigma $ must lie on the unit sphere because $\|\sigma \|=\|T\|=1$.
References
1. O'Neill, B. Elementary Differential Geometry, 1961, pg 71.
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Wikipedia
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Spherical law of cosines
In spherical trigonometry, the law of cosines (also called the cosine rule for sides[1]) is a theorem relating the sides and angles of spherical triangles, analogous to the ordinary law of cosines from plane trigonometry.
Given a unit sphere, a "spherical triangle" on the surface of the sphere is defined by the great circles connecting three points u, v, and w on the sphere (shown at right). If the lengths of these three sides are a (from u to v), b (from u to w), and c (from v to w), and the angle of the corner opposite c is C, then the (first) spherical law of cosines states:[2][1]
$\cos c=\cos a\cos b+\sin a\sin b\cos C\,$
Since this is a unit sphere, the lengths a, b, and c are simply equal to the angles (in radians) subtended by those sides from the center of the sphere. (For a non-unit sphere, the lengths are the subtended angles times the radius, and the formula still holds if a, b and c are reinterpreted as the subtended angles). As a special case, for C = π/2, then cos C = 0, and one obtains the spherical analogue of the Pythagorean theorem:
$\cos c=\cos a\cos b\,$
If the law of cosines is used to solve for c, the necessity of inverting the cosine magnifies rounding errors when c is small. In this case, the alternative formulation of the law of haversines is preferable.[3]
A variation on the law of cosines, the second spherical law of cosines,[4] (also called the cosine rule for angles[1]) states:
$\cos C=-\cos A\cos B+\sin A\sin B\cos c\,$
where A and B are the angles of the corners opposite to sides a and b, respectively. It can be obtained from consideration of a spherical triangle dual to the given one.
Proofs
First proof
Let u, v, and w denote the unit vectors from the center of the sphere to those corners of the triangle. The angles and distances do not change if the coordinate system is rotated, so we can rotate the coordinate system so that $\mathbf {u} $ is at the north pole and $\mathbf {v} $ is somewhere on the prime meridian (longitude of 0). With this rotation, the spherical coordinates for $\mathbf {v} $ are $(r,\theta ,\phi )=(1,a,0),$ where θ is the angle measured from the north pole not from the equator, and the spherical coordinates for $\mathbf {w} $ are $(r,\theta ,\phi )=(1,b,C).$ The Cartesian coordinates for $\mathbf {v} $ are $(x,y,z)=(\sin a,0,\cos a)$ and the Cartesian coordinates for $\mathbf {w} $ are $(x,y,z)=(\sin b\cos C,\sin b\sin C,\cos b).$ The value of $\cos c$ is the dot product of the two Cartesian vectors, which is $\sin a\sin b\cos C+\cos a\cos b.$
Second proof
Let u, v, and w denote the unit vectors from the center of the sphere to those corners of the triangle. We have u · u = 1, v · w = cos c, u · v = cos a, and u · w = cos b. The vectors u × v and u × w have lengths sin a and sin b respectively and the angle between them is C, so
sin a sin b cos C = (u × v) · (u × w) = (u · u)(v · w) − (u · v)(u · w) = cos c − cos a cos b,
using cross products, dot products, and the Binet–Cauchy identity (p × q) · (r × s) = (p · r)(q · s) − (p · s)(q · r).
Third proof
Let u, v, and w denote the unit vectors from the center of the sphere to those corners of the triangle. Consider the following rotational sequence where we first rotate the vector v to u by an angle $a,$ followed another rotation of vector u to w by an angle $b,$ after which we rotate the vector w back to v by an angle $c.$ The composition of these three rotations will form an identity transform. That is, the composite rotation maps the point v to itself. These three rotational operations can be represented by quaternions:
${\begin{aligned}q_{A}&=\cos {\frac {a}{2}}+\mathbf {A} \sin {\frac {a}{2}},\\q_{B}&=\cos {\frac {b}{2}}+\mathbf {B} \sin {\frac {b}{2}},\\q_{C}&=\cos {\frac {c}{2}}+\mathbf {C} \sin {\frac {c}{2}},\end{aligned}}$
where $\mathbf {A} ,$ $\mathbf {B} ,$ and $\mathbf {C} $ are the unit vectors representing the axes of rotations, as defined by the right-hand rule, respectively. The composition of these three rotations is unity, $q_{C}q_{B}q_{A}=1.$ Right multiplying both sides by conjugates $q_{A}^{*}q_{B}^{*},$ we have $q_{C}=q_{A}^{*}q_{B}^{*},$ where $ q_{A}^{*}=\cos {\frac {a}{2}}-\mathbf {A} \sin {\frac {a}{2}}$ and $ q_{B}^{*}=\cos {\frac {b}{2}}-\mathbf {B} \sin {\frac {b}{2}}.$ This gives us the identity[5][6]
$\cos {\frac {c}{2}}+\mathbf {C} \sin {\frac {c}{2}}=\left(\cos {\frac {a}{2}}-\mathbf {A} \sin {\frac {a}{2}}\right)\left(\cos {\frac {b}{2}}-\mathbf {B} \sin {\frac {b}{2}}\right).$
The quaternion product on the right-hand side of this identity is given by
$\left(\cos {\frac {a}{2}}\cos {\frac {b}{2}}-\mathbf {A} \cdot \mathbf {B} \sin {\frac {a}{2}}\sin {\frac {b}{2}}\right)-\left(\mathbf {A} \sin {\frac {a}{2}}\cos {\frac {b}{2}}+\mathbf {B} \cos {\frac {a}{2}}\sin {\frac {b}{2}}-\mathbf {A} \times \mathbf {B} \sin {\frac {a}{2}}\sin {\frac {b}{2}}\right).$
Equating the scalar parts on both sides of the identity, we have
$\cos {\frac {c}{2}}=\cos {\frac {a}{2}}\cos {\frac {b}{2}}-\mathbf {A} \cdot \mathbf {B} \sin {\frac {a}{2}}\sin {\frac {b}{2}}.$
Here $\mathbf {A} \cdot \mathbf {B} =\cos(\pi -C)=-\cos C.$ Since this identity is valid for any arc angles, suppressing the halves, we have
$\cos c=\cos a\cos b+\cos C\sin a\sin b.$
We can also recover the sine law by first noting that $\mathbf {A} \times \mathbf {B} =-\mathbf {u} \sin C$ and then equating the vector parts on both sides of the identity as
$\mathbf {C} \sin {\frac {c}{2}}=-\left(\mathbf {A} \sin {\frac {a}{2}}\cos {\frac {b}{2}}+\mathbf {B} \cos {\frac {a}{2}}\sin {\frac {b}{2}}+\mathbf {u} \sin C\sin {\frac {a}{2}}\sin {\frac {b}{2}}\right).$
The vector $\mathbf {u} $ is orthogonal to both the vectors $\mathbf {A} $ and $\mathbf {B} ,$ and as such $\mathbf {u} \cdot \mathbf {A} =\mathbf {u} \cdot \mathbf {B} =0.$ Taking dot product with respect to $\mathbf {u} $ on both sides, and suppressing the halves, we have $\mathbf {u} \cdot \mathbf {C} \sin c=-\sin C\sin a\sin b.$ Now $\mathbf {v} \times \mathbf {w} =-\mathbf {C} \sin c$ and so we have $\mathbf {u} \cdot (\mathbf {v} \times \mathbf {w} )=-\mathbf {u} \cdot \mathbf {C} \sin c=\sin C\sin a\sin b.$ Dividing each side by $\sin a\sin b\sin c,$ we have
${\frac {\sin C}{\sin c}}={\frac {\mathbf {u} \cdot (\mathbf {w} \times \mathbf {v} )}{\sin a\sin b\sin c}}.$
Since the right-hand side of the above expression is unchanged by cyclic permutation, we have
${\frac {\sin A}{\sin a}}={\frac {\sin B}{\sin b}}={\frac {\sin C}{\sin c}}.$
Rearrangements
The first and second spherical laws of cosines can be rearranged to put the sides (a, b, c) and angles (A, B, C) on opposite sides of the equations:
${\begin{aligned}\cos C&={\frac {\cos c-\cos a\cos b}{\sin a\sin b}}\\\cos c&={\frac {\cos C+\cos A\cos B}{\sin A\sin B}}\\\end{aligned}}$
Planar limit: small angles
For small spherical triangles, i.e. for small a, b, and c, the spherical law of cosines is approximately the same as the ordinary planar law of cosines,
$c^{2}\approx a^{2}+b^{2}-2ab\cos C\,.$
To prove this, we will use the small-angle approximation obtained from the Maclaurin series for the cosine and sine functions:
${\begin{aligned}\cos a&=1-{\frac {a^{2}}{2}}+O\left(a^{4}\right)\\\sin a&=a+O\left(a^{3}\right)\end{aligned}}$
Substituting these expressions into the spherical law of cosines nets:
$1-{\frac {c^{2}}{2}}+O\left(c^{4}\right)=1-{\frac {a^{2}}{2}}-{\frac {b^{2}}{2}}+{\frac {a^{2}b^{2}}{4}}+O\left(a^{4}\right)+O\left(b^{4}\right)+\cos(C)\left(ab+O\left(a^{3}b\right)+O\left(ab^{3}\right)+O\left(a^{3}b^{3}\right)\right)$
or after simplifying:
$c^{2}=a^{2}+b^{2}-2ab\cos C+O\left(c^{4}\right)+O\left(a^{4}\right)+O\left(b^{4}\right)+O\left(a^{2}b^{2}\right)+O\left(a^{3}b\right)+O\left(ab^{3}\right)+O\left(a^{3}b^{3}\right).$
The big O terms for a and b are dominated by O(a4) + O(b4) as a and b get small, so we can write this last expression as:
$c^{2}=a^{2}+b^{2}-2ab\cos C+O\left(a^{4}\right)+O\left(b^{4}\right)+O\left(c^{4}\right).$
History
Something equivalent to the spherical law of cosines was used (but not stated in general) by al-Khwārizmī (9th century), al-Battānī (9th century), and Nīlakaṇṭha (15th century).[7]
See also
• Half-side formula
• Hyperbolic law of cosines
• Solution of triangles
• Spherical law of sines
Notes
1. W. Gellert, S. Gottwald, M. Hellwich, H. Kästner, and H. Küstner, The VNR Concise Encyclopedia of Mathematics, 2nd ed., ch. 12 (Van Nostrand Reinhold: New York, 1989).
2. Romuald Ireneus 'Scibor-Marchocki, Spherical trigonometry, Elementary-Geometry Trigonometry web page (1997).
3. R. W. Sinnott, "Virtues of the Haversine", Sky and Telescope 68 (2), 159 (1984).
4. Reiman, István (1999). Geometria és határterületei. Szalay Könyvkiadó és Kereskedőház Kft. p. 83.
5. Brand, Louis (1947). "§186 Great Circle Arccs". Vector and Tensor Analysis. Wiley. pp. 416–417.
6. Kuipers, Jack B. (1999). "§10 Spherical Trignometry". Quaternions and Rotation Sequences. Princeton University Press. pp. 235–255.
7. Van Brummelen, Glen (2012). Heavenly mathematics: The forgotten art of spherical trigonometry. Princeton University Press. p. 98.
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Wikipedia
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Law of tangents
In trigonometry, the law of tangents or tangent rule[1] is a statement about the relationship between the tangents of two angles of a triangle and the lengths of the opposing sides.
Trigonometry
• Outline
• History
• Usage
• Functions (inverse)
• Generalized trigonometry
Reference
• Identities
• Exact constants
• Tables
• Unit circle
Laws and theorems
• Sines
• Cosines
• Tangents
• Cotangents
• Pythagorean theorem
Calculus
• Trigonometric substitution
• Integrals (inverse functions)
• Derivatives
In Figure 1, a, b, and c are the lengths of the three sides of the triangle, and α, β, and γ are the angles opposite those three respective sides. The law of tangents states that
${\frac {a-b}{a+b}}={\frac {\tan {\tfrac {1}{2}}(\alpha -\beta )}{\tan {\tfrac {1}{2}}(\alpha +\beta )}}.$
The law of tangents, although not as commonly known as the law of sines or the law of cosines, is equivalent to the law of sines, and can be used in any case where two sides and the included angle, or two angles and a side, are known.
Proof
To prove the law of tangents one can start with the law of sines:
${\frac {a}{\sin \alpha }}={\frac {b}{\sin \beta }}.$
Let
$d={\frac {a}{\sin \alpha }}={\frac {b}{\sin \beta }}$
so that
$a=d\sin \alpha \quad {\text{and}}\quad b=d\sin \beta .$
It follows that
${\frac {a-b}{a+b}}={\frac {d\sin \alpha -d\sin \beta }{d\sin \alpha +d\sin \beta }}={\frac {\sin \alpha -\sin \beta }{\sin \alpha +\sin \beta }}.$
Using the trigonometric identity, the factor formula for sines specifically
$\sin \alpha \pm \sin \beta =2\sin {\tfrac {1}{2}}(\alpha \pm \beta )\,\cos {\tfrac {1}{2}}(\alpha \mp \beta ),$
we get
${\frac {a-b}{a+b}}={\frac {2\sin {\tfrac {1}{2}}(\alpha -\beta )\,\cos {\tfrac {1}{2}}(\alpha +\beta )}{2\sin {\tfrac {1}{2}}(\alpha +\beta )\,\cos {\tfrac {1}{2}}(\alpha -\beta )}}={\frac {\sin {\tfrac {1}{2}}(\alpha -\beta )}{\cos {\tfrac {1}{2}}(\alpha -\beta )}}{\Bigg /}{\frac {\sin {\tfrac {1}{2}}(\alpha +\beta )}{\cos {\tfrac {1}{2}}(\alpha +\beta )}}={\frac {\tan {\tfrac {1}{2}}(\alpha -\beta )}{\tan {\tfrac {1}{2}}(\alpha +\beta )}}.$
As an alternative to using the identity for the sum or difference of two sines, one may cite the trigonometric identity
$\tan {\tfrac {1}{2}}(\alpha \pm \beta )={\frac {\sin \alpha \pm \sin \beta }{\cos \alpha +\cos \beta }}$
(see tangent half-angle formula).
Application
The law of tangents can be used to compute the missing side and angles of a triangle in which two sides a and b and the enclosed angle γ are given. From
$\tan {\tfrac {1}{2}}(\alpha -\beta )={\frac {a-b}{a+b}}\tan {\tfrac {1}{2}}(\alpha +\beta )={\frac {a-b}{a+b}}\cot {\tfrac {1}{2}}\gamma $
one can compute α − β; together with α + β = 180° − γ this yields α and β; the remaining side c can then be computed using the law of sines. In the time before electronic calculators were available, this method was preferable to an application of the law of cosines c = √a2 + b2 − 2ab cos γ, as this latter law necessitated an additional lookup in a logarithm table, in order to compute the square root. In modern times the law of tangents may have better numerical properties than the law of cosines: If γ is small, and a and b are almost equal, then an application of the law of cosines leads to a subtraction of almost equal values, incurring catastrophic cancellation.
Spherical version
On a sphere of unit radius, the sides of the triangle are arcs of great circles. Accordingly, their lengths can be expressed in radians or any other units of angular measure. Let A, B, C be the angles at the three vertices of the triangle and let a, b, c be the respective lengths of the opposite sides. The spherical law of tangents says[2]
${\frac {\tan {\tfrac {1}{2}}(A-B)}{\tan {\tfrac {1}{2}}(A+B)}}={\frac {\tan {\tfrac {1}{2}}(a-b)}{\tan {\tfrac {1}{2}}(a+b)}}.$
History
The law of tangents for planar triangles was described in the 11th century by Ibn Muʿādh al-Jayyānī.[3]
The law of tangents for spherical triangles was described in the 13th century by Persian mathematician Nasir al-Din al-Tusi (1201–1274), who also presented the law of sines for plane triangles in his five-volume work Treatise on the Quadrilateral.[3][4]
See also
• Law of sines
• Law of cosines
• Law of cotangents
• Mollweide's formula
• Half-side formula
• Tangent half-angle formula
Notes
1. See Eli Maor, Trigonometric Delights, Princeton University Press, 2002.
2. Daniel Zwillinger, CRC Standard Mathematical Tables and Formulae, 32nd Edition, CRC Press, 2011, page 219.
3. Marie-Thérèse Debarnot (1996). "Trigonometry". In Rushdī Rāshid, Régis Morelon (ed.). Encyclopedia of the history of Arabic science, Volume 2. Routledge. p. 182. ISBN 0-415-12411-5.
4. Q. Mushtaq, JL Berggren (2002). "Trigonometry". In C. E. Bosworth, M.S.Asimov (ed.). History of Civilizations of Central Asia, Volume 4, Part 2. Motilal Banarsidass. p. 190. ISBN 81-208-1596-3.
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Wikipedia
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Spherical lune
In spherical geometry, a spherical lune (or biangle) is an area on a sphere bounded by two half great circles which meet at antipodal points.[1] It is an example of a digon, {2}θ, with dihedral angle θ.[2] The word "lune" derives from luna, the Latin word for Moon.
For the plane geometry region, see Lune (geometry).
This article is about the surface. For the volume, see Spherical wedge.
Properties
Great circles are the largest possible circles (circumferences) of a sphere; each one divides the surface of the sphere into two equal halves. Two great circles always intersect at two polar opposite points.
Common examples of great circles are lines of longitude (meridians) on a sphere, which meet at the north and south poles.
A spherical lune has two planes of symmetry. It can be bisected into two lunes of half the angle, or it can be bisected by an equatorial line into two right spherical triangles.
Surface area
The surface area of a spherical lune is 2θ R2, where R is the radius of the sphere and θ is the dihedral angle in radians between the two half great circles.
When this angle equals 2π radians (360°) — i.e., when the second half great circle has moved a full circle, and the lune in between covers the sphere as a spherical monogon — the area formula for the spherical lune gives 4πR2, the surface area of the sphere.
Examples
A hosohedron is a tessellation of the sphere by lunes. A n-gonal regular hosohedron, {2,n} has n equal lunes of π/n radians. An n-hosohedron has dihedral symmetry Dnh, [n,2], (*22n) of order 4n. Each lune individually has cyclic symmetry C2v, [2], (*22) of order 4.
Each hosohedra can be divided by an equatorial bisector into two equal spherical triangles.
Family of regular hosohedra
n2345678910
Hosohedra
Bipyramidal
tiling
Astronomy
The visibly lighted portion of the Moon visible from the Earth is a spherical lune. The first of the two intersecting great circles is the terminator between the sunlit half of the Moon and the dark half. The second great circle is a terrestrial terminator that separates the half visible from the Earth from the unseen half. The spherical lune is a lighted crescent shape seen from Earth.
n-sphere lunes
Lunes can be defined on higher dimensional spheres as well.
In 4-dimensions a 3-sphere is a generalized sphere. It can contain regular digon lunes as {2}θ,φ, where θ and φ are two dihedral angles.
For example, a regular hosotope {2,p,q} has digon faces, {2}2π/p,2π/q, where its vertex figure is a spherical platonic solid, {p,q}. Each vertex of {p,q} defines an edge in the hosotope and adjacent pairs of those edges define lune faces. Or more specifically, the regular hosotope {2,4,3}, has 2 vertices, 8 180° arc edges in a cube, {4,3}, vertex figure between the two vertices, 12 lune faces, {2}π/4,π/3, between pairs of adjacent edges, and 6 hosohedral cells, {2,p}π/3.
References
1. Davis, Elwyn H. (1999). "Area of spherical triangles". The Mathematics Teacher. 92 (2): 150–153. doi:10.5951/MT.92.2.0150. JSTOR 27970882.
2. Weisstein, Eric W. "Spherical Lune". MathWorld.
• Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, Florida: CRC Press, p. 130, 1987.
• Harris, J. W. and Stocker, H. "Spherical Wedge." §4.8.6 in Handbook of Mathematics and Computational Science. New York: Springer-Verlag, p. 108, 1998.
• Gellert, W.; Gottwald, S.; Hellwich, M.; Kästner, H.; and Künstner, H. (Eds.). VNR Concise Encyclopedia of Mathematics, 2nd ed. New York: Van Nostrand Reinhold, p. 262, 1989.
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Wikipedia
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Spherical mean
In mathematics, the spherical mean of a function around a point is the average of all values of that function on a sphere of given radius centered at that point.
Definition
Consider an open set U in the Euclidean space Rn and a continuous function u defined on U with real or complex values. Let x be a point in U and r > 0 be such that the closed ball B(x, r) of center x and radius r is contained in U. The spherical mean over the sphere of radius r centered at x is defined as
${\frac {1}{\omega _{n-1}(r)}}\int \limits _{\partial B(x,r)}\!u(y)\,\mathrm {d} S(y)$
where ∂B(x, r) is the (n − 1)-sphere forming the boundary of B(x, r), dS denotes integration with respect to spherical measure and ωn−1(r) is the "surface area" of this (n − 1)-sphere.
Equivalently, the spherical mean is given by
${\frac {1}{\omega _{n-1}}}\int \limits _{\|y\|=1}\!u(x+ry)\,\mathrm {d} S(y)$
where ωn−1 is the area of the (n − 1)-sphere of radius 1.
The spherical mean is often denoted as
$\int \limits _{\partial B(x,r)}\!\!\!\!\!\!\!\!\!-\,u(y)\,\mathrm {d} S(y).$
The spherical mean is also defined for Riemannian manifolds in a natural manner.
Properties and uses
• From the continuity of $u$ it follows that the function
$r\to \int \limits _{\partial B(x,r)}\!\!\!\!\!\!\!\!\!-\,u(y)\,\mathrm {d} S(y)$
is continuous, and that its limit as $r\to 0$ is $u(x).$
• Spherical means can be used to solve the Cauchy problem for the wave equation $\partial _{t}^{2}u=c^{2}\,\Delta u$ in odd space dimension. The result, known as Kirchhoff's formula, is derived by using spherical means to reduce the wave equation in $\mathbb {R} ^{n}$ (for odd $n$) to the wave equation in $\mathbb {R} $, and then using d'Alembert's formula. The expression itself is presented in wave equation article.
• If $U$ is an open set in $\mathbb {R} ^{n}$ and $u$ is a C2 function defined on $U$, then $u$ is harmonic if and only if for all $x$ in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): U and all $r>0$ such that the closed ball $B(x,r)$ is contained in $U$ one has
$u(x)=\int \limits _{\partial B(x,r)}\!\!\!\!\!\!\!\!\!-\,u(y)\,\mathrm {d} S(y).$
This result can be used to prove the maximum principle for harmonic functions.
References
• Evans, Lawrence C. (1998). Partial differential equations. American Mathematical Society. ISBN 978-0-8218-0772-9.
• Sabelfeld, K. K.; Shalimova, I. A. (1997). Spherical means for PDEs. VSP. ISBN 978-90-6764-211-8.
• Sunada, Toshikazu (1981). "Spherical means and geodesic chains in a Riemannian manifold". Trans. Am. Math. Soc. 267 (2): 483–501. doi:10.1090/S0002-9947-1981-0626485-6.
External links
• Spherical mean at PlanetMath.
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Spherical measure
In mathematics — specifically, in geometric measure theory — spherical measure σn is the "natural" Borel measure on the n-sphere Sn. Spherical measure is often normalized so that it is a probability measure on the sphere, i.e. so that σn(Sn) = 1.
Definition of spherical measure
There are several ways to define spherical measure. One way is to use the usual "round" or "arclength" metric ρn on Sn; that is, for points x and y in Sn, ρn(x, y) is defined to be the (Euclidean) angle that they subtend at the centre of the sphere (the origin of Rn+1). Now construct n-dimensional Hausdorff measure Hn on the metric space (Sn, ρn) and define
$\sigma ^{n}={\frac {1}{H^{n}(\mathbf {S} ^{n})}}H^{n}.$
One could also have given Sn the metric that it inherits as a subspace of the Euclidean space Rn+1; the same spherical measure results from this choice of metric.
Another method uses Lebesgue measure λn+1 on the ambient Euclidean space Rn+1: for any measurable subset A of Sn, define σn(A) to be the (n + 1)-dimensional volume of the "wedge" in the ball Bn+1 that it subtends at the origin. That is,
$\sigma ^{n}(A):={\frac {1}{\alpha (n+1)}}\lambda ^{n+1}(\{tx\mid x\in A,t\in [0,1]\}),$
where
$\alpha (m):=\lambda ^{m}(\mathbf {B} _{1}^{m}(0)).$
The fact that all these methods define the same measure on Sn follows from an elegant result of Christensen: all these measures are obviously uniformly distributed on Sn, and any two uniformly distributed Borel regular measures on a separable metric space must be constant (positive) multiples of one another. Since all our candidate σn's have been normalized to be probability measures, they are all the same measure.
Relationship with other measures
The relationship of spherical measure to Hausdorff measure on the sphere and Lebesgue measure on the ambient space has already been discussed.
Spherical measure has a nice relationship to Haar measure on the orthogonal group. Let O(n) denote the orthogonal group acting on Rn and let θn denote its normalized Haar measure (so that θn(O(n)) = 1). The orthogonal group also acts on the sphere Sn−1. Then, for any x ∈ Sn−1 and any A ⊆ Sn−1,
$\theta ^{n}(\{g\in \mathrm {O} (n)\mid g(x)\in A\})=\sigma ^{n-1}(A).$
In the case that Sn is a topological group (that is, when n is 0, 1 or 3), spherical measure σn coincides with (normalized) Haar measure on Sn.
Isoperimetric inequality
There is an isoperimetric inequality for the sphere with its usual metric and spherical measure (see Ledoux & Talagrand, chapter 1):
If A ⊆ Sn−1 is any Borel set and B⊆ Sn−1 is a ρn-ball with the same σn-measure as A, then, for any r > 0,
$\sigma ^{n}(A_{r})\geq \sigma ^{n}(B_{r}),$
where Ar denotes the "inflation" of A by r, i.e.
$A_{r}:=\{x\in \mathbf {S} ^{n}\mid \rho _{n}(x,A)\leq r\}.$
In particular, if σn(A) ≥ 1/2 and n ≥ 2, then
$\sigma ^{n}(A_{r})\geq 1-{\sqrt {\frac {\pi }{8}}}\,\exp \left(-{\frac {(n-1)r^{2}}{2}}\right).$
References
• Christensen, Jens Peter Reus (1970). "On some measures analogous to Haar measure". Mathematica Scandinavica. 26: 103–106. ISSN 0025-5521. MR0260979
• Ledoux, Michel; Talagrand, Michel (1991). Probability in Banach spaces. Berlin: Springer-Verlag. pp. xii+480. ISBN 3-540-52013-9. MR1102015 (See chapter 1)
• Mattila, Pertti (1995). Geometry of sets and measures in Euclidean spaces: Fractals and rectifiability. Cambridge Studies in Advanced Mathematics No. 44. Cambridge: Cambridge University Press. pp. xii+343. ISBN 0-521-46576-1. MR1333890 (See chapter 3)
Measure theory
Basic concepts
• Absolute continuity of measures
• Lebesgue integration
• Lp spaces
• Measure
• Measure space
• Probability space
• Measurable space/function
Sets
• Almost everywhere
• Atom
• Baire set
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• equivalence relation
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Types of Measures
• Atomic
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Particular measures
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Applications & related
• Convex analysis
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Wikipedia
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Spherical multipole moments
In physics, spherical multipole moments are the coefficients in a series expansion of a potential that varies inversely with the distance R to a source, i.e., as ${\tfrac {1}{R}}.$ Examples of such potentials are the electric potential, the magnetic potential and the gravitational potential.
For clarity, we illustrate the expansion for a point charge, then generalize to an arbitrary charge density $\rho (\mathbf {r} ').$ Through this article, the primed coordinates such as $\mathbf {r} '$ refer to the position of charge(s), whereas the unprimed coordinates such as $\mathbf {r} $ refer to the point at which the potential is being observed. We also use spherical coordinates throughout, e.g., the vector $\mathbf {r} '$ has coordinates $(r',\theta ',\phi ')$ where $r'$ is the radius, $\theta '$ is the colatitude and $\phi '$ is the azimuthal angle.
Spherical multipole moments of a point charge
The electric potential due to a point charge located at $\mathbf {r'} $ is given by
$\Phi (\mathbf {r} )={\frac {q}{4\pi \varepsilon }}{\frac {1}{R}}={\frac {q}{4\pi \varepsilon }}{\frac {1}{\sqrt {r^{2}+r^{\prime 2}-2r'r\cos \gamma }}}.$
where $R\ {\stackrel {\mathrm {def} }{=}}\ \left|\mathbf {r} -\mathbf {r'} \right|$ is the distance between the charge position and the observation point and $\gamma $ is the angle between the vectors $\mathbf {r} $ and $\mathbf {r'} $. If the radius $r$ of the observation point is greater than the radius $r'$ of the charge, we may factor out 1/r and expand the square root in powers of $(r'/r)<1$ using Legendre polynomials
$\Phi (\mathbf {r} )={\frac {q}{4\pi \varepsilon r}}\sum _{\ell =0}^{\infty }\left({\frac {r'}{r}}\right)^{\ell }P_{\ell }(\cos \gamma )$
This is exactly analogous to the axial multipole expansion.
We may express $\cos \gamma $ in terms of the coordinates of the observation point and charge position using the spherical law of cosines (Fig. 2)
$\cos \gamma =\cos \theta \cos \theta '+\sin \theta \sin \theta '\cos(\phi -\phi ')$
Substituting this equation for $\cos \gamma $ into the Legendre polynomials and factoring the primed and unprimed coordinates yields the important formula known as the spherical harmonic addition theorem
$P_{\ell }(\cos \gamma )={\frac {4\pi }{2\ell +1}}\sum _{m=-\ell }^{\ell }Y_{\ell m}(\theta ,\phi )Y_{\ell m}^{*}(\theta ',\phi ')$
where the $Y_{\ell m}$ functions are the spherical harmonics. Substitution of this formula into the potential yields
$\Phi (\mathbf {r} )={\frac {q}{4\pi \varepsilon r}}\sum _{\ell =0}^{\infty }\left({\frac {r'}{r}}\right)^{\ell }\left({\frac {4\pi }{2\ell +1}}\right)\sum _{m=-\ell }^{\ell }Y_{\ell m}(\theta ,\phi )Y_{\ell m}^{*}(\theta ',\phi ')$
which can be written as
$\Phi (\mathbf {r} )={\frac {1}{4\pi \varepsilon }}\sum _{\ell =0}^{\infty }\sum _{m=-\ell }^{\ell }\left({\frac {Q_{\ell m}}{r^{\ell +1}}}\right){\sqrt {\frac {4\pi }{2\ell +1}}}Y_{\ell m}(\theta ,\phi )$
where the multipole moments are defined
$Q_{\ell m}\ {\stackrel {\mathrm {def} }{=}}\ q\left(r'\right)^{\ell }{\sqrt {\frac {4\pi }{2\ell +1}}}Y_{\ell m}^{*}(\theta ',\phi ').$
As with axial multipole moments, we may also consider the case when the radius $r$ of the observation point is less than the radius $r'$ of the charge. In that case, we may write
$\Phi (\mathbf {r} )={\frac {q}{4\pi \varepsilon r'}}\sum _{\ell =0}^{\infty }\left({\frac {r}{r'}}\right)^{\ell }\left({\frac {4\pi }{2\ell +1}}\right)\sum _{m=-\ell }^{\ell }Y_{\ell m}(\theta ,\phi )Y_{\ell m}^{*}(\theta ',\phi ')$
which can be written as
$\Phi (\mathbf {r} )={\frac {1}{4\pi \varepsilon }}\sum _{\ell =0}^{\infty }\sum _{m=-\ell }^{\ell }I_{\ell m}r^{\ell }{\sqrt {\frac {4\pi }{2\ell +1}}}Y_{\ell m}(\theta ,\phi )$
where the interior spherical multipole moments are defined as the complex conjugate of irregular solid harmonics
$I_{\ell m}\ {\stackrel {\mathrm {def} }{=}}\ {\frac {q}{\left(r'\right)^{\ell +1}}}{\sqrt {\frac {4\pi }{2\ell +1}}}Y_{\ell m}^{*}(\theta ',\phi ')$
The two cases can be subsumed in a single expression if $r_{<}$ and $r_{>}$ are defined to be the lesser and greater, respectively, of the two radii $r$ and $r'$; the potential of a point charge then takes the form, which is sometimes referred to as Laplace expansion
$\Phi (\mathbf {r} )={\frac {q}{4\pi \varepsilon }}\sum _{\ell =0}^{\infty }{\frac {r_{<}^{\ell }}{r_{>}^{\ell +1}}}\left({\frac {4\pi }{2\ell +1}}\right)\sum _{m=-\ell }^{\ell }Y_{\ell m}(\theta ,\phi )Y_{\ell m}^{*}(\theta ',\phi ')$
Exterior spherical multipole moments
It is straightforward to generalize these formulae by replacing the point charge $q$ with an infinitesimal charge element $\rho (\mathbf {r} ')d\mathbf {r} '$ and integrating. The functional form of the expansion is the same. In the exterior case, where $r>r'$, the result is:
$\Phi (\mathbf {r} )={\frac {1}{4\pi \varepsilon }}\sum _{\ell =0}^{\infty }\sum _{m=-\ell }^{\ell }\left({\frac {Q_{\ell m}}{r^{\ell +1}}}\right){\sqrt {\frac {4\pi }{2\ell +1}}}Y_{\ell m}(\theta ,\phi )\,,$
where the general multipole moments are defined
$Q_{\ell m}\ {\stackrel {\mathrm {def} }{=}}\ \int d\mathbf {r} '\rho (\mathbf {r} ')\left(r'\right)^{\ell }{\sqrt {\frac {4\pi }{2\ell +1}}}Y_{\ell m}^{*}(\theta ',\phi ').$
Note
The potential Φ(r) is real, so that the complex conjugate of the expansion is equally valid. Taking of the complex conjugate leads to a definition of the multipole moment which is proportional to Yℓm, not to its complex conjugate. This is a common convention, see molecular multipoles for more on this.
Interior spherical multipole moments
Similarly, the interior multipole expansion has the same functional form. In the interior case, where $r'>r$, the result is:
$\Phi (\mathbf {r} )={\frac {1}{4\pi \varepsilon }}\sum _{\ell =0}^{\infty }\sum _{m=-\ell }^{\ell }I_{\ell m}r^{\ell }{\sqrt {\frac {4\pi }{2\ell +1}}}Y_{\ell m}(\theta ,\phi ),$
with the interior multipole moments defined as
$I_{\ell m}\ {\stackrel {\mathrm {def} }{=}}\ \int d\mathbf {r} '{\frac {\rho (\mathbf {r} ')}{\left(r'\right)^{\ell +1}}}{\sqrt {\frac {4\pi }{2\ell +1}}}Y_{\ell m}^{*}(\theta ',\phi ').$
Interaction energies of spherical multipoles
A simple formula for the interaction energy of two non-overlapping but concentric charge distributions can be derived. Let the first charge distribution $\rho _{1}(\mathbf {r} ')$ be centered on the origin and lie entirely within the second charge distribution $\rho _{2}(\mathbf {r} ')$. The interaction energy between any two static charge distributions is defined by
$U\ {\stackrel {\mathrm {def} }{=}}\ \int d\mathbf {r} \rho _{2}(\mathbf {r} )\Phi _{1}(\mathbf {r} ).$
The potential $\Phi _{1}(\mathbf {r} )$ of the first (central) charge distribution may be expanded in exterior multipoles
$\Phi (\mathbf {r} )={\frac {1}{4\pi \varepsilon }}\sum _{\ell =0}^{\infty }\sum _{m=-\ell }^{\ell }Q_{1\ell m}\left({\frac {1}{r^{\ell +1}}}\right){\sqrt {\frac {4\pi }{2\ell +1}}}Y_{\ell m}(\theta ,\phi )$
where $Q_{1\ell m}$ represents the $\ell m$ exterior multipole moment of the first charge distribution. Substitution of this expansion yields the formula
$U={\frac {1}{4\pi \varepsilon }}\sum _{\ell =0}^{\infty }\sum _{m=-\ell }^{\ell }Q_{1\ell m}\int d\mathbf {r} \ \rho _{2}(\mathbf {r} )\left({\frac {1}{r^{\ell +1}}}\right){\sqrt {\frac {4\pi }{2\ell +1}}}Y_{\ell m}(\theta ,\phi )$
Since the integral equals the complex conjugate of the interior multipole moments $I_{2\ell m}$ of the second (peripheral) charge distribution, the energy formula reduces to the simple form
$U={\frac {1}{4\pi \varepsilon }}\sum _{\ell =0}^{\infty }\sum _{m=-\ell }^{\ell }Q_{1\ell m}I_{2\ell m}^{*}$
For example, this formula may be used to determine the electrostatic interaction energies of the atomic nucleus with its surrounding electronic orbitals. Conversely, given the interaction energies and the interior multipole moments of the electronic orbitals, one may find the exterior multipole moments (and, hence, shape) of the atomic nucleus.
Special case of axial symmetry
The spherical multipole expansion takes a simple form if the charge distribution is axially symmetric (i.e., is independent of the azimuthal angle $\phi '$). By carrying out the $\phi '$ integrations that define $Q_{\ell m}$ and $I_{\ell m}$, it can be shown the multipole moments are all zero except when $m=0$. Using the mathematical identity
$P_{\ell }(\cos \theta )\ {\stackrel {\mathrm {def} }{=}}\ {\sqrt {\frac {4\pi }{2\ell +1}}}Y_{\ell 0}(\theta ,\phi )$
the exterior multipole expansion becomes
$\Phi (\mathbf {r} )={\frac {1}{4\pi \varepsilon }}\sum _{\ell =0}^{\infty }\left({\frac {Q_{\ell }}{r^{\ell +1}}}\right)P_{\ell }(\cos \theta )$
where the axially symmetric multipole moments are defined
$Q_{\ell }\ {\stackrel {\mathrm {def} }{=}}\ \int d\mathbf {r} '\rho (\mathbf {r} ')\left(r'\right)^{\ell }P_{\ell }(\cos \theta ')$
In the limit that the charge is confined to the $z$-axis, we recover the exterior axial multipole moments.
Similarly the interior multipole expansion becomes
$\Phi (\mathbf {r} )={\frac {1}{4\pi \varepsilon }}\sum _{\ell =0}^{\infty }I_{\ell }r^{\ell }P_{\ell }(\cos \theta )$
where the axially symmetric interior multipole moments are defined
$I_{\ell }\ {\stackrel {\mathrm {def} }{=}}\ \int d\mathbf {r} '{\frac {\rho (\mathbf {r} ')}{\left(r'\right)^{\ell +1}}}P_{\ell }(\cos \theta ')$
In the limit that the charge is confined to the $z$-axis, we recover the interior axial multipole moments.
See also
• Solid harmonics
• Laplace expansion
• Multipole expansion
• Legendre polynomials
• Axial multipole moments
• Cylindrical multipole moments
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Wikipedia
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Automorphic number
In mathematics, an automorphic number (sometimes referred to as a circular number) is a natural number in a given number base $b$ whose square "ends" in the same digits as the number itself.
Definition and properties
Given a number base $b$, a natural number $n$ with $k$ digits is an automorphic number if $n$ is a fixed point of the polynomial function $f(x)=x^{2}$ over $\mathbb {Z} /b^{k}\mathbb {Z} $, the ring of integers modulo $b^{k}$. As the inverse limit of $\mathbb {Z} /b^{k}\mathbb {Z} $ is $\mathbb {Z} _{b}$, the ring of $b$-adic integers, automorphic numbers are used to find the numerical representations of the fixed points of $f(x)=x^{2}$ over $\mathbb {Z} _{b}$.
For example, with $b=10$, there are four 10-adic fixed points of $f(x)=x^{2}$, the last 10 digits of which are one of these
$\ldots 0000000000$
$\ldots 0000000001$
$\ldots 8212890625$ (sequence A018247 in the OEIS)
$\ldots 1787109376$ (sequence A018248 in the OEIS)
Thus, the automorphic numbers in base 10 are 0, 1, 5, 6, 25, 76, 376, 625, 9376, 90625, 109376, 890625, 2890625, 7109376, 12890625, 87109376, 212890625, 787109376, 1787109376, 8212890625, 18212890625, 81787109376, 918212890625, 9918212890625, 40081787109376, 59918212890625, ... (sequence A003226 in the OEIS).
A fixed point of $f(x)$ is a zero of the function $g(x)=f(x)-x$. In the ring of integers modulo $b$, there are $2^{\omega (b)}$ zeroes to $g(x)=x^{2}-x$, where the prime omega function $\omega (b)$ is the number of distinct prime factors in $b$. An element $x$ in $\mathbb {Z} /b\mathbb {Z} $ is a zero of $g(x)=x^{2}-x$ if and only if $x\equiv 0{\bmod {p}}^{v_{p}(b)}$ or $x\equiv 1{\bmod {p}}^{v_{p}(b)}$ for all $p|b$. Since there are two possible values in $\lbrace 0,1\rbrace $, and there are $\omega (b)$ such $p|b$, there are $2^{\omega (b)}$ zeroes of $g(x)=x^{2}-x$, and thus there are $2^{\omega (b)}$ fixed points of $f(x)=x^{2}$. According to Hensel's lemma, if there are $k$ zeroes or fixed points of a polynomial function modulo $b$, then there are $k$ corresponding zeroes or fixed points of the same function modulo any power of $b$, and this remains true in the inverse limit. Thus, in any given base $b$ there are $2^{\omega (b)}$ $b$-adic fixed points of $f(x)=x^{2}$.
As 0 is always a zero-divisor, 0 and 1 are always fixed points of $f(x)=x^{2}$, and 0 and 1 are automorphic numbers in every base. These solutions are called trivial automorphic numbers. If $b$ is a prime power, then the ring of $b$-adic numbers has no zero-divisors other than 0, so the only fixed points of $f(x)=x^{2}$ are 0 and 1. As a result, nontrivial automorphic numbers, those other than 0 and 1, only exist when the base $b$ has at least two distinct prime factors.
Automorphic numbers in base b
All $b$-adic numbers are represented in base $b$, using A−Z to represent digit values 10 to 35.
$b$ Prime factors of $b$ Fixed points in $\mathbb {Z} /b\mathbb {Z} $ of $f(x)=x^{2}$ $b$-adic fixed points of $f(x)=x^{2}$ Automorphic numbers in base $b$
62, 30, 1, 3, 4
$\ldots 0000000000$
$\ldots 0000000001$
$\ldots 2221350213$
$\ldots 3334205344$
0, 1, 3, 4, 13, 44, 213, 344, 5344, 50213, 205344, 350213, 1350213, 4205344, 21350213, 34205344, 221350213, 334205344, 2221350213, 3334205344, ...
102, 50, 1, 5, 6
$\ldots 0000000000$
$\ldots 0000000001$
$\ldots 8212890625$
$\ldots 1787109376$
0, 1, 5, 6, 25, 76, 376, 625, 9376, 90625, 109376, 890625, 2890625, 7109376, 12890625, 87109376, 212890625, 787109376, 1787109376, 8212890625, ...
122, 30, 1, 4, 9
$\ldots 0000000000$
$\ldots 0000000001$
$\ldots 21{\text{B}}61{\text{B}}3854$
$\ldots 9{\text{A}}05{\text{A}}08369$
0, 1, 4, 9, 54, 69, 369, 854, 3854, 8369, B3854, 1B3854, A08369, 5A08369, 61B3854, B61B3854, 1B61B3854, A05A08369, 21B61B3854, 9A05A08369, ...
142, 70, 1, 7, 8
$\ldots 0000000000$
$\ldots 0000000001$
$\ldots 7337{\text{A}}{\text{A}}0{\text{C}}37$
$\ldots 6{\text{A}}{\text{A}}633{\text{D}}1{\text{A}}8$
0, 1, 7, 8, 37, A8, 1A8, C37, D1A8, 3D1A8, A0C37, 33D1A8, AA0C37, 633D1A8, 7AA0C37, 37AA0C37, A633D1A8, 337AA0C37, AA633D1A8, 6AA633D1A8, 7337AA0C37, ...
153, 50, 1, 6, 10
$\ldots 0000000000$
$\ldots 0000000001$
$\ldots 624{\text{D}}4{\text{B}}{\text{D}}{\text{A}}86$
$\ldots 8{\text{C}}{\text{A}}1{\text{A}}3146{\text{A}}$
0, 1, 6, A, 6A, 86, 46A, A86, 146A, DA86, 3146A, BDA86, 4BDA86, A3146A, 1A3146A, D4BDA86, 4D4BDA86, A1A3146A, 24D4BDA86, CA1A3146A, 624D4BDA86, 8CA1A3146A, ...
182, 30, 1, 9, 10
...000000
...000001
...4E1249
...D3GFDA
202, 50, 1, 5, 16
...000000
...000001
...1AB6B5
...I98D8G
213, 70, 1, 7, 15
...000000
...000001
...86H7G7
...CE3D4F
222, 110, 1, 11, 12
...000000
...000001
...8D185B
...D8KDGC
242, 30, 1, 9, 16
...000000
...000001
...E4D0L9
...9JAN2G
262, 130, 1, 13, 14
...0000
...0001
...1G6D
...O9JE
282, 70, 1, 8, 21
...0000
...0001
...AAQ8
...HH1L
302, 3, 50, 1, 6, 10, 15, 16, 21, 25
...0000
...0001
...B2J6
...H13A
...1Q7F
...S3MG
...CSQL
...IRAP
333, 110, 1, 12, 22
...0000
...0001
...1KPM
...VC7C
342, 170, 1, 17, 18
...0000
...0001
...248H
...VTPI
355, 70, 1, 15, 21
...0000
...0001
...5MXL
...TC1F
362, 30, 1, 9, 28
...0000
...0001
...DN29
...MCXS
Extensions
Automorphic numbers can be extended to any such polynomial function of degree $n$ $ f(x)=\sum _{i=0}^{n}a_{i}x^{i}$ with b-adic coefficients $a_{i}$. These generalised automorphic numbers form a tree.
a-automorphic numbers
An $a$-automorphic number occurs when the polynomial function is $f(x)=ax^{2}$
For example, with $b=10$ and $a=2$, as there are two fixed points for $f(x)=2x^{2}$ in $\mathbb {Z} /10\mathbb {Z} $ ($x=0$ and $x=8$), according to Hensel's lemma there are two 10-adic fixed points for $f(x)=2x^{2}$,
$\ldots 0000000000$
$\ldots 0893554688$
so the 2-automorphic numbers in base 10 are 0, 8, 88, 688, 4688...
Trimorphic numbers
A trimorphic number or spherical number occurs when the polynomial function is $f(x)=x^{3}$.[1] All automorphic numbers are trimorphic. The terms circular and spherical were formerly used for the slightly different case of a number whose powers all have the same last digit as the number itself.[2]
For base $b=10$, the trimorphic numbers are:
0, 1, 4, 5, 6, 9, 24, 25, 49, 51, 75, 76, 99, 125, 249, 251, 375, 376, 499, 501, 624, 625, 749, 751, 875, 999, 1249, 3751, 4375, 4999, 5001, 5625, 6249, 8751, 9375, 9376, 9999, ... (sequence A033819 in the OEIS)
For base $b=12$, the trimorphic numbers are:
0, 1, 3, 4, 5, 7, 8, 9, B, 15, 47, 53, 54, 5B, 61, 68, 69, 75, A7, B3, BB, 115, 253, 368, 369, 4A7, 5BB, 601, 715, 853, 854, 969, AA7, BBB, 14A7, 2369, 3853, 3854, 4715, 5BBB, 6001, 74A7, 8368, 8369, 9853, A715, BBBB, ...
Programming example
def hensels_lemma(polynomial_function, base: int, power: int):
"""Hensel's lemma."""
if power == 0:
return [0]
if power > 0:
roots = hensels_lemma(polynomial_function, base, power - 1)
new_roots = []
for root in roots:
for i in range(0, base):
new_i = i * base ** (power - 1) + root
new_root = polynomial_function(new_i) % pow(base, power)
if new_root == 0:
new_roots.append(new_i)
return new_roots
base = 10
digits = 10
def automorphic_polynomial(x):
return x ** 2 - x
for i in range(1, digits + 1):
print(hensels_lemma(automorphic_polynomial, base, i))
See also
• Arithmetic dynamics
• Kaprekar number
• p-adic number
• p-adic analysis
• Zero-divisor
References
1. See Gérard Michon's article at
2. "spherical number". Oxford English Dictionary (Online ed.). Oxford University Press. (Subscription or participating institution membership required.)
• examples of 1-automorphic numbers at PlanetMath.
External links
• Weisstein, Eric W. "Automorphic number". MathWorld.
• Weisstein, Eric W. "Trimorphic Number". MathWorld.
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Spherical geometry
Spherical geometry or spherics (from Ancient Greek σφαιρικά) is the geometry of the two-dimensional surface of a sphere[lower-alpha 1] or the n-dimensional surface of higher dimensional spheres.
Geometry
Projecting a sphere to a plane
• Outline
• History (Timeline)
Branches
• Euclidean
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• Elliptic
• Spherical
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• Non-Archimedean geometry
• Projective
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Long studied for its practical applications to astronomy, navigation, and geodesy, spherical geometry and the metrical tools of spherical trigonometry are in many respects analogous to Euclidean plane geometry and trigonometry, but also have some important differences.
The sphere can be studied either extrinsically as a surface embedded in 3-dimensional Euclidean space (part of the study of solid geometry), or intrinsically using methods that only involve the surface itself without reference to any surrounding space.
Principles
In plane (Euclidean) geometry, the basic concepts are points and (straight) lines. In spherical geometry, the basic concepts are point and great circle. However, two great circles on a plane intersect in two antipodal points, unlike coplanar lines in Elliptic geometry.
In the extrinsic 3-dimensional picture, a great circle is the intersection of the sphere with any plane through the center. In the intrinsic approach, a great circle is a geodesic; a shortest path between any two of its points provided they are close enough. Or, in the (also intrinsic) axiomatic approach analogous to Euclid's axioms of plane geometry, "great circle" is simply an undefined term, together with postulates stipulating the basic relationships between great circles and the also-undefined "points". This is the same as Euclid's method of treating point and line as undefined primitive notions and axiomatizing their relationships.
Great circles in many ways play the same logical role in spherical geometry as lines in Euclidean geometry, e.g., as the sides of (spherical) triangles. This is more than an analogy; spherical and plane geometry and others can all be unified under the umbrella of geometry built from distance measurement, where "lines" are defined to mean shortest paths (geodesics). Many statements about the geometry of points and such "lines" are equally true in all those geometries provided lines are defined that way, and the theory can be readily extended to higher dimensions. Nevertheless, because its applications and pedagogy are tied to solid geometry, and because the generalization loses some important properties of lines in the plane, spherical geometry ordinarily does not use the term "line" at all to refer to anything on the sphere itself. If developed as a part of solid geometry, use is made of points, straight lines and planes (in the Euclidean sense) in the surrounding space.
In spherical geometry, angles are defined between great circles, resulting in a spherical trigonometry that differs from ordinary trigonometry in many respects; for example, the sum of the interior angles of a spherical triangle exceeds 180 degrees.
Relation to similar geometries
Because a sphere and a plane differ geometrically, (intrinsic) spherical geometry has some features of a non-Euclidean geometry and is sometimes described as being one. However, spherical geometry was not considered a full-fledged non-Euclidean geometry sufficient to resolve the ancient problem of whether the parallel postulate is a logical consequence of the rest of Euclid's axioms of plane geometry, because it requires another axiom to be modified. The resolution was found instead in elliptic geometry, to which spherical geometry is closely related, and hyperbolic geometry; each of these new geometries makes a different change to the parallel postulate.
The principles of any of these geometries can be extended to any number of dimensions.
An important geometry related to that of the sphere is that of the real projective plane; it is obtained by identifying antipodal points (pairs of opposite points) on the sphere. Locally, the projective plane has all the properties of spherical geometry, but it has different global properties. In particular, it is non-orientable, or one-sided, and unlike the sphere it cannot be drawn as a surface in 3-dimensional space without intersecting itself.
Concepts of spherical geometry may also be applied to the oblong sphere, though minor modifications must be implemented on certain formulas.
History
Greek antiquity
The earliest mathematical work of antiquity to come down to our time is On the rotating sphere (Περὶ κινουμένης σφαίρας, Peri kinoumenes sphairas) by Autolycus of Pitane, who lived at the end of the fourth century BC.[1]
Spherical trigonometry was studied by early Greek mathematicians such as Theodosius of Bithynia, a Greek astronomer and mathematician who wrote the Sphaerics, a book on the geometry of the sphere,[2] and Menelaus of Alexandria, who wrote a book on spherical trigonometry called Sphaerica and developed Menelaus' theorem.[3][4]
Islamic world
See also: Mathematics in medieval Islam
The Book of Unknown Arcs of a Sphere written by the Islamic mathematician Al-Jayyani is considered to be the first treatise on spherical trigonometry. The book contains formulae for right-handed triangles, the general law of sines, and the solution of a spherical triangle by means of the polar triangle.[5]
The book On Triangles by Regiomontanus, written around 1463, is the first pure trigonometrical work in Europe. However, Gerolamo Cardano noted a century later that much of its material on spherical trigonometry was taken from the twelfth-century work of the Andalusi scholar Jabir ibn Aflah.[6]
Euler's work
Leonhard Euler published a series of important memoirs on spherical geometry:
• L. Euler, Principes de la trigonométrie sphérique tirés de la méthode des plus grands et des plus petits, Mémoires de l'Académie des Sciences de Berlin 9 (1753), 1755, p. 233–257; Opera Omnia, Series 1, vol. XXVII, p. 277–308.
• L. Euler, Eléments de la trigonométrie sphéroïdique tirés de la méthode des plus grands et des plus petits, Mémoires de l'Académie des Sciences de Berlin 9 (1754), 1755, p. 258–293; Opera Omnia, Series 1, vol. XXVII, p. 309–339.
• L. Euler, De curva rectificabili in superficie sphaerica, Novi Commentarii academiae scientiarum Petropolitanae 15, 1771, pp. 195–216; Opera Omnia, Series 1, Volume 28, pp. 142–160.
• L. Euler, De mensura angulorum solidorum, Acta academiae scientiarum imperialis Petropolitinae 2, 1781, p. 31–54; Opera Omnia, Series 1, vol. XXVI, p. 204–223.
• L. Euler, Problematis cuiusdam Pappi Alexandrini constructio, Acta academiae scientiarum imperialis Petropolitinae 4, 1783, p. 91–96; Opera Omnia, Series 1, vol. XXVI, p. 237–242.
• L. Euler, Geometrica et sphaerica quaedam, Mémoires de l'Académie des Sciences de Saint-Pétersbourg 5, 1815, p. 96–114; Opera Omnia, Series 1, vol. XXVI, p. 344–358.
• L. Euler, Trigonometria sphaerica universa, ex primis principiis breviter et dilucide derivata, Acta academiae scientiarum imperialis Petropolitinae 3, 1782, p. 72–86; Opera Omnia, Series 1, vol. XXVI, p. 224–236.
• L. Euler, Variae speculationes super area triangulorum sphaericorum, Nova Acta academiae scientiarum imperialis Petropolitinae 10, 1797, p. 47–62; Opera Omnia, Series 1, vol. XXIX, p. 253–266.
Properties
Spherical geometry has the following properties:[7]
• Any two great circles intersect in two diametrically opposite points, called antipodal points.
• Any two points that are not antipodal points determine a unique great circle.
• There is a natural unit of angle measurement (based on a revolution), a natural unit of length (based on the circumference of a great circle) and a natural unit of area (based on the area of the sphere).
• Each great circle is associated with a pair of antipodal points, called its poles which are the common intersections of the set of great circles perpendicular to it. This shows that a great circle is, with respect to distance measurement on the surface of the sphere, a circle: the locus of points all at a specific distance from a center.
• Each point is associated with a unique great circle, called the polar circle of the point, which is the great circle on the plane through the centre of the sphere and perpendicular to the diameter of the sphere through the given point.
As there are two arcs determined by a pair of points, which are not antipodal, on the great circle they determine, three non-collinear points do not determine a unique triangle. However, if we only consider triangles whose sides are minor arcs of great circles, we have the following properties:
• The angle sum of a triangle is greater than 180° and less than 540°.
• The area of a triangle is proportional to the excess of its angle sum over 180°.
• Two triangles with the same angle sum are equal in area.
• There is an upper bound for the area of triangles.
• The composition (product) of two reflections-across-a-great-circle may be considered as a rotation about either of the points of intersection of their axes.
• Two triangles are congruent if and only if they correspond under a finite product of such reflections.
• Two triangles with corresponding angles equal are congruent (i.e., all similar triangles are congruent).
Relation to Euclid's postulates
If "line" is taken to mean great circle, spherical geometry obeys two of Euclid's postulates: the second postulate ("to produce [extend] a finite straight line continuously in a straight line") and the fourth postulate ("that all right angles are equal to one another"). However, it violates the other three. Contrary to the first postulate ("that between any two points, there is a unique line segment joining them"), there is not a unique shortest route between any two points (antipodal points such as the north and south poles on a spherical globe are counterexamples); contrary to the third postulate, a sphere does not contain circles of arbitrarily great radius; and contrary to the fifth (parallel) postulate, there is no point through which a line can be drawn that never intersects a given line.[8]
A statement that is equivalent to the parallel postulate is that there exists a triangle whose angles add up to 180°. Since spherical geometry violates the parallel postulate, there exists no such triangle on the surface of a sphere. The sum of the angles of a triangle on a sphere is 180°(1 + 4f), where f is the fraction of the sphere's surface that is enclosed by the triangle. For any positive value of f, this exceeds 180°.
See also
• Spherical astronomy
• Spherical conic
• Spherical distance
• Spherical polyhedron
• Half-side formula
• Lénárt sphere
• Versor
Notes
1. In this context the word "sphere" refers only to the 2-dimensional surface and other terms like "ball" (or "solid sphere") are used for the surface together with its 3-dimensional interior.
References
1. Rosenfeld, B.A (1988). A history of non-Euclidean geometry : evolution of the concept of a geometric space. New York: Springer-Verlag. p. 2. ISBN 0-387-96458-4.
2. "Theodosius of Bithynia – Dictionary definition of Theodosius of Bithynia". HighBeam Research. Retrieved 25 March 2015.
3. O'Connor, John J.; Robertson, Edmund F., "Menelaus of Alexandria", MacTutor History of Mathematics Archive, University of St Andrews
4. "Menelaus of Alexandria Facts, information, pictures". HighBeam Research. Retrieved 25 March 2015.
5. School of Mathematical and Computational Sciences University of St Andrews
6. "Victor J. Katz-Princeton University Press". Archived from the original on 2016-10-01. Retrieved 2009-03-01.
7. Merserve, pp. 281-282 harvnb error: no target: CITEREFMerserve (help)
8. Gowers, Timothy, Mathematics: A Very Short Introduction, Oxford University Press, 2002: pp. 94 and 98.
Further reading
• Meserve, Bruce E. (1983) [1959], Fundamental Concepts of Geometry, Dover, ISBN 0-486-63415-9
• Papadopoulos, Athanase (2015), Euler, la géométrie sphérique et le calcul des variations. In: Leonhard Euler : Mathématicien, physicien et théoricien de la musique (dir. X. Hascher et A. Papadopoulos), CNRS Editions, Paris, ISBN 978-2-271-08331-9
• Van Brummelen, Glen (2013). Heavenly Mathematics: The Forgotten Art of Spherical Trigonometry. Princeton University Press. ISBN 9780691148922. Retrieved 31 December 2014.
• Roshdi Rashed and Athanase Papadopoulos (2017) Menelaus' Spherics: Early Translation and al-Mahani'/alHarawi's version. Critical edition of Menelaus' Spherics from the Arabic manuscripts, with historical and mathematical commentaries, De Gruyter Series: Scientia Graeco-Arabica 21 ISBN 978-3-11-057142-4
External links
Wikimedia Commons has media related to Spherical geometry.
• The Geometry of the Sphere Archived 2011-06-21 at the Wayback Machine Rice University
• Weisstein, Eric W. "Spherical Geometry". MathWorld.
• Navigation Spreadsheets: Navigation Triangles
• Sphaerica - geometry software for constructing on the sphere
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Spherical sector
In geometry, a spherical sector,[1] also known as a spherical cone,[2] is a portion of a sphere or of a ball defined by a conical boundary with apex at the center of the sphere. It can be described as the union of a spherical cap and the cone formed by the center of the sphere and the base of the cap. It is the three-dimensional analogue of the sector of a circle.
Volume
If the radius of the sphere is denoted by r and the height of the cap by h, the volume of the spherical sector is
$V={\frac {2\pi r^{2}h}{3}}\,.$
This may also be written as
$V={\frac {2\pi r^{3}}{3}}(1-\cos \varphi )\,,$
where φ is half the cone angle, i.e., φ is the angle between the rim of the cap and the direction to the middle of the cap as seen from the sphere center.
The volume V of the sector is related to the area A of the cap by:
$V={\frac {rA}{3}}\,.$
Area
The curved surface area of the spherical sector (on the surface of the sphere, excluding the cone surface) is
$A=2\pi rh\,.$
It is also
$A=\Omega r^{2}$
where Ω is the solid angle of the spherical sector in steradians, the SI unit of solid angle. One steradian is defined as the solid angle subtended by a cap area of A = r2.
Derivation
Further information: double integral and triple integral
The volume can be calculated by integrating the differential volume element
$dV=\rho ^{2}\sin \phi \,d\rho \,d\phi \,d\theta $
over the volume of the spherical sector,
$V=\int _{0}^{2\pi }\int _{0}^{\varphi }\int _{0}^{r}\rho ^{2}\sin \phi \,d\rho \,d\phi \,d\theta =\int _{0}^{2\pi }d\theta \int _{0}^{\varphi }\sin \phi \,d\phi \int _{0}^{r}\rho ^{2}d\rho ={\frac {2\pi r^{3}}{3}}(1-\cos \varphi )\,,$
where the integrals have been separated, because the integrand can be separated into a product of functions each with one dummy variable.
The area can be similarly calculated by integrating the differential spherical area element
$dA=r^{2}\sin \phi \,d\phi \,d\theta $
over the spherical sector, giving
$A=\int _{0}^{2\pi }\int _{0}^{\varphi }r^{2}\sin \phi \,d\phi \,d\theta =r^{2}\int _{0}^{2\pi }d\theta \int _{0}^{\varphi }\sin \phi \,d\phi =2\pi r^{2}(1-\cos \varphi )\,,$
where φ is inclination (or elevation) and θ is azimuth (right). Notice r is a constant. Again, the integrals can be separated.
See also
• Circular sector — the analogous 2D figure.
• Spherical cap
• Spherical segment
• Spherical wedge
References
1. Weisstein, Eric W. "Spherical sector". MathWorld.
2. Weisstein, Eric W. "Spherical cone". MathWorld.
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Spherical shell
In geometry, a spherical shell is a generalization of an annulus to three dimensions. It is the region of a ball between two concentric spheres of differing radii.[1]
Volume
The volume of a spherical shell is the difference between the enclosed volume of the outer sphere and the enclosed volume of the inner sphere:
$V={\frac {4}{3}}\pi R^{3}-{\frac {4}{3}}\pi r^{3}$
$V={\frac {4}{3}}\pi \left(R^{3}-r^{3}\right)$
$V={\frac {4}{3}}\pi (R-r)\left(R^{2}+Rr+r^{2}\right)$
where r is the radius of the inner sphere and R is the radius of the outer sphere.
Approximation
An approximation for the volume of a thin spherical shell is the surface area of the inner sphere multiplied by the thickness t of the shell:[2]
$V\approx 4\pi r^{2}t,$
when t is very small compared to r ($t\ll r$).
The total surface area of the spherical shell is $4\pi r^{2}$.
See also
• Spherical pressure vessel
• Ball
• Solid torus
• Bubble
• Sphere
References
1. W., Weisstein, Eric. "Spherical Shell". mathworld.wolfram.com. Wolfram Research, Inc. Archived from the original on 2 August 2016. Retrieved 7 January 2017.{{cite web}}: CS1 maint: multiple names: authors list (link)
2. Znamenski, Andrey Varlamov, Lev Aslamazov; scientific editor, A.A. Abrikosov, Jr. ; translators, A.A. Abrikosov, Jr., J. Vydryg, & D. (2012). The wonders of physics (3rd ed.). Singapore: World Scientific. p. 78. ISBN 978-9814374156. Archived from the original on 20 December 2017. Retrieved 7 January 2017. {{cite book}}: |first1= has generic name (help)CS1 maint: multiple names: authors list (link)
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Wikipedia
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Spherical space form conjecture
In geometric topology, the spherical space form conjecture (now a theorem) states that a finite group acting on the 3-sphere is conjugate to a group of isometries of the 3-sphere.
Spherical space form conjecture
FieldGeometric topology
Conjectured byHeinz Hopf
Conjectured in1926
First proof byGrigori Perelman
First proof in2006
Implied byGeometrization conjecture
Equivalent toPoincaré conjecture
Thurston elliptization conjecture
History
The conjecture was posed by Heinz Hopf in 1926 after determining the fundamental groups of three-dimensional spherical space forms as a generalization of the Poincaré conjecture to the non-simply connected case.[1][2]
Status
The conjecture is implied by Thurston's geometrization conjecture, which was proven by Grigori Perelman in 2003. The conjecture was independently proven for groups whose actions have fixed points—this special case is known as the Smith conjecture. It is also proven for various groups acting without fixed points, such as cyclic groups whose orders are a power of two (George Livesay, Robert Myers) and cyclic groups of order 3 (J. Hyam Rubinstein).[3]
See also
• Killing–Hopf theorem
References
1. Hopf, Heinz (1926), "Zum Clifford-Kleinschen Raumproblem", Mathematische Annalen, 95 (1): 313–339, doi:10.1007/BF01206614
2. Hambleton, Ian (2015), "Topological spherical space forms", Handbook of Group Actions, Clay Math. Proc., vol. 3, Beijing-Boston: ALM, pp. 151–172
3. Hass, Joel (2005), "Minimal surfaces and the topology of three-manifolds", Global theory of minimal surfaces, Clay Math. Proc., vol. 2, Providence, R.I.: Amer. Math. Soc., pp. 705–724, MR 2167285
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Spherical design
A spherical design, part of combinatorial design theory in mathematics, is a finite set of N points on the d-dimensional unit d-sphere Sd such that the average value of any polynomial f of degree t or less on the set equals the average value of f on the whole sphere (that is, the integral of f over Sd divided by the area or measure of Sd). Such a set is often called a spherical t-design to indicate the value of t, which is a fundamental parameter. The concept of a spherical design is due to Delsarte, Goethals, and Seidel,[1] although these objects were understood as particular examples of cubature formulas earlier.
Spherical designs can be of value in approximation theory, in statistics for experimental design, in combinatorics, and in geometry. The main problem is to find examples, given d and t, that are not too large; however, such examples may be hard to come by. Spherical t-designs have also recently been appropriated in quantum mechanics in the form of quantum t-designs with various applications to quantum information theory and quantum computing.
Existence of spherical designs
The existence and structure of spherical designs on the circle were studied in depth by Hong.[2] Shortly thereafter, Seymour and Zaslavsky[3] proved that such designs exist of all sufficiently large sizes; that is, given positive integers d and t, there is a number N(d,t) such that for every N ≥ N(d,t) there exists a spherical t-design of N points in dimension d. However, their proof gave no idea of how big N(d,t) is.
Mimura constructively found conditions in terms of the number of points and the dimension which characterize exactly when spherical 2-designs exist. Maximally sized collections of equiangular lines (up to identification of lines as antipodal points on the sphere) are examples of minimal sized spherical 5-designs. There are many sporadic small spherical designs; many of them are related to finite group actions on the sphere.
In 2013, Bondarenko, Radchenko, and Viazovska[4] obtained the asymptotic upper bound $N(d,t)<C_{d}t^{d}$ for all positive integers d and t. This asymptotically matches the lower bound given originally by Delsarte, Goethals, and Seidel. The value of Cd is currently unknown, while exact values of $N(d,t)$ are known in relatively few cases.
See also
• Thomson problem
External links
• Spherical t-designs for different values of N and t can be found precomputed at Neil Sloane's website.
Notes
1. Delsarte, Goethals & Seidel 1977.
2. Hong 1982.
3. Seymour & Zaslavsky 1984.
4. Bondarenko, Radchenko & Viazovska 2013.
References
• Bondarenko, Andriy; Radchenko, Danylo; Viazovska, Maryna (2013), "Optimal asymptotic bounds for spherical designs", Annals of Mathematics, Second Series, 178 (2): 443–452, arXiv:1009.4407, doi:10.4007/annals.2013.178.2.2, MR 3071504, S2CID 2490453.
• Mimura, Yoshio (1990), "A construction of spherical 2-design", Graphs and Combinatorics, 6 (4): 369–372, doi:10.1007/BF01787704, S2CID 28942727.
• Delsarte, P.; Goethals, J. M.; Seidel, J. J. (1977), "Spherical codes and designs", Geometriae Dedicata, 6 (3): 363–388, doi:10.1007/BF03187604, MR 0485471, S2CID 125833142. Reprinted in Seidel, J. J. (1991), Geometry and combinatorics: Selected works of J. J. Seidel, Boston, MA: Academic Press, Inc., ISBN 0-12-189420-7, MR 1116326.
• Hong, Yiming (1982), "On spherical t-designs in R2", European Journal of Combinatorics, 3 (3): 255–258, doi:10.1016/S0195-6698(82)80036-X, MR 0679209.
• Seymour, P. D.; Zaslavsky, Thomas (1984), "Averaging sets: a generalization of mean values and spherical designs", Advances in Mathematics, 52 (3): 213–240, doi:10.1016/0001-8708(84)90022-7, MR 0744857.
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Spherical basis
In pure and applied mathematics, particularly quantum mechanics and computer graphics and their applications, a spherical basis is the basis used to express spherical tensors. The spherical basis closely relates to the description of angular momentum in quantum mechanics and spherical harmonic functions.
"Spherical tensor" redirects here. For the concept related to operators, see tensor operator.
While spherical polar coordinates are one orthogonal coordinate system for expressing vectors and tensors using polar and azimuthal angles and radial distance, the spherical basis are constructed from the standard basis and use complex numbers.
In three dimensions
A vector A in 3D Euclidean space R3 can be expressed in the familiar Cartesian coordinate system in the standard basis ex, ey, ez, and coordinates Ax, Ay, Az:
$\mathbf {A} =A_{x}\mathbf {e} _{x}+A_{y}\mathbf {e} _{y}+A_{z}\mathbf {e} _{z}$
(1)
or any other coordinate system with associated basis set of vectors. From this extend the scalars to allow multiplication by complex numbers, so that we are now working in $\mathbb {C} ^{3}$ rather than $\mathbb {R} ^{3}$.
Basis definition
In the spherical bases denoted e+, e−, e0, and associated coordinates with respect to this basis, denoted A+, A−, A0, the vector A is:
$\mathbf {A} =A_{+}\mathbf {e} _{+}+A_{-}\mathbf {e} _{-}+A_{0}\mathbf {e} _{0}$
(2)
where the spherical basis vectors can be defined in terms of the Cartesian basis using complex-valued coefficients in the xy plane:[1]
${\begin{aligned}\mathbf {e} _{+}&=-{\frac {1}{\sqrt {2}}}\mathbf {e} _{x}-{\frac {i}{\sqrt {2}}}\mathbf {e} _{y}\\\mathbf {e} _{-}&=+{\frac {1}{\sqrt {2}}}\mathbf {e} _{x}-{\frac {i}{\sqrt {2}}}\mathbf {e} _{y}\\\end{aligned}}\quad \rightleftharpoons \quad \mathbf {e} _{\pm }=\mp {\frac {1}{\sqrt {2}}}\left(\mathbf {e} _{x}\pm i\mathbf {e} _{y}\right)\,$
(3A)
in which $i$ denotes the imaginary unit, and one normal to the plane in the z direction:
$\mathbf {e} _{0}=\mathbf {e} _{z}$
The inverse relations are:
${\begin{aligned}\mathbf {e} _{x}&=-{\frac {1}{\sqrt {2}}}\mathbf {e} _{+}+{\frac {1}{\sqrt {2}}}\mathbf {e_{-}} \\\mathbf {e} _{y}&=+{\frac {i}{\sqrt {2}}}\mathbf {e} _{+}+{\frac {i}{\sqrt {2}}}\mathbf {e_{-}} \\\mathbf {e} _{z}&=\mathbf {e} _{0}\end{aligned}}$
(3B)
Commutator definition
While giving a basis in a 3-dimensional space is a valid definition for a spherical tensor, it only covers the case for when the rank $k$ is 1. For higher ranks, one may use either the commutator, or rotation definition of a spherical tensor. The commutator definition is given below, any operator $T_{q}^{(k)}$ that satisfies the following relations is a spherical tensor:
$[J_{\pm },T_{q}^{(k)}]=\hbar {\sqrt {(k\mp q)(k\pm q+1)}}T_{q\pm 1}^{(k)}$
$[J_{z},T_{q}^{(k)}]=\hbar qT_{q}^{(k)}$
Rotation definition
Analogously to how the spherical harmonics transform under a rotation, a general spherical tensor transforms as follows, when the states transform under the unitary Wigner D-matrix ${\mathcal {D}}(R)$, where R is a (3×3 rotation) group element in SO(3). That is, these matrices represent the rotation group elements. With the help of its Lie algebra, one can show these two definitions are equivalent.
${\mathcal {D}}(R)T_{q}^{(k)}{\mathcal {D}}^{\dagger }(R)=\sum _{q'=-k}^{k}T_{q'}^{(k)}{\mathcal {D}}_{q'q}^{(k)}$
See also: Wigner D-matrix
Coordinate vectors
Main article: Coordinate vector
For the spherical basis, the coordinates are complex-valued numbers A+, A0, A−, and can be found by substitution of (3B) into (1), or directly calculated from the inner product ⟨, ⟩ (5):
${\begin{aligned}A_{+}&=\left\langle \mathbf {A} ,\mathbf {e} _{+}\right\rangle =-{\frac {A_{x}}{\sqrt {2}}}+{\frac {iA_{y}}{\sqrt {2}}}\\A_{-}&=\left\langle \mathbf {A} ,\mathbf {e} _{-}\right\rangle =+{\frac {A_{x}}{\sqrt {2}}}+{\frac {iA_{y}}{\sqrt {2}}}\\\end{aligned}}\quad \rightleftharpoons \quad A_{\pm }=\left\langle \mathbf {e} _{\pm },\mathbf {A} \right\rangle ={\frac {1}{\sqrt {2}}}\left(\mp A_{x}+iA_{y}\right)$
(4A)
$A_{0}=\left\langle \mathbf {e} _{0},\mathbf {A} \right\rangle =\left\langle \mathbf {e} _{z},\mathbf {A} \right\rangle =A_{z}$
with inverse relations:
${\begin{aligned}A_{x}&=-{\frac {1}{\sqrt {2}}}A_{+}+{\frac {1}{\sqrt {2}}}A_{-}\\A_{y}&=-{\frac {i}{\sqrt {2}}}A_{+}-{\frac {i}{\sqrt {2}}}A_{-}\\A_{z}&=A_{0}\end{aligned}}$
(4B)
In general, for two vectors with complex coefficients in the same real-valued orthonormal basis ei, with the property ei·ej = δij, the inner product is:
$\left\langle \mathbf {a} ,\mathbf {b} \right\rangle =\mathbf {a} \cdot \mathbf {b} ^{\star }=\sum _{j}a_{j}b_{j}^{\star }$
(5)
where · is the usual dot product and the complex conjugate * must be used to keep the magnitude (or "norm") of the vector positive definite.
Properties (three dimensions)
Orthonormality
The spherical basis is an orthonormal basis, since the inner product ⟨, ⟩ (5) of every pair vanishes meaning the basis vectors are all mutually orthogonal:
$\left\langle \mathbf {e} _{+},\mathbf {e} _{-}\right\rangle =\left\langle \mathbf {e} _{-},\mathbf {e} _{0}\right\rangle =\left\langle \mathbf {e} _{0},\mathbf {e} _{+}\right\rangle =0$
and each basis vector is a unit vector:
$\left\langle \mathbf {e} _{+},\mathbf {e} _{+}\right\rangle =\left\langle \mathbf {e} _{-},\mathbf {e} _{-}\right\rangle =\left\langle \mathbf {e} _{0},\mathbf {e} _{0}\right\rangle =1$
hence the need for the normalizing factors of $1/\!{\sqrt {2}}$.
Change of basis matrix
See also: change of basis
The defining relations (3A) can be summarized by a transformation matrix U:
${\begin{pmatrix}\mathbf {e} _{+}\\\mathbf {e} _{-}\\\mathbf {e} _{0}\end{pmatrix}}=\mathbf {U} {\begin{pmatrix}\mathbf {e} _{x}\\\mathbf {e} _{y}\\\mathbf {e} _{z}\end{pmatrix}}\,,\quad \mathbf {U} ={\begin{pmatrix}-{\frac {1}{\sqrt {2}}}&-{\frac {i}{\sqrt {2}}}&0\\+{\frac {1}{\sqrt {2}}}&-{\frac {i}{\sqrt {2}}}&0\\0&0&1\end{pmatrix}}\,,$
with inverse:
${\begin{pmatrix}\mathbf {e} _{x}\\\mathbf {e} _{y}\\\mathbf {e} _{z}\end{pmatrix}}=\mathbf {U} ^{-1}{\begin{pmatrix}\mathbf {e} _{+}\\\mathbf {e} _{-}\\\mathbf {e} _{0}\end{pmatrix}}\,,\quad \mathbf {U} ^{-1}={\begin{pmatrix}-{\frac {1}{\sqrt {2}}}&+{\frac {1}{\sqrt {2}}}&0\\+{\frac {i}{\sqrt {2}}}&+{\frac {i}{\sqrt {2}}}&0\\0&0&1\end{pmatrix}}\,.$
It can be seen that U is a unitary matrix, in other words its Hermitian conjugate U† (complex conjugate and matrix transpose) is also the inverse matrix U−1.
For the coordinates:
${\begin{pmatrix}A_{+}\\A_{-}\\A_{0}\end{pmatrix}}=\mathbf {U} ^{\mathrm {*} }{\begin{pmatrix}A_{x}\\A_{y}\\A_{z}\end{pmatrix}}\,,\quad \mathbf {U} ^{\mathrm {*} }={\begin{pmatrix}-{\frac {1}{\sqrt {2}}}&+{\frac {i}{\sqrt {2}}}&0\\+{\frac {1}{\sqrt {2}}}&+{\frac {i}{\sqrt {2}}}&0\\0&0&1\end{pmatrix}}\,,$
and inverse:
${\begin{pmatrix}A_{x}\\A_{y}\\A_{z}\end{pmatrix}}=(\mathbf {U} ^{\mathrm {*} })^{-1}{\begin{pmatrix}A_{+}\\A_{-}\\A_{0}\end{pmatrix}}\,,\quad (\mathbf {U} ^{\mathrm {*} })^{-1}={\begin{pmatrix}-{\frac {1}{\sqrt {2}}}&+{\frac {1}{\sqrt {2}}}&0\\-{\frac {i}{\sqrt {2}}}&-{\frac {i}{\sqrt {2}}}&0\\0&0&1\end{pmatrix}}\,.$
Cross products
Taking cross products of the spherical basis vectors, we find an obvious relation:
$\mathbf {e} _{q}\times \mathbf {e} _{q}={\boldsymbol {0}}$
where q is a placeholder for +, −, 0, and two less obvious relations:
$\mathbf {e} _{\pm }\times \mathbf {e} _{\mp }=\pm i\mathbf {e} _{0}$
$\mathbf {e} _{\pm }\times \mathbf {e} _{0}=\pm i\mathbf {e} _{\pm }$
Inner product in the spherical basis
The inner product between two vectors A and B in the spherical basis follows from the above definition of the inner product:
$\left\langle \mathbf {A} ,\mathbf {B} \right\rangle =A_{+}B_{+}^{\star }+A_{-}B_{-}^{\star }+A_{0}B_{0}^{\star }$
See also
• Wigner–Eckart theorem
• Wigner D matrix
• 3D rotation group
References
1. W.J. Thompson (2008). Angular Momentum. John Wiley & Sons. p. 311. ISBN 9783527617838.
General
• S. S. M. Wong (2008). Introductory Nuclear Physics (2nd ed.). John Wiley & Sons. ISBN 978-35-276-179-13.
External links
Tensors
Glossary of tensor theory
Scope
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• Coordinate system
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Notation
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Tensor
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Operations
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Related
abstractions
• Affine connection
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Notable tensors
Mathematics
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Wikipedia
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Spherical polyhedron
In geometry, a spherical polyhedron or spherical tiling is a tiling of the sphere in which the surface is divided or partitioned by great arcs into bounded regions called spherical polygons. Much of the theory of symmetrical polyhedra is most conveniently derived in this way.
The most familiar spherical polyhedron is the soccer ball, thought of as a spherical truncated icosahedron. The next most popular spherical polyhedron is the beach ball, thought of as a hosohedron.
Some "improper" polyhedra, such as hosohedra and their duals, dihedra, exist as spherical polyhedra, but their flat-faced analogs are degenerate. The example hexagonal beach ball, {2, 6}, is a hosohedron, and {6, 2} is its dual dihedron.
History
The first known man-made polyhedra are spherical polyhedra carved in stone. Many have been found in Scotland, and appear to date from the neolithic period (the New Stone Age).
During the 10th Century, the Islamic scholar Abū al-Wafā' Būzjānī (Abu'l Wafa) wrote the first serious study of spherical polyhedra.
Two hundred years ago, at the start of the 19th Century, Poinsot used spherical polyhedra to discover the four regular star polyhedra.
In the middle of the 20th Century, Coxeter used them to enumerate all but one of the uniform polyhedra, through the construction of kaleidoscopes (Wythoff construction).
Examples
All regular polyhedra, semiregular polyhedra, and their duals can be projected onto the sphere as tilings:
Schläfli
symbol
{p,q} t{p,q} r{p,q} t{q,p} {q,p} rr{p,q} tr{p,q} sr{p,q}
Vertex
configuration
pq q.2p.2p p.q.p.q p.2q.2q qp q.4.p.4 4.2q.2p 3.3.q.3.p
Tetrahedral
symmetry
(3 3 2)
33
3.6.6
3.3.3.3
3.6.6
33
3.4.3.4
4.6.6
3.3.3.3.3
V3.6.6
V3.3.3.3
V3.6.6
V3.4.3.4
V4.6.6
V3.3.3.3.3
Octahedral
symmetry
(4 3 2)
43
3.8.8
3.4.3.4
4.6.6
34
3.4.4.4
4.6.8
3.3.3.3.4
V3.8.8
V3.4.3.4
V4.6.6
V3.4.4.4
V4.6.8
V3.3.3.3.4
Icosahedral
symmetry
(5 3 2)
53
3.10.10
3.5.3.5
5.6.6
35
3.4.5.4
4.6.10
3.3.3.3.5
V3.10.10
V3.5.3.5
V5.6.6
V3.4.5.4
V4.6.10
V3.3.3.3.5
Dihedral
example p=6
(2 2 6)
62
2.12.12
2.6.2.6
6.4.4
26
2.4.6.4
4.4.12
3.3.3.6
n 2 3 4 5 6 7 8 10 ...
n-Prism
(2 2 p)
...
n-Bipyramid
(2 2 p)
...
n-Antiprism ...
n-Trapezohedron ...
Improper cases
Spherical tilings allow cases that polyhedra do not, namely hosohedra: figures as {2,n}, and dihedra: figures as {n,2}. Generally, regular hosohedra and regular dihedra are used.
Family of regular hosohedra · *n22 symmetry mutations of regular hosohedral tilings: nn
Space SphericalEuclidean
Tiling name (Monogonal)
Henagonal hosohedron
Digonal hosohedron (Triangular)
Trigonal hosohedron
(Tetragonal)
Square hosohedron
Pentagonal hosohedron Hexagonal hosohedron Heptagonal hosohedron Octagonal hosohedron Enneagonal hosohedron Decagonal hosohedron Hendecagonal hosohedron Dodecagonal hosohedron ... Apeirogonal hosohedron
Tiling image ...
Schläfli symbol {2,1}{2,2}{2,3}{2,4}{2,5}{2,6}{2,7}{2,8}{2,9}{2,10}{2,11}{2,12}...{2,∞}
Coxeter diagram ...
Faces and edges 123456789101112...∞
Vertices 2...2
Vertex config. 22.223242526272829210211212...2∞
Family of regular dihedra · *n22 symmetry mutations of regular dihedral tilings: nn
Space SphericalEuclidean
Tiling name (Hengonal)
Monogonal dihedron
Digonal dihedron (Triangular)
Trigonal dihedron
(Tetragonal)
Square dihedron
Pentagonal dihedron Hexagonal dihedron ... Apeirogonal dihedron
Tiling image ...
Schläfli symbol {1,2}{2,2}{3,2}{4,2}{5,2}{6,2}...{∞,2}
Coxeter diagram ...
Faces 2 {1}2 {2}2 {3}2 {4}2 {5}2 {6}...2 {∞}
Edges and vertices 123456...∞
Vertex config. 1.12.23.34.45.56.6...∞.∞
Relation to tilings of the projective plane
Spherical polyhedra having at least one inversive symmetry are related to projective polyhedra[1] (tessellations of the real projective plane) – just as the sphere has a 2-to-1 covering map of the projective plane, projective polyhedra correspond under 2-fold cover to spherical polyhedra that are symmetric under reflection through the origin.
The best-known examples of projective polyhedra are the regular projective polyhedra, the quotients of the centrally symmetric Platonic solids, as well as two infinite classes of even dihedra and hosohedra:[2]
• Hemi-cube, {4,3}/2
• Hemi-octahedron, {3,4}/2
• Hemi-dodecahedron, {5,3}/2
• Hemi-icosahedron, {3,5}/2
• Hemi-dihedron, {2p,2}/2, p>=1
• Hemi-hosohedron, {2,2p}/2, p>=1
See also
Wikimedia Commons has media related to Spherical polyhedra.
• Spherical geometry
• Spherical trigonometry
• Polyhedron
• Projective polyhedron
• Toroidal polyhedron
• Conway polyhedron notation
References
1. McMullen, Peter; Schulte, Egon (2002). "6C. Projective Regular Polytopes". Abstract Regular Polytopes. Cambridge University Press. pp. 162–5. ISBN 0-521-81496-0.
2. Coxeter, H.S.M. (1969). "§21.3 Regular maps'". Introduction to Geometry (2nd ed.). Wiley. pp. 386–8. ISBN 978-0-471-50458-0. MR 0123930.
Further reading
• Poinsot, L. (1810). "Memoire sur les polygones et polyèdres". J. De l'École Polytechnique. 9: 16–48.
• Coxeter, H.S.M.; Longuet-Higgins, M.S.; Miller, J.C.P. (1954). "Uniform polyhedra". Phil. Trans. 246 A (916): 401–50. JSTOR 91532.
• Coxeter, H.S.M. (1973). Regular Polytopes (3rd ed.). Dover. ISBN 0-486-61480-8.
Tessellation
Periodic
• Pythagorean
• Rhombille
• Schwarz triangle
• Rectangle
• Domino
• Uniform tiling and honeycomb
• Coloring
• Convex
• Kisrhombille
• Wallpaper group
• Wythoff
Aperiodic
• Ammann–Beenker
• Aperiodic set of prototiles
• List
• Einstein problem
• Socolar–Taylor
• Gilbert
• Penrose
• Pentagonal
• Pinwheel
• Quaquaversal
• Rep-tile and Self-tiling
• Sphinx
• Socolar
• Truchet
Other
• Anisohedral and Isohedral
• Architectonic and catoptric
• Circle Limit III
• Computer graphics
• Honeycomb
• Isotoxal
• List
• Packing
• Problems
• Domino
• Wang
• Heesch's
• Squaring
• Dividing a square into similar rectangles
• Prototile
• Conway criterion
• Girih
• Regular Division of the Plane
• Regular grid
• Substitution
• Voronoi
• Voderberg
By vertex type
Spherical
• 2n
• 33.n
• V33.n
• 42.n
• V42.n
Regular
• 2∞
• 36
• 44
• 63
Semi-
regular
• 32.4.3.4
• V32.4.3.4
• 33.42
• 33.∞
• 34.6
• V34.6
• 3.4.6.4
• (3.6)2
• 3.122
• 42.∞
• 4.6.12
• 4.82
Hyper-
bolic
• 32.4.3.5
• 32.4.3.6
• 32.4.3.7
• 32.4.3.8
• 32.4.3.∞
• 32.5.3.5
• 32.5.3.6
• 32.6.3.6
• 32.6.3.8
• 32.7.3.7
• 32.8.3.8
• 33.4.3.4
• 32.∞.3.∞
• 34.7
• 34.8
• 34.∞
• 35.4
• 37
• 38
• 3∞
• (3.4)3
• (3.4)4
• 3.4.62.4
• 3.4.7.4
• 3.4.8.4
• 3.4.∞.4
• 3.6.4.6
• (3.7)2
• (3.8)2
• 3.142
• 3.162
• (3.∞)2
• 3.∞2
• 42.5.4
• 42.6.4
• 42.7.4
• 42.8.4
• 42.∞.4
• 45
• 46
• 47
• 48
• 4∞
• (4.5)2
• (4.6)2
• 4.6.12
• 4.6.14
• V4.6.14
• 4.6.16
• V4.6.16
• 4.6.∞
• (4.7)2
• (4.8)2
• 4.8.10
• V4.8.10
• 4.8.12
• 4.8.14
• 4.8.16
• 4.8.∞
• 4.102
• 4.10.12
• 4.122
• 4.12.16
• 4.142
• 4.162
• 4.∞2
• (4.∞)2
• 54
• 55
• 56
• 5∞
• 5.4.6.4
• (5.6)2
• 5.82
• 5.102
• 5.122
• (5.∞)2
• 64
• 65
• 66
• 68
• 6.4.8.4
• (6.8)2
• 6.82
• 6.102
• 6.122
• 6.162
• 73
• 74
• 77
• 7.62
• 7.82
• 7.142
• 83
• 84
• 86
• 88
• 8.62
• 8.122
• 8.162
• ∞3
• ∞4
• ∞5
• ∞∞
• ∞.62
• ∞.82
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Wikipedia
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Plancherel theorem for spherical functions
In mathematics, the Plancherel theorem for spherical functions is an important result in the representation theory of semisimple Lie groups, due in its final form to Harish-Chandra. It is a natural generalisation in non-commutative harmonic analysis of the Plancherel formula and Fourier inversion formula in the representation theory of the group of real numbers in classical harmonic analysis and has a similarly close interconnection with the theory of differential equations. It is the special case for zonal spherical functions of the general Plancherel theorem for semisimple Lie groups, also proved by Harish-Chandra. The Plancherel theorem gives the eigenfunction expansion of radial functions for the Laplacian operator on the associated symmetric space X; it also gives the direct integral decomposition into irreducible representations of the regular representation on L2(X). In the case of hyperbolic space, these expansions were known from prior results of Mehler, Weyl and Fock.
The main reference for almost all this material is the encyclopedic text of Helgason (1984).
History
The first versions of an abstract Plancherel formula for the Fourier transform on a unimodular locally compact group G were due to Segal and Mautner.[1] At around the same time, Harish-Chandra[2][3] and Gelfand & Naimark[4][5] derived an explicit formula for SL(2,R) and complex semisimple Lie groups, so in particular the Lorentz groups. A simpler abstract formula was derived by Mautner for a "topological" symmetric space G/K corresponding to a maximal compact subgroup K. Godement gave a more concrete and satisfactory form for positive definite spherical functions, a class of special functions on G/K. Since when G is a semisimple Lie group these spherical functions φλ were naturally labelled by a parameter λ in the quotient of a Euclidean space by the action of a finite reflection group, it became a central problem to determine explicitly the Plancherel measure in terms of this parametrization. Generalizing the ideas of Hermann Weyl from the spectral theory of ordinary differential equations, Harish-Chandra[6][7] introduced his celebrated c-function c(λ) to describe the asymptotic behaviour of the spherical functions φλ and proposed c(λ)−2 dλ as the Plancherel measure. He verified this formula for the special cases when G is complex or real rank one, thus in particular covering the case when G/K is a hyperbolic space. The general case was reduced to two conjectures about the properties of the c-function and the so-called spherical Fourier transform. Explicit formulas for the c-function were later obtained for a large class of classical semisimple Lie groups by Bhanu-Murthy. In turn these formulas prompted Gindikin and Karpelevich to derive a product formula[8] for the c-function, reducing the computation to Harish-Chandra's formula for the rank 1 case. Their work finally enabled Harish-Chandra to complete his proof of the Plancherel theorem for spherical functions in 1966.[9]
In many special cases, for example for complex semisimple group or the Lorentz groups, there are simple methods to develop the theory directly. Certain subgroups of these groups can be treated by techniques generalising the well-known "method of descent" due to Jacques Hadamard. In particular Flensted-Jensen (1978) gave a general method for deducing properties of the spherical transform for a real semisimple group from that of its complexification.
One of the principal applications and motivations for the spherical transform was Selberg's trace formula. The classical Poisson summation formula combines the Fourier inversion formula on a vector group with summation over a cocompact lattice. In Selberg's analogue of this formula, the vector group is replaced by G/K, the Fourier transform by the spherical transform and the lattice by a cocompact (or cofinite) discrete subgroup. The original paper of Selberg (1956) implicitly invokes the spherical transform; it was Godement (1957) who brought the transform to the fore, giving in particular an elementary treatment for SL(2,R) along the lines sketched by Selberg.
Spherical functions
Main article: Zonal spherical function
Let G be a semisimple Lie group and K a maximal compact subgroup of G. The Hecke algebra Cc(K \G/K), consisting of compactly supported K-biinvariant continuous functions on G, acts by convolution on the Hilbert space H=L2(G / K). Because G / K is a symmetric space, this *-algebra is commutative. The closure of its (the Hecke algebra's) image in the operator norm is a non-unital commutative C* algebra ${\mathfrak {A}}$, so by the Gelfand isomorphism can be identified with the continuous functions vanishing at infinity on its spectrum X.[10] Points in the spectrum are given by continuous *-homomorphisms of ${\mathfrak {A}}$ into C, i.e. characters of ${\mathfrak {A}}$.
If S' denotes the commutant of a set of operators S on H, then ${\mathfrak {A}}^{\prime }$ can be identified with the commutant of the regular representation of G on H. Now ${\mathfrak {A}}$ leaves invariant the subspace H0 of K-invariant vectors in H. Moreover, the abelian von Neumann algebra it generates on H0 is maximal Abelian. By spectral theory, there is an essentially unique[11] measure μ on the locally compact space X and a unitary transformation U between H0 and L2(X, μ) which carries the operators in ${\mathfrak {A}}$ onto the corresponding multiplication operators.
The transformation U is called the spherical Fourier transform or sometimes just the spherical transform and μ is called the Plancherel measure. The Hilbert space H0 can be identified with L2(K\G/K), the space of K-biinvariant square integrable functions on G.
The characters χλ of ${\mathfrak {A}}$ (i.e. the points of X) can be described by positive definite spherical functions φλ on G, via the formula
$\chi _{\lambda }(\pi (f))=\int _{G}f(g)\cdot \varphi _{\lambda }(g)\,dg.$
for f in Cc(K\G/K), where π(f) denotes the convolution operator in ${\mathfrak {A}}$ and the integral is with respect to Haar measure on G.
The spherical functions φλ on G are given by Harish-Chandra's formula:
$\varphi _{\lambda }(g)=\int _{K}\lambda ^{\prime }(gk)^{-1}\,dk.$
In this formula:
• the integral is with respect to Haar measure on K;
• λ is an element of A* =Hom(A,T) where A is the Abelian vector subgroup in the Iwasawa decomposition G =KAN of G;
• λ' is defined on G by first extending λ to a character of the solvable subgroup AN, using the group homomorphism onto A, and then setting
$\lambda '(kx)=\Delta _{AN}(x)^{1/2}\lambda (x)$
for k in K and x in AN, where ΔAN is the modular function of AN.
• Two different characters λ1 and λ2 give the same spherical function if and only if λ1 = λ2·s, where s is in the Weyl group of A
$W=N_{K}(A)/C_{K}(A),$
the quotient of the normaliser of A in K by its centraliser, a finite reflection group.
It follows that
• X can be identified with the quotient space A*/W.
Spherical principal series
The spherical function φλ can be identified with the matrix coefficient of the spherical principal series of G. If M is the centralizer of A in K, this is defined as the unitary representation πλ of G induced by the character of B = MAN given by the composition of the homomorphism of MAN onto A and the character λ. The induced representation is defined on functions f on G with
$f(gb)=\Delta (b)^{1/2}\lambda (b)f(g)$
for b in B by
$\pi (g)f(x)=f(g^{-1}x),$
where
$\|f\|^{2}=\int _{K}|f(k)|^{2}\,dk<\infty .$
The functions f can be identified with functions in L2(K / M) and
$\chi _{\lambda }(g)=(\pi (g)1,1).$
As Kostant (1969) proved, the representations of the spherical principal series are irreducible and two representations πλ and πμ are unitarily equivalent if and only if μ = σ(λ) for some σ in the Weyl group of A.
Example: SL(2, C)
The group G = SL(2,C) acts transitively on the quaternionic upper half space
${\mathfrak {H}}^{3}=\{x+yi+tj\mid t>0\}$
by Möbius transformations. The complex matrix
$g={\begin{pmatrix}a&b\\c&d\end{pmatrix}}$
acts as
$g(w)=(aw+b)(cw+d)^{-1}.$
The stabiliser of the point j is the maximal compact subgroup K = SU(2), so that ${\mathfrak {H}}^{3}=G/K.$ It carries the G-invariant Riemannian metric
$ds^{2}=r^{-2}\left(dx^{2}+dy^{2}+dr^{2}\right)$
with associated volume element
$dV=r^{-3}\,dx\,dy\,dr$
and Laplacian operator
$\Delta =-r^{2}(\partial _{x}^{2}+\partial _{y}^{2}+\partial _{r}^{2})+r\partial _{r}.$
Every point in ${\mathfrak {H}}^{3}$ can be written as k(etj) with k in SU(2) and t determined up to a sign. The Laplacian has the following form on functions invariant under SU(2), regarded as functions of the real parameter t:
$\Delta =-\partial _{t}^{2}-2\coth t\partial _{t}.$
The integral of an SU(2)-invariant function is given by
$\int f\,dV=\int _{-\infty }^{\infty }f(t)\,\sinh ^{2}t\,dt.$
Identifying the square integrable SU(2)-invariant functions with L2(R) by the unitary transformation Uf(t) = f(t) sinh t, Δ is transformed into the operator
$U^{*}\Delta U=-{d^{2} \over dt^{2}}+1.$
By the Plancherel theorem and Fourier inversion formula for R, any SU(2)-invariant function f can be expressed in terms of the spherical functions
$\Phi _{\lambda }(t)={\sin \lambda t \over \lambda \sinh t},$
by the spherical transform
${\tilde {f}}(\lambda )=\int f\Phi _{-\lambda }\,dV$
and the spherical inversion formula
$f(x)=\int {\tilde {f}}(\lambda )\Phi _{\lambda }(x)\lambda ^{2}\,d\lambda .$
Taking $f=f_{2}^{*}\star f_{1}$ with fi in Cc(G / K) and $f^{*}(g)={\overline {f(g^{-1})}}$, and evaluating at i yields the Plancherel formula
$\int _{G}f_{1}{\overline {f_{2}}}\,dg=\int {\tilde {f}}_{1}(\lambda ){\overline {{\tilde {f}}_{2}(\lambda )}}\,\lambda ^{2}\,d\lambda .$
For biinvariant functions this establishes the Plancherel theorem for spherical functions: the map
${\begin{cases}U:L^{2}(K\backslash G/K)\to L^{2}(\mathbb {R} ,\lambda ^{2}\,d\lambda )\\U:f\longmapsto {\tilde {f}}\end{cases}}$
is unitary and sends the convolution operator defined by $f\in L^{1}(K\backslash G/K)$ into the multiplication operator defined by ${\tilde {f}}$.
The spherical function Φλ is an eigenfunction of the Laplacian:
$\Delta \Phi _{\lambda }=(\lambda ^{2}+1)\Phi _{\lambda }.$
Schwartz functions on R are the spherical transforms of functions f belonging to the Harish-Chandra Schwartz space
${\mathcal {S}}=\left\{f\left|\sup _{t}\left|(1+t^{2})^{N}(I+\Delta )^{M}f(t)\sinh(t)\right|<\infty \right\}\right..$
By the Paley-Wiener theorem, the spherical transforms of smooth SU(2)-invariant functions of compact support are precisely functions on R which are restrictions of holomorphic functions on C satisfying an exponential growth condition
$|F(\lambda )|\leq Ce^{R\left|\operatorname {Im} \lambda \right|}.$
As a function on G, Φλ is the matrix coefficient of the spherical principal series defined on L2(C), where C is identified with the boundary of ${\mathfrak {H}}^{3}$. The representation is given by the formula
$\pi _{\lambda }(g^{-1})\xi (z)=|cz+d|^{-2-i\lambda }\xi (g(z)).$
The function
$\xi _{0}(z)=\pi ^{-1}\left(1+|z|^{2}\right)^{-2}$
is fixed by SU(2) and
$\Phi _{\lambda }(g)=(\pi _{\lambda }(g)\xi _{0},\xi _{0}).$
The representations πλ are irreducible and unitarily equivalent only when the sign of λ is changed. The map W of $L^{2}({\mathfrak {H}}^{3})$ onto L2([0,∞) × C) (with measure λ2 dλ on the first factor) given by
$Wf(\lambda ,z)=\int _{G/K}f(g)\pi _{\lambda }(g)\xi _{0}(z)\,dg$
is unitary and gives the decomposition of $L^{2}({\mathfrak {H}}^{3})$ as a direct integral of the spherical principal series.
Example: SL(2, R)
The group G = SL(2,R) acts transitively on the Poincaré upper half plane
${\mathfrak {H}}^{2}=\{x+ri\mid r>0\}$
by Möbius transformations. The real matrix
$g={\begin{pmatrix}a&b\\c&d\end{pmatrix}}$
acts as
$g(w)=(aw+b)(cw+d)^{-1}.$
The stabiliser of the point i is the maximal compact subgroup K = SO(2), so that ${\mathfrak {H}}^{2}$ = G / K. It carries the G-invariant Riemannian metric
$ds^{2}=r^{-2}\left(dx^{2}+dr^{2}\right)$
with associated area element
$dA=r^{-2}\,dx\,dr$
and Laplacian operator
$\Delta =-r^{2}(\partial _{x}^{2}+\partial _{r}^{2}).$
Every point in ${\mathfrak {H}}^{2}$ can be written as k( et i ) with k in SO(2) and t determined up to a sign. The Laplacian has the following form on functions invariant under SO(2), regarded as functions of the real parameter t:
$\Delta =-\partial _{t}^{2}-\coth t\partial _{t}.$
The integral of an SO(2)-invariant function is given by
$\int f\,dA=\int _{-\infty }^{\infty }f(t)\left|\sinh t\right|dt.$
There are several methods for deriving the corresponding eigenfunction expansion for this ordinary differential equation including:
1. the classical spectral theory of ordinary differential equations applied to the hypergeometric equation (Mehler, Weyl, Fock);
2. variants of Hadamard's method of descent, realising 2-dimensional hyperbolic space as the quotient of 3-dimensional hyperbolic space by the free action of a 1-parameter subgroup of SL(2,C);
3. Abel's integral equation, following Selberg and Godement;
4. orbital integrals (Harish-Chandra, Gelfand & Naimark).
The second and third technique will be described below, with two different methods of descent: the classical one due Hadamard, familiar from treatments of the heat equation[12] and the wave equation[13] on hyperbolic space; and Flensted-Jensen's method on the hyperboloid.
Hadamard's method of descent
If f(x,r) is a function on ${\mathfrak {H}}^{2}$ and
$M_{1}f(x,y,r)=r^{1/2}\cdot f(x,r)$
then
$\Delta _{3}M_{1}f=M_{1}\left(\Delta _{2}+{\tfrac {3}{4}}\right)f,$
where Δn is the Laplacian on ${\mathfrak {H}}^{n}$.
Since the action of SL(2,C) commutes with Δ3, the operator M0 on S0(2)-invariant functions obtained by averaging M1f by the action of SU(2) also satisfies
$\Delta _{3}M_{0}=M_{0}\left(\Delta _{2}+{\tfrac {3}{4}}\right).$
The adjoint operator M1* defined by
$M_{1}^{*}F(x,r)=r^{1/2}\int _{-\infty }^{\infty }F(x,y,r)\,dy$
satisfies
$\int _{{\mathfrak {H}}^{3}}(M_{1}f)\cdot F\,dV=\int _{{\mathfrak {H}}^{2}}f\cdot (M_{1}^{*}F)\,dA.$
The adjoint M0*, defined by averaging M*f over SO(2), satisfies
$\int _{{\mathfrak {H}}^{3}}(M_{0}f)\cdot F\,dV=\int _{{\mathfrak {H}}^{2}}f\cdot (M_{0}^{*}F)\,dA$
for SU(2)-invariant functions F and SO(2)-invariant functions f. It follows that
$M_{i}^{*}\Delta _{3}=\left(\Delta _{2}+{\tfrac {3}{4}}\right)M_{i}^{*}.$
The function
$f_{\lambda }=M_{1}^{*}\Phi _{\lambda }$
is SO(2)-invariant and satisfies
$\Delta _{2}f_{\lambda }=\left(\lambda ^{2}+{\tfrac {1}{4}}\right)f_{\lambda }.$
On the other hand,
$b(\lambda )=f_{\lambda }(i)=\int {\sin \lambda t \over \lambda \sinh t}\,dt={\pi \over \lambda }\tanh {\pi \lambda \over 2},$
since the integral can be computed by integrating $e^{i\lambda t}/\sinh t$ around the rectangular indented contour with vertices at ±R and ±R + πi. Thus the eigenfunction
$\phi _{\lambda }=b(\lambda )^{-1}M_{1}\Phi _{\lambda }$
satisfies the normalisation condition φλ(i) = 1. There can only be one such solution either because the Wronskian of the ordinary differential equation must vanish or by expanding as a power series in sinh r.[14] It follows that
$\varphi _{\lambda }(e^{t}i)={\frac {1}{2\pi }}\int _{0}^{2\pi }\left(\cosh t-\sinh t\cos \theta \right)^{-1-i\lambda }\,d\theta .$
Similarly it follows that
$\Phi _{\lambda }=M_{1}\phi _{\lambda }.$
If the spherical transform of an SO(2)-invariant function on ${\mathfrak {H}}^{2}$ is defined by
${\tilde {f}}(\lambda )=\int f\varphi _{-\lambda }\,dA,$
then
${(M_{1}^{*}F)}^{\sim }(\lambda )={\tilde {F}}(\lambda ).$
Taking f=M1*F, the SL(2, C) inversion formula for F immediately yields
$f(x)=\int _{-\infty }^{\infty }\varphi _{\lambda }(x){\tilde {f}}(\lambda ){\lambda \pi \over 2}\tanh \left({\pi \lambda \over 2}\right)\,d\lambda ,$
the spherical inversion formula for SO(2)-invariant functions on ${\mathfrak {H}}^{2}$.
As for SL(2,C), this immediately implies the Plancherel formula for fi in Cc(SL(2,R) / SO(2)):
$\int _{{\mathfrak {H}}^{2}}f_{1}{\overline {f_{2}}}\,dA=\int _{-\infty }^{\infty }{\tilde {f}}_{1}{\overline {\tilde {f_{2}}}}{\lambda \pi \over 2}\tanh \left({\pi \lambda \over 2}\right)\,d\lambda .$
The spherical function φλ is an eigenfunction of the Laplacian:
$\Delta _{2}\varphi _{\lambda }=\left(\lambda ^{2}+{\tfrac {1}{4}}\right)\varphi _{\lambda }.$
Schwartz functions on R are the spherical transforms of functions f belonging to the Harish-Chandra Schwartz space
${\mathcal {S}}=\left\{f\left|\sup _{t}\left|(1+t^{2})^{N}(I+\Delta )^{M}f(t)\varphi _{0}(t)\right|<\infty \right\}\right..$
The spherical transforms of smooth SO(2)-invariant functions of compact support are precisely functions on R which are restrictions of holomorphic functions on C satisfying an exponential growth condition
$|F(\lambda )|\leq Ce^{R|\Im \lambda |}.$
Both these results can be deduced by descent from the corresponding results for SL(2,C),[15] by verifying directly that the spherical transform satisfies the given growth conditions[16][17] and then using the relation $(M_{1}^{*}F)^{\sim }={\tilde {F}}$.
As a function on G, φλ is the matrix coefficient of the spherical principal series defined on L2(R), where R is identified with the boundary of ${\mathfrak {H}}^{2}$. The representation is given by the formula
$\pi _{\lambda }(g^{-1})\xi (x)=|cx+d|^{-1-i\lambda }\xi (g(x)).$
The function
$\xi _{0}(x)=\pi ^{-1}\left(1+|x|^{2}\right)^{-1}$
is fixed by SO(2) and
$\Phi _{\lambda }(g)=(\pi _{\lambda }(g)\xi _{0},\xi _{0}).$
The representations πλ are irreducible and unitarily equivalent only when the sign of λ is changed. The map $W:L^{2}({\mathfrak {H}}^{2})\to L^{2}([0,\infty )\times \mathbb {R} )$ with measure $ {\frac {\lambda \pi }{2}}\tanh \left({\frac {\pi \lambda }{2}}\right)\,d\lambda $ on the first factor, is given by the formula
$Wf(\lambda ,x)=\int _{G/K}f(g)\pi _{\lambda }(g)\xi _{0}(x)\,dg$
is unitary and gives the decomposition of $L^{2}({\mathfrak {H}}^{2})$ as a direct integral of the spherical principal series.
Flensted–Jensen's method of descent
Hadamard's method of descent relied on functions invariant under the action of 1-parameter subgroup of translations in the y parameter in ${\mathfrak {H}}^{3}$. Flensted–Jensen's method uses the centraliser of SO(2) in SL(2,C) which splits as a direct product of SO(2) and the 1-parameter subgroup K1 of matrices
$g_{t}={\begin{pmatrix}\cosh t&i\sinh t\\-i\sinh t&\cosh t\end{pmatrix}}.$
The symmetric space SL(2,C)/SU(2) can be identified with the space H3 of positive 2×2 matrices A with determinant 1
$A={\begin{pmatrix}a+b&x+iy\\x-iy&a-b\end{pmatrix}}$
with the group action given by
$g\cdot A=gAg^{*}.$
Thus
$g_{t}\cdot A={\begin{pmatrix}a\cosh 2t+y\sinh 2t+b&x+i(y\cosh 2t+a\sinh 2t)\\x-i(y\cosh 2t+a\sinh 2t)&a\cosh 2t+y\sinh 2t-b\end{pmatrix}}.$
So on the hyperboloid $a^{2}=1+b^{2}+x^{2}+y^{2}$, gt only changes the coordinates y and a. Similarly the action of SO(2) acts by rotation on the coordinates (b,x) leaving a and y unchanged. The space H2 of real-valued positive matrices A with y = 0 can be identified with the orbit of the identity matrix under SL(2,R). Taking coordinates (b,x,y) in H3 and (b,x) on H2 the volume and area elements are given by
$dV=(1+r^{2})^{-1/2}\,db\,dx\,dy,\,\,\,dA=(1+r^{2})^{-1/2}\,db\,dx,$
where r2 equals b2 + x2 + y2 or b2 + x2, so that r is related to hyperbolic distance from the origin by $r=\sinh t$.
The Laplacian operators are given by the formula
$\Delta _{n}=-L_{n}-R_{n}^{2}-(n-1)R_{n},\,$
where
$L_{2}=\partial _{b}^{2}+\partial _{x}^{2},\,\,\,R_{2}=b\partial _{b}+x\partial _{x}$
and
$L_{3}=\partial _{b}^{2}+\partial _{x}^{2}+\partial _{y}^{2},\,\,\,R_{3}=b\partial _{b}+x\partial _{x}+y\partial _{y}.\,$
For an SU(2)-invariant function F on H3 and an SO(2)-invariant function on H2, regarded as functions of r or t,
$\int _{H^{3}}F\,dV=4\pi \int _{-\infty }^{\infty }F(t)\sinh ^{2}t\,dt,\,\,\,\int _{H^{2}}f\,dV=2\pi \int _{-\infty }^{\infty }f(t)\sinh t\,dt.$
If f(b,x) is a function on H2, Ef is defined by
$Ef(b,x,y)=f(b,x).\,$
Thus
$\Delta _{3}Ef=E(\Delta _{2}-R_{2})f.\,$
If f is SO(2)-invariant, then, regarding f as a function of r or t,
$(-\Delta _{2}+R_{2})f=\partial _{t}^{2}f+\coth t\partial _{t}f+r\partial _{r}f=\partial _{t}^{2}f+(\coth t+\tanh t)\partial _{t}f.$
On the other hand,
$\partial _{t}^{2}+(\coth t+\tanh t)\partial _{t}=\partial _{t}^{2}+2\coth(2t)\partial _{t}.$
Thus, setting Sf(t) = f(2t),
$(\Delta _{2}-R_{2})Sf=4S\Delta _{2}f,$
leading to the fundamental descent relation of Flensted-Jensen for M0 = ES:
$\Delta _{3}M_{0}f=4M_{0}\Delta _{2}f.$
The same relation holds with M0 by M, where Mf is obtained by averaging M0f over SU(2).
The extension Ef is constant in the y variable and therefore invariant under the transformations gs. On the other hand, for F a suitable function on H3, the function QF defined by
$QF=\int _{K_{1}}F\circ g_{s}\,ds$
is independent of the y variable. A straightforward change of variables shows that
$\int _{H^{3}}F\,dV=\int _{H^{2}}(1+b^{2}+x^{2})^{1/2}QF\,dA.$
Since K1 commutes with SO(2), QF is SO(2)--invariant if F is, in particular if F is SU(2)-invariant. In this case QF is a function of r or t, so that M*F can be defined by
$M^{*}F(t)=QF(t/2).$
The integral formula above then yields
$\int _{H^{3}}F\,dV=\int _{H^{2}}M^{*}F\,dA$
and hence, since for f SO(2)-invariant,
$M^{*}((Mf)\cdot F)=f\cdot (M^{*}F),$
the following adjoint formula:
$\int _{H^{3}}(Mf)\cdot F\,dV=\int _{H^{2}}f\cdot (M*F)\,dV.$
As a consequence
$M^{*}\Delta _{3}=4\Delta _{2}M^{*}.$
Thus, as in the case of Hadamard's method of descent.
$M^{*}\Phi _{2\lambda }=b(\lambda )\varphi _{\lambda }$
with
$b(\lambda )=M^{*}\Phi _{2\lambda }(0)=\pi \tanh \pi \lambda $
and
$\Phi _{2\lambda }=M\varphi _{\lambda }.$
It follows that
${(M^{*}F)}^{\sim }(\lambda )={\tilde {F}}(2\lambda ).$
Taking f=M*F, the SL(2,C) inversion formula for F then immediately yields
$f(x)=\int _{-\infty }^{\infty }\varphi _{\lambda }(x){\tilde {f}}(\lambda )\,{\lambda \pi \over 2}\tanh \left({\frac {\pi \lambda }{2}}\right)\,d\lambda ,$
Abel's integral equation
The spherical function φλ is given by
$\varphi _{\lambda }(g)=\int _{K}\alpha '(kg)\,dk,$
so that
${\tilde {f}}(\lambda )=\int _{S}f(s)\alpha '(s)\,ds,$
Thus
${\tilde {f}}(\lambda )=\int _{-\infty }^{\infty }\int _{0}^{\infty }f\!\left({\frac {a^{2}+a^{-2}+b^{2}}{2}}\right)a^{-i\lambda /2}\,da\,db,$
so that defining F by
$F(u)=\int _{-\infty }^{\infty }f\!\left(u+{\frac {t^{2}}{2}}\right)\,dt,$
the spherical transform can be written
${\tilde {f}}(\lambda )=\int _{0}^{\infty }F\!\left({\frac {a^{2}+a^{-2}}{2}}\right)a^{-i\lambda }\,da=\int _{0}^{\infty }F(\cosh t)e^{-it\lambda }\,dt.$
The relation between F and f is classically inverted by the Abel integral equation:
$f(x)={\frac {-1}{2\pi }}\int _{-\infty }^{\infty }F'\!\left(x+{t^{2} \over 2}\right)\,dt.$
In fact[18]
$\int _{-\infty }^{\infty }F'\!\left(x+{\frac {t^{2}}{2}}\right)\,dt=\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }f'\!\left(x+{\frac {t^{2}+u^{2}}{2}}\right)\,dt\,du={2\pi }\int _{0}^{\infty }f'\!\left(x+{\frac {r^{2}}{2}}\right)r\,dr=2\pi f(x).$
The relation between F and ${\tilde {f}}$ is inverted by the Fourier inversion formula:
$F(\cosh t)={2 \over \pi }\int _{0}^{\infty }{\tilde {f}}(i\lambda )\cos(\lambda t)\,d\lambda .$
Hence
$f(i)={1 \over 2\pi ^{2}}\int _{0}^{\infty }{\tilde {f}}(\lambda )\lambda \,d\lambda \int _{-\infty }^{\infty }{\sin \lambda t/2 \over \sinh t}\cosh {t \over 2}\,dt={1 \over 2\pi ^{2}}\int _{-\infty }^{\infty }{\tilde {f}}(\lambda ){\lambda \pi \over 2}\tanh \left({\pi \lambda \over 2}\right)\,d\lambda .$
This gives the spherical inversion for the point i. Now for fixed g in SL(2,R) define[19]
$f_{1}(w)=\int _{K}f(gkw)\,dk,$
another rotation invariant function on ${\mathfrak {H}}^{2}$ with f1(i)=f(g(i)). On the other hand, for biinvariant functions f,
$\pi _{\lambda }(f)\xi _{0}={\tilde {f}}(\lambda )\xi _{0}$
so that
${\tilde {f}}_{1}(\lambda )={\tilde {f}}(\lambda )\cdot \varphi _{\lambda }(w),$
where w = g(i). Combining this with the above inversion formula for f1 yields the general spherical inversion formula:
$f(w)={1 \over \pi ^{2}}\int _{0}^{\infty }{\tilde {f}}(\lambda )\varphi _{\lambda }(w){\lambda \pi \over 2}\tanh \left({\pi \lambda \over 2}\right)\,d\lambda .$
Other special cases
All complex semisimple Lie groups or the Lorentz groups SO0(N,1) with N odd can be treated directly by reduction to the usual Fourier transform.[15][20] The remaining real Lorentz groups can be deduced by Flensted-Jensen's method of descent, as can other semisimple Lie groups of real rank one.[21] Flensted-Jensen's method of descent also applies to the treatment of real semisimple Lie groups for which the Lie algebras are normal real forms of complex semisimple Lie algebras.[15] The special case of SL(N,R) is treated in detail in Jorgenson & Lang (2001); this group is also the normal real form of SL(N,C).
The approach of Flensted-Jensen (1978) applies to a wide class of real semisimple Lie groups of arbitrary real rank and yields the explicit product form of the Plancherel measure on ${\mathfrak {a}}$* without using Harish-Chandra's expansion of the spherical functions φλ in terms of his c-function, discussed below. Although less general, it gives a simpler approach to the Plancherel theorem for this class of groups.
Complex semisimple Lie groups
If G is a complex semisimple Lie group, it is the complexification of its maximal compact subgroup U, a compact semisimple Lie group. If ${\mathfrak {g}}$ and ${\mathfrak {u}}$ are their Lie algebras, then ${\mathfrak {g}}={\mathfrak {u}}\oplus i{\mathfrak {u}}.$ Let T be a maximal torus in U with Lie algebra ${\mathfrak {t}}.$ Then setting
$A=\exp i{\mathfrak {t}},\qquad P=\exp i{\mathfrak {u}},$
there is the Cartan decomposition:
$G=P\cdot U=UAU.$
The finite-dimensional irreducible representations πλ of U are indexed by certain λ in ${\mathfrak {t}}^{*}$.[22] The corresponding character formula and dimension formula of Hermann Weyl give explicit formulas for
$\chi _{\lambda }(e^{X})=\operatorname {Tr} \pi _{\lambda }(e^{X}),(X\in {\mathfrak {t}}),\qquad d(\lambda )=\dim \pi _{\lambda }.$
These formulas, initially defined on ${\mathfrak {t}}^{*}\times {\mathfrak {t}}$ and ${\mathfrak {t}}^{*}$, extend holomorphic to their complexifications. Moreover,
$\chi _{\lambda }(e^{X})={\sum _{\sigma \in W}{\rm {sign}}(\sigma )e^{i\lambda (\sigma X)} \over \delta (e^{X})},$
where W is the Weyl group $W=N_{U}(T)/T$ and δ(eX) is given by a product formula (Weyl's denominator formula) which extends holomorphically to the complexification of ${\mathfrak {t}}$. There is a similar product formula for d(λ), a polynomial in λ.
On the complex group G, the integral of a U-biinvariant function F can be evaluated as
$\int _{G}F(g)\,dg={1 \over |W|}\int _{\mathfrak {a}}F(e^{X})\,|\delta (e^{X})|^{2}\,dX.$
where ${\mathfrak {a}}=i{\mathfrak {t}}$.
The spherical functions of G are labelled by λ in ${\mathfrak {a}}=i{\mathfrak {t}}^{*}$ and given by the Harish-Chandra-Berezin formula[23]
$\Phi _{\lambda }(e^{X})={\chi _{\lambda }(e^{X}) \over d(\lambda )}.$
They are the matrix coefficients of the irreducible spherical principal series of G induced from the character of the Borel subgroup of G corresponding to λ; these representations are irreducible and can all be realized on L2(U/T).
The spherical transform of a U-biinvariant function F is given by
${\tilde {F}}(\lambda )=\int _{G}F(g)\Phi _{-\lambda }(g)\,dg$
and the spherical inversion formula by
$F(g)={1 \over |W|}\int _{{\mathfrak {a}}^{*}}{\tilde {F}}(\lambda )\Phi _{\lambda }(g)|d(\lambda )|^{2}\,d\,\lambda =\int _{{\mathfrak {a}}_{+}^{*}}{\tilde {F}}(\lambda )\Phi _{\lambda }(g)|d(\lambda )|^{2}\,d\,\lambda ,$
where ${\mathfrak {a}}_{+}^{*}$ is a Weyl chamber. In fact the result follows from the Fourier inversion formula on ${\mathfrak {a}}$ since[24]
$d(\lambda )\delta (e^{X})\Phi _{\lambda }(e^{X})=\sum _{\sigma \in W}{\rm {sign}}(\sigma )e^{i\lambda (X)},$
so that ${\overline {d(\lambda )}}{\tilde {F}}(\lambda )$ is just the Fourier transform of $F(e^{X})\delta (e^{X})$.
Note that the symmetric space G/U has as compact dual[25] the compact symmetric space U x U / U, where U is the diagonal subgroup. The spherical functions for the latter space, which can be identified with U itself, are the normalized characters χλ/d(λ) indexed by lattice points in the interior of ${\mathfrak {a}}_{+}^{*}$ and the role of A is played by T. The spherical transform of f of a class function on U is given by
${\tilde {f}}(\lambda )=\int _{U}f(u){{\overline {\chi _{\lambda }(u)}} \over d(\lambda )}\,du$
and the spherical inversion formula now follows from the theory of Fourier series on T:
$f(u)=\sum _{\lambda }{\tilde {f}}(\lambda ){\chi _{\lambda }(u) \over d(\lambda )}d(\lambda )^{2}.$
There is an evident duality between these formulas and those for the non-compact dual.[26]
Real semisimple Lie groups
Let G0 be a normal real form of the complex semisimple Lie group G, the fixed points of an involution σ, conjugate linear on the Lie algebra of G. Let τ be a Cartan involution of G0 extended to an involution of G, complex linear on its Lie algebra, chosen to commute with σ. The fixed point subgroup of τσ is a compact real form U of G, intersecting G0 in a maximal compact subgroup K0. The fixed point subgroup of τ is K, the complexification of K0. Let G0= K0·P0 be the corresponding Cartan decomposition of G0 and let A be a maximal Abelian subgroup of P0. Flensted-Jensen (1978) proved that
$G=KA_{+}U,$
where A+ is the image of the closure of a Weyl chamber in ${\mathfrak {a}}$ under the exponential map. Moreover,
$K\backslash G/U=A_{+}.$
Since
$K_{0}\backslash G_{0}/K_{0}=A_{+}$
it follows that there is a canonical identification between K \ G / U, K0 \ G0 /K0 and A+. Thus K0-biinvariant functions on G0 can be identified with functions on A+ as can functions on G that are left invariant under K and right invariant under U. Let f be a function in $C_{c}^{\infty }(K_{0}\backslash G_{0}/K_{0})$ and define Mf in $C_{c}^{\infty }(U\backslash G/U)$ by
$Mf(a)=\int _{U}f(ua^{2})\,du.$
Here a third Cartan decomposition of G = UAU has been used to identify U \ G / U with A+.
Let Δ be the Laplacian on G0/K0 and let Δc be the Laplacian on G/U. Then
$4M\Delta =\Delta _{c}M.$
For F in $C_{c}^{\infty }(U\backslash G/U)$, define M*F in $C_{c}^{\infty }(K_{0}\backslash G_{0}/K_{0})$ by
$M^{*}F(a^{2})=\int _{K}F(ga)\,dg.$
Then M and M* satisfy the duality relations
$\int _{G/U}(Mf)\cdot F=\int _{G_{0}/K_{0}}f\cdot (M^{*}F).$
In particular
$M^{*}\Delta _{c}=4\Delta M^{*}.$
There is a similar compatibility for other operators in the center of the universal enveloping algebra of G0. It follows from the eigenfunction characterisation of spherical functions that $M^{*}\Phi _{2\lambda }$ is proportional to φλ on G0, the constant of proportionality being given by
$b(\lambda )=M^{*}\Phi _{2\lambda }(1)=\int _{K}\Phi _{2\lambda }(k)\,dk.$
Moreover, in this case[27]
$(M^{*}F)^{\sim }(\lambda )={\tilde {F}}(2\lambda ).$
If f = M*F, then the spherical inversion formula for F on G implies that for f on G0:[28][29]
$f(g)=\int _{{\mathfrak {a}}_{+}^{*}}{\tilde {f}}(\lambda )\varphi _{\lambda }(g)\,\,2^{{\rm {dim}}\,A}\cdot |b(\lambda )|\cdot |d(2\lambda )|^{2}\,d\lambda ,$
since
$f(g)=M^{*}F(g)=\int _{{\mathfrak {a}}_{+}^{*}}{\tilde {F}}(2\lambda )M^{*}\Phi _{2\lambda }(g)2^{{\rm {dim}}\,A}|d(2\lambda )|^{2}\,d\lambda =\int _{{\mathfrak {a}}_{+}^{*}}{\tilde {f}}(\lambda )\varphi _{\lambda }(g)\,\,b(\lambda )2^{{\rm {dim}}\,A}|d(2\lambda )|^{2}\,d\lambda .$
The direct calculation of the integral for b(λ), generalising the computation of Godement (1957) for SL(2,R), was left as an open problem by Flensted-Jensen (1978).[30] An explicit product formula for b(λ) was known from the prior determination of the Plancherel measure by Harish-Chandra (1966), giving[31][32]
$b(\lambda )=C\cdot d(2\lambda )^{-1}\cdot \prod _{\alpha >0}\tanh {\pi (\alpha ,\lambda ) \over (\alpha ,\alpha )},$
where α ranges over the positive roots of the root system in ${\mathfrak {a}}$ and C is a normalising constant, given as a quotient of products of Gamma functions.
Harish-Chandra's Plancherel theorem
Let G be a noncompact connected real semisimple Lie group with finite center. Let ${\mathfrak {g}}$ denote its Lie algebra. Let K be a maximal compact subgroup given as the subgroup of fixed points of a Cartan involution σ. Let ${\mathfrak {g}}_{\pm }$ be the ±1 eigenspaces of σ in ${\mathfrak {g}}$, so that ${\mathfrak {k}}={\mathfrak {g}}_{+}$ is the Lie algebra of K and ${\mathfrak {p}}={\mathfrak {g}}_{-}$ give the Cartan decomposition
${\mathfrak {g}}={\mathfrak {k}}+{\mathfrak {p}},\,\,G=\exp {\mathfrak {p}}\cdot K.$
Let ${\mathfrak {a}}$ be a maximal Abelian subalgebra of ${\mathfrak {p}}$ and for α in ${\mathfrak {a}}^{*}$ let
${\mathfrak {g}}_{\alpha }=\{X\in {\mathfrak {g}}:[H,X]=\alpha (H)X\,\,(H\in {\mathfrak {a}})\}.$
If α ≠ 0 and ${\mathfrak {g}}_{\alpha }\neq (0)$, then α is called a restricted root and $m_{\alpha }=\dim {\mathfrak {g}}_{\alpha }$ is called its multiplicity. Let A = exp ${\mathfrak {a}}$, so that G = KAK.The restriction of the Killing form defines an inner product on ${\mathfrak {p}}$ and hence ${\mathfrak {a}}$, which allows ${\mathfrak {a}}^{*}$ to be identified with ${\mathfrak {a}}$. With respect to this inner product, the restricted roots Σ give a root system. Its Weyl group can be identified with $W=N_{K}(A)/C_{K}(A)$. A choice of positive roots defines a Weyl chamber ${\mathfrak {a}}_{+}^{*}$. The reduced root system Σ0 consists of roots α such that α/2 is not a root.
Defining the spherical functions φ λ as above for λ in ${\mathfrak {a}}^{*}$, the spherical transform of f in Cc∞(K \ G / K) is defined by
${\tilde {f}}(\lambda )=\int _{G}f(g)\varphi _{-\lambda }(g)\,dg.$
The spherical inversion formula states that
$f(g)=\int _{{\mathfrak {a}}_{+}^{*}}{\tilde {f}}(\lambda )\varphi _{\lambda }(g)\,|c(\lambda )|^{-2}\,d\lambda ,$
where Harish-Chandra's c-function c(λ) is defined by[33]
$c(\lambda )=c_{0}\cdot \prod _{\alpha \in \Sigma _{0}^{+}}{\frac {2^{-i(\lambda ,\alpha _{0})}\Gamma (i(\lambda ,\alpha _{0}))}{\Gamma \!\left({\frac {1}{2}}\left[{\frac {1}{2}}m_{\alpha }+1+i(\lambda ,\alpha _{0})\right]\right)\Gamma \!\left({\frac {1}{2}}\left[{\frac {1}{2}}m_{\alpha }+m_{2\alpha }+i(\lambda ,\alpha _{0})\right]\right)}}$
with $\alpha _{0}=(\alpha ,\alpha )^{-1}\alpha $ and the constant c0 chosen so that c(−iρ) = 1 where
$\rho ={\frac {1}{2}}\sum _{\alpha \in \Sigma ^{+}}m_{\alpha }\alpha .$
The Plancherel theorem for spherical functions states that the map
$W:f\mapsto {\tilde {f}},\,\,\,\ L^{2}(K\backslash G/K)\rightarrow L^{2}({\mathfrak {a}}_{+}^{*},|c(\lambda )|^{-2}\,d\lambda )$
is unitary and transforms convolution by $f\in L^{1}(K\backslash G/K)$ into multiplication by ${\tilde {f}}$.
Harish-Chandra's spherical function expansion
Since G = KAK, functions on G/K that are invariant under K can be identified with functions on A, and hence ${\mathfrak {a}}$, that are invariant under the Weyl group W. In particular since the Laplacian Δ on G/K commutes with the action of G, it defines a second order differential operator L on ${\mathfrak {a}}$, invariant under W, called the radial part of the Laplacian. In general if X is in ${\mathfrak {a}}$, it defines a first order differential operator (or vector field) by
$Xf(y)=\left.{\frac {d}{dt}}f(y+tX)\right|_{t=0}.$
L can be expressed in terms of these operators by the formula[34]
$L=\Delta _{\mathfrak {a}}-\sum _{\alpha >0}m_{\alpha }\,\coth \alpha \,A_{\alpha },$
where Aα in ${\mathfrak {a}}$ is defined by
$(A_{\alpha },X)=\alpha (X)$
and
$\Delta _{\mathfrak {a}}=-\sum X_{i}^{2}$
is the Laplacian on ${\mathfrak {a}}$, corresponding to any choice of orthonormal basis (Xi).
Thus
$L=L_{0}-\sum _{\alpha >0}m_{\alpha }\,(\coth \alpha -1)A_{\alpha },$
where
$L_{0}=\Delta _{\mathfrak {a}}-\sum _{\alpha >0}A_{\alpha },$
so that L can be regarded as a perturbation of the constant-coefficient operator L0.
Now the spherical function φλ is an eigenfunction of the Laplacian:
$\Delta \varphi _{\lambda }=\left(\left\|\lambda \right\|^{2}+\left\|\rho \right\|^{2}\right)\varphi _{\lambda }$
and therefore of L, when viewed as a W-invariant function on ${\mathfrak {a}}$.
Since eiλ–ρ and its transforms under W are eigenfunctions of L0 with the same eigenvalue, it is natural look for a formula for φλ in terms of a perturbation series
$f_{\lambda }=e^{i\lambda -\rho }\sum _{\mu \in \Lambda }a_{\mu }(\lambda )e^{-\mu },$
with Λ the cone of all non-negative integer combinations of positive roots, and the transforms of fλ under W. The expansion
$\coth x-1=2\sum _{m>0}e^{-2mx},$
leads to a recursive formula for the coefficients aμ(λ). In particular they are uniquely determined and the series and its derivatives converges absolutely on ${\mathfrak {a}}_{+}$, a fundamental domain for W. Remarkably it turns out that fλ is also an eigenfunction of the other G-invariant differential operators on G/K, each of which induces a W-invariant differential operator on ${\mathfrak {a}}$.
It follows that φλ can be expressed in terms as a linear combination of fλ and its transforms under W:[35]
$\varphi _{\lambda }=\sum _{s\in W}c(s\lambda )f_{s\lambda }.$
Here c(λ) is Harish-Chandra's c-function. It describes the asymptotic behaviour of φλ in ${\mathfrak {a}}_{+}$, since[36]
$\varphi _{\lambda }(e^{t}X)\sim c(\lambda )e^{(i\lambda -\rho )Xt}$
for X in ${\mathfrak {a}}_{+}$ and t > 0 large.
Harish-Chandra obtained a second integral formula for φλ and hence c(λ) using the Bruhat decomposition of G:[37]
$G=\bigcup _{s\in W}BsB,$
where B = MAN and the union is disjoint. Taking the Coxeter element s0 of W, the unique element mapping ${\mathfrak {a}}_{+}$ onto $-{\mathfrak {a}}_{+}$, it follows that σ(N) has a dense open orbit G/B = K/M whose complement is a union of cells of strictly smaller dimension and therefore has measure zero. It follows that the integral formula for φλ initially defined over K/M
$\varphi _{\lambda }(g)=\int _{K/M}\lambda '(gk)^{-1}\,dk.$
can be transferred to σ(N):[38]
$\varphi _{\lambda }(e^{X})=e^{i\lambda -\rho }\int _{\sigma (N)}{{\overline {\lambda '(n)}} \over \lambda '(e^{X}ne^{-X})}\,dn,$
for X in ${\mathfrak {a}}$.
Since
$\lim _{t\to \infty }e^{tX}ne^{-tX}=1$
for X in ${\mathfrak {a}}_{+}$, the asymptotic behaviour of φλ can be read off from this integral, leading to the formula:[39]
$c(\lambda )=\int _{\sigma (N)}{\overline {\lambda '(n)}}\,dn.$
Harish-Chandra's c-function
The many roles of Harish-Chandra's c-function in non-commutative harmonic analysis are surveyed in Helgason (2000). Although it was originally introduced by Harish-Chandra in the asymptotic expansions of spherical functions, discussed above, it was also soon understood to be intimately related to intertwining operators between induced representations, first studied in this context by Bruhat (1957) harvtxt error: no target: CITEREFBruhat1957 (help). These operators exhibit the unitary equivalence between πλ and πsλ for s in the Weyl group and a c-function cs(λ) can be attached to each such operator: namely the value at 1 of the intertwining operator applied to ξ0, the constant function 1, in L2(K/M).[40] Equivalently, since ξ0 is up to scalar multiplication the unique vector fixed by K, it is an eigenvector of the intertwining operator with eigenvalue cs(λ). These operators all act on the same space L2(K/M), which can be identified with the representation induced from the 1-dimensional representation defined by λ on MAN. Once A has been chosen, the compact subgroup M is uniquely determined as the centraliser of A in K. The nilpotent subgroup N, however, depends on a choice of a Weyl chamber in ${\mathfrak {a}}^{*}$, the various choices being permuted by the Weyl group W = M ' / M, where M ' is the normaliser of A in K. The standard intertwining operator corresponding to (s, λ) is defined on the induced representation by[41]
$A(s,\lambda )F(k)=\int _{\sigma (N)\cap s^{-1}Ns}F(ksn)\,dn,$
where σ is the Cartan involution. It satisfies the intertwining relation
$A(s,\lambda )\pi _{\lambda }(g)=\pi _{s\lambda }(g)A(s,\lambda ).$
The key property of the intertwining operators and their integrals is the multiplicative cocycle property[42]
$A(s_{1}s_{2},\lambda )=A(s_{1},s_{2}\lambda )A(s_{2},\lambda ),$
whenever
$\ell (s_{1}s_{2})=\ell (s_{1})+\ell (s_{2})$
for the length function on the Weyl group associated with the choice of Weyl chamber. For s in W, this is the number of chambers crossed by the straight line segment between X and sX for any point X in the interior of the chamber. The unique element of greatest length s0, namely the number of positive restricted roots, is the unique element that carries the Weyl chamber ${\mathfrak {a}}_{+}^{*}$ onto $-{\mathfrak {a}}_{+}^{*}$. By Harish-Chandra's integral formula, it corresponds to Harish-Chandra's c-function:
$c(\lambda )=c_{s_{0}}(\lambda ).$
The c-functions are in general defined by the equation
$A(s,\lambda )\xi _{0}=c_{s}(\lambda )\xi _{0},$
where ξ0 is the constant function 1 in L2(K/M). The cocycle property of the intertwining operators implies a similar multiplicative property for the c-functions:
$c_{s_{1}s_{2}}(\lambda )=c_{s_{1}}(s_{2}\lambda )c_{s_{2}}(\lambda )$
provided
$\ell (s_{1}s_{2})=\ell (s_{1})+\ell (s_{2}).$
This reduces the computation of cs to the case when s = sα, the reflection in a (simple) root α, the so-called "rank-one reduction" of Gindikin & Karpelevich (1962). In fact the integral involves only the closed connected subgroup Gα corresponding to the Lie subalgebra generated by ${\mathfrak {g}}_{\pm \alpha }$ where α lies in Σ0+.[43] Then Gα is a real semisimple Lie group with real rank one, i.e. dim Aα = 1, and cs is just the Harish-Chandra c-function of Gα. In this case the c-function can be computed directly by various means:
• by noting that φλ can be expressed in terms of the hypergeometric function for which the asymptotic expansion is known from the classical formulas of Gauss for the connection coefficients;[6][44]
• by directly computing the integral, which can be expressed as an integral in two variables and hence a product of two beta functions.[45][46]
This yields the following formula:
$c_{s_{\alpha }}(\lambda )=c_{0}{\frac {2^{-i(\lambda ,\alpha _{0})}\Gamma (i(\lambda ,\alpha _{0}))}{\Gamma \!\left({\frac {1}{2}}\left({\frac {1}{2}}m_{\alpha }+1+i(\lambda ,\alpha _{0})\right)\right)\Gamma \!\left({\frac {1}{2}}\left({\frac {1}{2}}m_{\alpha }+m_{2\alpha }+i(\lambda ,\alpha _{0})\right)\right)}},$
where
$c_{0}=2^{m_{\alpha }/2+m_{2\alpha }}\Gamma \!\left({\tfrac {1}{2}}(m_{\alpha }+m_{2\alpha }+1)\right).$
The general Gindikin–Karpelevich formula for c(λ) is an immediate consequence of this formula and the multiplicative properties of cs(λ).
Paley–Wiener theorem
The Paley-Wiener theorem generalizes the classical Paley-Wiener theorem by characterizing the spherical transforms of smooth K-bivariant functions of compact support on G. It is a necessary and sufficient condition that the spherical transform be W-invariant and that there is an R > 0 such that for each N there is an estimate
$|{\tilde {f}}(\lambda )|\leq C_{N}(1+|\lambda |)^{-N}e^{R\left|\operatorname {Im} \lambda \right|}.$
In this case f is supported in the closed ball of radius R about the origin in G/K.
This was proved by Helgason and Gangolli (Helgason (1970) pg. 37).
The theorem was later proved by Flensted-Jensen (1986) independently of the spherical inversion theorem, using a modification of his method of reduction to the complex case.[47]
Rosenberg's proof of inversion formula
Rosenberg (1977) noticed that the Paley-Wiener theorem and the spherical inversion theorem could be proved simultaneously, by a trick which considerably simplified previous proofs.
The first step of his proof consists in showing directly that the inverse transform, defined using Harish-Chandra's c-function, defines a function supported in the closed ball of radius R about the origin if the Paley-Wiener estimate is satisfied. This follows because the integrand defining the inverse transform extends to a meromorphic function on the complexification of ${\mathfrak {a}}^{*}$; the integral can be shifted to ${\mathfrak {a}}^{*}+i\mu t$ for μ in ${\mathfrak {a}}_{+}^{*}$ and t > 0. Using Harish-Chandra's expansion of φλ and the formulas for c(λ) in terms of Gamma functions, the integral can be bounded for t large and hence can be shown to vanish outside the closed ball of radius R about the origin.[48]
This part of the Paley-Wiener theorem shows that
$T(f)=\int _{{\mathfrak {a}}_{+}^{*}}{\tilde {f}}(\lambda )|c(\lambda )|^{-2}\,d\lambda $
defines a distribution on G/K with support at the origin o. A further estimate for the integral shows that it is in fact given by a measure and that therefore there is a constant C such that
$T(f)=Cf(o).$
By applying this result to
$f_{1}(g)=\int _{K}f(x^{-1}kg)\,dk,$
it follows that
$Cf=\int _{{\mathfrak {a}}_{+}^{*}}{\tilde {f}}(\lambda )\varphi _{\lambda }|c(\lambda )|^{-2}\,d\lambda .$
A further scaling argument allows the inequality C = 1 to be deduced from the Plancherel theorem and Paley-Wiener theorem on ${\mathfrak {a}}$.[49][50]
Schwartz functions
The Harish-Chandra Schwartz space can be defined as[51]
${\mathcal {S}}(K\backslash G/K)=\left\{f\in C^{\infty }(G/K)^{K}:\sup _{x}\left|(1+d(x,o))^{m}(\Delta +I)^{n}f(x)\right|<\infty \right\}.$
Under the spherical transform it is mapped onto ${\mathcal {S}}({\mathfrak {a}}^{*})^{W},$ the space of W-invariant Schwartz functions on ${\mathfrak {a}}^{*}.$
The original proof of Harish-Chandra was a long argument by induction.[6][7][52] Anker (1991) found a short and simple proof, allowing the result to be deduced directly from versions of the Paley-Wiener and spherical inversion formula. He proved that the spherical transform of a Harish-Chandra Schwartz function is a classical Schwartz function. His key observation was then to show that the inverse transform was continuous on the Paley-Wiener space endowed with classical Schwartz space seminorms, using classical estimates.
Notes
1. Helgason 1984, pp. 492–493, historical notes on the Plancherel theorem for spherical functions
2. Harish-Chandra 1951
3. Harish-Chandra 1952
4. Gelfand & Naimark 1948
5. Guillemin & Sternberg 1977
6. Harish-Chandra 1958a
7. Harish-Chandra 1958b
8. Gindikin & Karpelevich 1962
9. Harish-Chandra 1966, section 21
10. The spectrum coincides with that of the commutative Banach *-algebra of integrable K-biinvariant functions on G under convolution, a dense *-subalgebra of ${\mathfrak {A}}$.
11. The measure class of μ in the sense of the Radon–Nikodym theorem is unique.
12. Davies 1990 harvnb error: no target: CITEREFDavies1990 (help)
13. Lax & Phillips 1976
14. Helgason 1984, p. 38
15. Flensted-Jensen 1978
16. Anker 1991
17. Jorgenson & Lang 2001
18. Helgason 1984, p. 41
19. Helgason 1984, p. 46
20. Takahashi 1963
21. Loeb 1979
22. These are indexed by highest weights shifted by half the sum of the positive roots.
23. Helgason 1984, pp. 423–433
24. Flensted-Jensen 1978, p. 115
25. Helgason 1978
26. The spherical inversion formula for U is equivalent to the statement that the functions $\chi _{\lambda }d(\lambda )^{-1/2}$ form an orthonormal basis for the class functions.
27. Flensted-Jensen 1978, p. 133
28. Flensted-Jensen 1978, p. 133
29. Helgason 1984, p. 490–491
30. b(λ) can be written as integral over A0 where K = K0 A0 K0 is the Cartan decomposition of K. The integral then becomes an alternating sum of multidimensional Godement-type integrals, whose combinatorics is governed by that of the Cartan-Helgason theorem for U/K0. An equivalent computation that arises in the theory of the Radon transform has been discussed by Beerends (1987), Stade (1999) and Gindikin (2008).
31. Helgason 1984
32. Beerends 1987, p. 4–5
33. Helgason, p. 447 harvnb error: no target: CITEREFHelgason (help)
34. Helgason 1984, p. 267
35. Helgason 1984, p. 430
36. Helgason 1984, p. 435
37. Helgason 1978, p. 403
38. Helgason 1984, p. 436
39. Helgason 1984, p. 447
40. Knapp 2001, Chapter VII
41. Knapp 2001, p. 177
42. Knapp 2001, p. 182
43. Helgason 1978, p. 407
44. Helgason 1984, p. 484
45. Helgason 1978, p. 414
46. Helgason 1984, p. 437
47. The second statement on supports follows from Flensted-Jensen's proof by using the explicit methods associated with Kostant polynomials instead of the results of Mustapha Rais.
48. Helgason 1984, pp. 452–453
49. Rosenberg 1977
50. Helgason 1984, p. 588–589
51. Anker 1991, p. 347
52. Helgason 1984, p. 489
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• Lax, Peter D.; Phillips, Ralph (1976), Scattering theory for automorphic functions, Annals of Mathematics Studies, vol. 87, Princeton University Press, ISBN 0-691-08184-0
• Loeb, Jacques (1979), L'analyse harmonique sur les espaces symétriques de rang 1. Une réduction aux espaces hyperboliques réels de dimension impaire, Lecture Notes in Math, vol. 739, Springer, pp. 623–646
• Rosenberg, Jonathan (1977), "A quick proof of Harish-Chandra's Plancherel theorem for spherical functions on a semisimple Lie group", Proceedings of the American Mathematical Society, 63 (1): 143–149, doi:10.1090/S0002-9939-1977-0507231-8, JSTOR 2041084
• Selberg, Atle (1956), "Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series", J. Indian Math. Soc., 20: 47–87
• Stade, E. (1999), "The hyperbolic tangent and generalized Mellin inversion", Rocky Mountain Journal of Mathematics, 29 (2): 691–707, doi:10.1216/rmjm/1181071659
• Takahashi, R. (1963), "Sur les représentations unitaires des groupes de Lorentz généralisés", Bull. Soc. Math. France (in French), 91: 289–433, doi:10.24033/bsmf.1598
Functional analysis (topics – glossary)
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Theorems
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Spherical variety
In algebraic geometry, given a reductive algebraic group G and a Borel subgroup B, a spherical variety is a G-variety with an open dense B-orbit. It is sometimes also assumed to be normal. Examples are flag varieties, symmetric spaces and (affine or projective) toric varieties.
There is also a notion of real spherical varieties.
A projective spherical variety is a Mori dream space.[1]
Spherical embeddings are classified by so-called colored fans, a generalization of fans for toric varieties; this is known as Luna-Vust Theory.
In his seminal paper, Luna (2001) developed a framework to classify complex spherical subgroups of reductive groups; he reduced the classification of spherical subgroups to wonderful subgroups. He further worked out the case of groups of type A and conjectured that combinatorial objects consisting of "homogeneous spherical data" classify spherical subgroups. This is known as the Luna Conjecture. This classification is now complete according to Luna's program; see contributions of Bravi, Cupit-Foutou, Losev and Pezzini.
As conjectured by Knop, every "smooth" affine spherical variety is uniquely determined by its weight monoid. This uniqueness result was proven by Losev.
Knop (2013) has been developing a program to classify spherical varieties in arbitrary characteristic.
References
1. Brion, Michel (2007). "The total coordinate ring of a wonderful variety". Journal of Algebra. 313 (1): 61–99. arXiv:math/0603157. doi:10.1016/j.jalgebra.2006.12.022. S2CID 15154549.
• Paolo Bravi, Wonderful varieties of type E, Representation theory 11 (2007), 174–191.
• Paolo Bravi and Stéphanie Cupit-Foutou, Classification of strict wonderful varieties, Annales de l'Institut Fourier (2010), Volume 60, Issue 2, 641–681.
• Paolo Bravi and Guido Pezzini, Wonderful varieties of type D, Representation theory 9 (2005), pp. 578–637.
• Paolo Bravi and Guido Pezzini, Wonderful subgroups of reductive groups and spherical systems, J. Algebra 409 (2014), 101–147.
• Paolo Bravi and Guido Pezzini, The spherical systems of the wonderful reductive subgroups, J. Lie Theory 25 (2015), 105–123.
• Paolo Bravi and Guido Pezzini, Primitive wonderful varieties, Arxiv 1106.3187.
• Stéphanie Cupit-Foutou, Wonderful Varieties. a geometrical realization, Arxiv 0907.2852.
• Michel Brion, "Introduction to actions of algebraic groups"
• Knop, Friedrich (2013), "Localization of spherical varieties", Algebra & Number Theory, 8 (3): 703–728, arXiv:1303.2561, doi:10.2140/ant.2014.8.703, S2CID 119293458
• Losev, Ivan (2006). "Proof of the Knop conjecture". arXiv:math/0612561.
• Losev, Ivan (2009). "Uniqueness properties for spherical varieties". arXiv:0904.2937 [math.AG].
• Luna, Dominique (2001), "Variétés sphériques de type A", Publications Mathématiques de l'Institut des Hautes Études Scientifiques, 94: 161–226, doi:10.1007/s10240-001-8194-0, S2CID 123850545
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Legendre wavelet
In functional analysis, compactly supported wavelets derived from Legendre polynomials are termed Legendre wavelets or spherical harmonic wavelets.[1] Legendre functions have widespread applications in which spherical coordinate system is appropriate.[2][3][4] As with many wavelets there is no nice analytical formula for describing these harmonic spherical wavelets. The low-pass filter associated to Legendre multiresolution analysis is a finite impulse response (FIR) filter.
Wavelets associated to FIR filters are commonly preferred in most applications.[3] An extra appealing feature is that the Legendre filters are linear phase FIR (i.e. multiresolution analysis associated with linear phase filters). These wavelets have been implemented on MATLAB (wavelet toolbox). Although being compactly supported wavelet, legdN are not orthogonal (but for N = 1).[5]
Legendre multiresolution filters
Associated Legendre polynomials are the colatitudinal part of the spherical harmonics which are common to all separations of Laplace's equation in spherical polar coordinates.[2] The radial part of the solution varies from one potential to another, but the harmonics are always the same and are a consequence of spherical symmetry. Spherical harmonics $P_{n}(z)$ are solutions of the Legendre $2^{nd}$-order differential equation, n integer:
$\left(1-z^{2}\right){\frac {d^{2}y}{dz^{2}}}-2z{\frac {dy}{dz}}+n(n+1)y=0.$
$P_{n}(\cos(\theta ))$ polynomials can be used to define the smoothing filter $H(\omega )$ of a multiresolution analysis (MRA).[6] Since the appropriate boundary conditions for an MRA are $|H(0)|=1$ and $|H(\pi )|=0$, the smoothing filter of an MRA can be defined so that the magnitude of the low-pass $|H(\omega )|$ can be associated to Legendre polynomials according to: $\nu =2n+1.$
$|H_{\nu }(\omega )|=\left|{\frac {P_{\nu }\left(\cos \left({\frac {\omega }{2}}\right)\right)}{P_{\nu }\cos(0)}}\right|$
Illustrative examples of filter transfer functions for a Legendre MRA are shown in figure 1, for $\nu =1,3,5.$ A low-pass behaviour is exhibited for the filter H, as expected. The number of zeroes within $-\pi <\omega <\pi $ is equal to the degree of the Legendre polynomial. Therefore, the roll-off of side-lobes with frequency is easily controlled by the parameter $\nu $.
The low-pass filter transfer function is given by
$H_{\nu }(\omega )=-e^{-j\nu {\frac {\omega -\pi }{2}}}P_{\nu }\left(\cos \left({\tfrac {\omega }{2}}\right)\right)$
The transfer function of the high-pass analysing filter $G_{\nu }(\omega )$ is chosen according to Quadrature mirror filter condition,[6][7] yielding:
$H_{\nu }(\omega )=-e^{-j{(\nu -2)}{\frac {\omega }{2}}}P_{\nu }\left(\sin \left({\tfrac {\omega }{2}}\right)\right)$
Indeed, $|G_{\nu }(0)|=0$ and $|G_{\nu }(\pi )|=1$, as expected.
Legendre multiresolution filter coefficients
A suitable phase assignment is done so as to properly adjust the transfer function $H_{\nu }(\omega )$ to the form
$H_{\nu }(\omega )={\frac {1}{\sqrt {2}}}\sum _{k\in Z}h_{k}^{\nu }e^{-j\omega k}$
The filter coefficients $\{h_{k}\}_{k\in \mathbb {Z} }$ are given by:
$h_{k}^{\nu }=-{\frac {\sqrt {2}}{2^{2\nu }}}{\binom {2k}{k}}{\binom {2\nu -2k}{\nu -k}}$
from which the symmetry:
${h_{k}^{\nu }}={h_{\nu -k}^{\nu }},$
follows. There are just $\nu +1$ non-zero filter coefficients on $H_{n}(\omega )$, so that the Legendre wavelets have compact support for every odd integer $\nu $.
Table I - Smoothing Legendre FIR filter coefficients for $\nu =1,3,5$ ($N$ is the wavelet order.)
$\nu =1(N=1)$ $\nu =3(N=2)$ $\nu =5(N=3)$
$h_{0}$ $-{\tfrac {\sqrt {2}}{2}}$ $-5{\tfrac {\sqrt {2}}{16}}$ $-63{\tfrac {\sqrt {2}}{256}}$
$h_{1}$ $-{\tfrac {\sqrt {2}}{2}}$ $-3{\tfrac {\sqrt {2}}{16}}$ $-35{\tfrac {\sqrt {2}}{256}}$
$h_{2}$ $-3{\tfrac {\sqrt {2}}{16}}$ $-30{\tfrac {\sqrt {2}}{256}}$
$h_{3}$ $-5{\tfrac {\sqrt {2}}{16}}$ $-30{\tfrac {\sqrt {2}}{256}}$
$h_{4}$ $-35{\tfrac {\sqrt {2}}{256}}$
$h_{5}$ $-63{\tfrac {\sqrt {2}}{256}}$
N.B. The minus signal can be suppressed.
MATLAB implementation of Legendre wavelets
Legendre wavelets can be easily loaded into the MATLAB wavelet toolbox—The m-files to allow the computation of Legendre wavelet transform, details and filter are (freeware) available. The finite support width Legendre family is denoted by legd (short name). Wavelets: 'legdN'. The parameter N in the legdN family is found according to $2N=\nu +1$ (length of the MRA filters).
Legendre wavelets can be derived from the low-pass reconstruction filter by an iterative procedure (the cascade algorithm). The wavelet has compact support and finite impulse response AMR filters (FIR) are used (table 1). The first wavelet of the Legendre's family is exactly the well-known Haar wavelet. Figure 2 shows an emerging pattern that progressively looks like the wavelet's shape.
The Legendre wavelet shape can be visualised using the wavemenu command of MATLAB. Figure 3 shows legd8 wavelet displayed using MATLAB. Legendre Polynomials are also associated with windows families.[8]
Legendre wavelet packets
Wavelet packets (WP) systems derived from Legendre wavelets can also be easily accomplished. Figure 5 illustrates the WP functions derived from legd2.
References
1. Lira et al
2. Gradshteyn, Izrail Solomonovich; Ryzhik, Iosif Moiseevich; Geronimus, Yuri Veniaminovich; Tseytlin, Michail Yulyevich; Jeffrey, Alan (2015) [October 2014]. Zwillinger, Daniel; Moll, Victor Hugo (eds.). Table of Integrals, Series, and Products. Translated by Scripta Technica, Inc. (8 ed.). Academic Press, Inc. ISBN 978-0-12-384933-5. LCCN 2014010276.
3. Colomer and Colomer
4. Ramm and Zaslavsky
5. Herley and Vetterli
6. Mallat
7. Vetterli and Herley
8. Jaskula
Bibliography
• M.M.S. Lira, H.M. de Oliveira, M.A. Carvalho Jr, R.M.C.Souza, Compactly Supported Wavelets Derived from Legendre Polynomials: Spherical Harmonic Wavelets, In: Computational Methods in Circuits and Systems Applications, N.E. Mastorakis, I.A. Stahopulos, C. Manikopoulos, G.E. Antoniou, V.M. Mladenov, I.F. Gonos Eds., WSEAS press, pp. 211–215, 2003. ISBN 960-8052-88-2. Available at ee.ufpe.br
• A. A. Colomer and A. A. Colomer, Adaptive ECG Data Compression Using Discrete Legendre Transform, Digital Signal Processing, 7, 1997, pp. 222–228.
• A.G. Ramm, A.I. Zaslavsky, X-Ray Transform, the Legendre Transform, and Envelopes, J. of Math. Analysis and Appl., 183, pp. 528–546, 1994.
• C. Herley, M. Vetterli, Orthogonalization of Compactly Supported Wavelet Bases, IEEE Digital Signal Process. Workshop, 13-16 Sep., pp. 1.7.1-1.7.2, 1992.
• S. Mallat, A Theory for Multiresolution Signal Decomposition: The Wavelet Representation, IEEE Transactions on Pattern Analysis and Machine Intelligence, 11, July pp. 674–693, 1989.
• M. Vetterli, C. Herly, Wavelets and Filter Banks: Theory and Design, IEEE Trans. on Acoustics, Speech, and Signal Processing, 40, 9, p. 2207, 1992.
• M. Jaskula, New Windows Family Based on Modified Legendre Polynomials, IEEE Instrum. And Measurement Technol. Conf., Anchorage, AK, May, 2002, pp. 553–556.
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Spherical wedge
In geometry, a spherical wedge or ungula is a portion of a ball bounded by two plane semidisks and a spherical lune (termed the wedge's base). The angle between the radii lying within the bounding semidisks is the dihedral α. If AB is a semidisk that forms a ball when completely revolved about the z-axis, revolving AB only through a given α produces a spherical wedge of the same angle α.[1] Beman (2008)[2] remarks that "a spherical wedge is to the sphere of which it is a part as the angle of the wedge is to a perigon." [A] A spherical wedge of α = π radians (180°) is called a hemisphere, while a spherical wedge of α = 2π radians (360°) constitutes a complete ball.
The volume of a spherical wedge can be intuitively related to the AB definition in that while the volume of a ball of radius r is given by 4/3πr3, the volume a spherical wedge of the same radius r is given by[3]
$V={\frac {\alpha }{2\pi }}\cdot {\tfrac {4}{3}}\pi r^{3}={\tfrac {2}{3}}\alpha r^{3}\,.$
Extrapolating the same principle and considering that the surface area of a sphere is given by 4πr2, it can be seen that the surface area of the lune corresponding to the same wedge is given by[A]
$A={\frac {\alpha }{2\pi }}\cdot 4\pi r^{2}=2\alpha r^{2}\,.$
Hart (2009)[3] states that the "volume of a spherical wedge is to the volume of the sphere as the number of degrees in the [angle of the wedge] is to 360".[A] Hence, and through derivation of the spherical wedge volume formula, it can be concluded that, if Vs is the volume of the sphere and Vw is the volume of a given spherical wedge,
${\frac {V_{\mathrm {w} }}{V_{\mathrm {s} }}}={\frac {\alpha }{2\pi }}\,.$
Also, if Sl is the area of a given wedge's lune, and Ss is the area of the wedge's sphere,[4][A]
${\frac {S_{\mathrm {l} }}{S_{\mathrm {s} }}}={\frac {\alpha }{2\pi }}\,.$
See also
• Spherical cap
• Spherical segment
• Ungula
Notes
A. ^ A distinction is sometimes drawn between the terms "sphere" and "ball", where a sphere is regarded as being merely the outer surface of a solid ball. It is common to use the terms interchangeably, as the commentaries of both Beman (2008) and Hart (2008) do.
References
1. Morton, P. (1830). Geometry, Plane, Solid, and Spherical, in Six Books. Baldwin & Cradock. p. 180.
2. Beman, D. W. (2008). New Plane and Solid Geometry. BiblioBazaar. p. 338. ISBN 0-554-44701-0.
3. Hart, C. A. (2009). Solid Geometry. BiblioBazaar. p. 465. ISBN 1-103-11804-8.
4. Avallone, E. A.; Baumeister, T.; Sadegh, A.; Marks, L. S. (2006). Marks' Standard Handbook for Mechanical Engineers. McGraw-Hill Professional. p. 43. ISBN 0-07-142867-4.
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Spherical segment
In geometry, a spherical segment is the solid defined by cutting a sphere or a ball with a pair of parallel planes. It can be thought of as a spherical cap with the top truncated, and so it corresponds to a spherical frustum.
The surface of the spherical segment (excluding the bases) is called spherical zone.
If the radius of the sphere is called R, the radii of the spherical segment bases are r1 and r2, and the height of the segment (the distance from one parallel plane to the other) called h, then the volume of the spherical segment is
$V={\frac {\pi h}{6}}\left(3r_{1}^{2}+3r_{2}^{2}+h^{2}\right).$
The curved surface area of the spherical zone—which excludes the top and bottom bases—is given by
$A=2\pi Rh.$
See also
• Spherical cap
• Spherical wedge
• Spherical sector
References
• Kern, William F.; Bland, James R. (1938). Solid Mensuration with Proofs. p. 95–97.
External links
Wikimedia Commons has media related to Spherical segments.
• Weisstein, Eric W. "Spherical segment". MathWorld.
• Weisstein, Eric W. "Spherical zone". MathWorld.
• Summary of spherical formulas
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Spherically complete field
In mathematics, a field K with an absolute value is called spherically complete if the intersection of every decreasing sequence of balls (in the sense of the metric induced by the absolute value) is nonempty:
$B_{1}\supseteq B_{2}\supseteq \cdots \Rightarrow \bigcap _{n\in {\mathbf {N} }}B_{n}\neq \emptyset .$
The definition can be adapted also to a field K with a valuation v taking values in an arbitrary ordered abelian group: (K,v) is spherically complete if every collection of balls that is totally ordered by inclusion has a nonempty intersection.
Spherically complete fields are important in nonarchimedean functional analysis, since many results analogous to theorems of classical functional analysis require the base field to be spherically complete.
Examples
• Any locally compact field is spherically complete. This includes, in particular, the fields Qp of p-adic numbers, and any of their finite extensions.
• Every spherically complete field is complete. On the other hand, Cp, the completion of the algebraic closure of Qp, is not spherically complete.[1]
• Any field of Hahn series is spherically complete.
References
1. Robert, p. 143
Schneider, Peter (2001). Nonarchimedean Functional Analysis. Springer. ISBN 3-540-42533-0.
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Sphericity (graph theory)
In graph theory, the sphericity of a graph is a graph invariant defined to be the smallest dimension of Euclidean space required to realize the graph as an intersection graph of unit spheres. The sphericity of a graph is a generalization of the boxicity and cubicity invariants defined by F.S. Roberts in the late 1960s.[1][2] The concept of sphericity was first introduced by Hiroshi Maehara in the early 1980s.[3]
Definition
Let $G$ be a graph. Then the sphericity of $G$, denoted by $\operatorname {sph} (G)$, is the smallest integer $n$ such that $G$ can be realized as an intersection graph of unit spheres in $n$-dimensional Euclidean space $\mathbb {R} ^{n}$.[4]
Sphericity can also be defined using the language of space graphs as follows. For a finite set of points in some $n$-dimensional Euclidean space, a space graph is built by connecting pairs of points with a line segment when their Euclidean distance is less than some specified constant. Then the sphericity of a graph $G$ is the minimum $n$ such that $G$ is isomorphic to a space graph in $\mathbb {R} ^{n}$.[3]
Graphs of sphericity 1 are known as interval graphs or indifference graphs. Graphs of sphericity 2 are known as unit disk graphs.
Bounds
The sphericity of certain graph classes can be computed exactly. The following sphericities were given by Maehara on page 56 of his original paper on the topic.
Graph Description Sphericity Notes
$K_{1}$ Complete graph 0
$K_{n}$ Complete graph 1 $n>1$
$P_{n}$ Path graph 1 $n>1$
$C_{n}$ Circuit graph 2 $n>3$
$K_{m(2)}$ Complete m-partite graph on m sets of size 2 2 $m>1$
The most general known upper bound on sphericity is as follows. Assuming the graph is not complete, then $\operatorname {sph} (G)\leq |G|-\omega (G)$ where $\omega (G)$ is the clique number of $G$ and $|G|$ denotes the number of vertices of $G.$[3]
References
1. Roberts, F. S. (1969). On the boxicity and cubicity of a graph. In W. T. Tutte (Ed.), Recent Progress in Combinatorics (pp. 301–310). San Diego, CA: Academic Press. ISBN 978-0-12-705150-5
2. Fishburn, Peter C (1983-12-01). "On the sphericity and cubicity of graphs". Journal of Combinatorial Theory, Series B. 35 (3): 309–318. doi:10.1016/0095-8956(83)90057-6. ISSN 0095-8956.
3. Maehara, Hiroshi (1984-01-01). "Space graphs and sphericity". Discrete Applied Mathematics. 7 (1): 55–64. doi:10.1016/0166-218X(84)90113-6. ISSN 0166-218X.
4. Maehara, Hiroshi (1986-03-01). "On the sphericity of the graphs of semiregular polyhedra". Discrete Mathematics. 58 (3): 311–315. doi:10.1016/0012-365X(86)90150-0. ISSN 0012-365X.
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Sphericon
In solid geometry, the sphericon is a solid that has a continuous developable surface with two congruent, semi-circular edges, and four vertices that define a square. It is a member of a special family of rollers that, while being rolled on a flat surface, bring all the points of their surface to contact with the surface they are rolling on. It was discovered independently by carpenter Colin Roberts (who named it) in the UK in 1969,[1] by dancer and sculptor Alan Boeding of MOMIX in 1979,[2] and by inventor David Hirsch, who patented it in Israel in 1980.[3]
Construction
The sphericon may be constructed from a bicone (a double cone) with an apex angle of 90 degrees, by splitting the bicone along a plane through both apexes, rotating one of the two halves by 90 degrees, and reattaching the two halves.[4] Alternatively, the surface of a sphericon can be formed by cutting and gluing a paper template in the form of four circular sectors (with central angles $\pi /{\sqrt {2}}$) joined edge-to-edge.[5]
Geometric properties
The surface area of a sphericon with radius $r$ is given by
$S=2{\sqrt {2}}\pi r^{2}$.
The volume is given by
$V={\frac {2}{3}}\pi r^{3}$,
exactly half the volume of a sphere with the same radius.
History
Around 1969, Colin Roberts (a carpenter from the UK) made a sphericon out of wood while attempting to carve a Möbius strip without a hole.[1]
In 1979, David Hirsch invented a device for generating a meander motion. The device consisted of two perpendicular half discs joined at their axes of symmetry. While examining various configurations of this device, he discovered that the form created by joining the two half discs, exactly at their diameter centers, is actually a skeletal structure of a solid made of two half bicones, joined at their square cross-sections with an offset angle of 90 degrees, and that the two objects have exactly the same meander motion. Hirsch filed a patent in Israel in 1980, and a year later, a pull toy named Wiggler Duck, based on Hirsch's device, was introduced by Playskool Company.
In 1999, Colin Roberts sent Ian Stewart a package containing a letter and two sphericon models. In response, Stewart wrote an article "Cone with a Twist" in his Mathematical Recreations column of Scientific American.[1] This sparked quite a bit of interest in the shape, and has been used by Tony Phillips to develop theories about mazes.[6] Robert's name for the shape, the sphericon, was taken by Hirsch as the name for his company, Sphericon Ltd.[7]
In popular culture
In 1979, modern dancer Alan Boeding designed his "Circle Walker" sculpture from two crosswise semicircles, a skeletal version of the sphericon. He began dancing with a scaled-up version of the sculpture in 1980 as part of an MFA program in sculpture at Indiana University, and after he joined the MOMIX dance company in 1984 the piece became incorporated into the company's performances.[2][8] The company's later piece "Dream Catcher" is based around a similar Boeding sculpture whose linked teardrop shapes incorporate the skeleton and rolling motion of the oloid, a similar rolling shape formed from two perpendicular circles each passing through the center of the other.[9]
In 2008, renowned British woodturner David Springett published the book "Woodturning Full Circle", which explains how sphericons (and other unusual solid forms, such as streptohedrons) can be made on a wood lathe.[10]
References
1. Stewart, Ian (October 1999). "Mathematical Recreations: Cone with a Twist". Scientific American. 281 (4): 116–117. JSTOR 26058451. Archived from the original on 2019-02-12.
2. Boeding, Alan (April 27, 1988), "Circle dancing", The Christian Science Monitor
3. David Haran Hirsch (1980): "Patent no. 59720: A device for generating a meander motion; Patent drawings; Patent application form; Patent claims
4. Paul J. Roberts (2010). "The Sphericon". Archived from the original on 2012-07-23.
5. A mesh at www.pjroberts.com/sphericon, archived by web.archive.org
6. Michele Emmer (2005). The Visual Mind II. MIT Press. pp. 667–685. ISBN 978-0-262-05076-0.
7. ""Sphericon Ltd. - Israel-Export" (pdf)" (PDF).
8. Green, Judith (May 2, 1991), "hits and misses at Momix: it's not quite dance, but it's sometimes art", Dance review, San Jose Mercury News
9. Anderson, Jack (February 8, 2001), "Leaping Lizards and Odd Denizens of the Desert", Dance Review, The New York Times
10. Springett, David, Woodturning Full Circle
External links
Look up sphericon in Wiktionary, the free dictionary.
• Sphericon construction animation at the National Curve Bank website.
• Paper model of a sphericon Make a sphericon
• Sphericon variations using regular polygons with different numbers of sides
• A Sphericon in Motion showing the characteristic wobbly motion as it rolls across a flat surface
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Spherinder
In four-dimensional geometry, the spherinder, or spherical cylinder or spherical prism, is a geometric object, defined as the Cartesian product of a 3-ball (or solid 2-sphere) of radius r1 and a line segment of length 2r2:
$D=\{(x,y,z,w)|x^{2}+y^{2}+z^{2}\leq r_{1}^{2},\ w^{2}\leq r_{2}^{2}\}$
Like the duocylinder, it is also analogous to a cylinder in 3-space, which is the Cartesian product of a disk with a line segment.
It can be seen in 3-dimensional space by stereographic projection as two concentric spheres, in a similar way that a tesseract (cubic prism) can be projected as two concentric cubes, and how a circular cylinder can be projected into 2-dimensional space as two concentric circles.
Relation to other shapes
In 3-space, a cylinder can be considered intermediate between a cube and a sphere. In 4-space there are three intermediate forms between the tesseract and the hypersphere. Altogether, they are the:
• tesseract (1-ball × 1-ball × 1-ball × 1-ball), whose hypersurface is eight cubes connected at 24 squares
• cubinder (2-ball × 1-ball × 1-ball)
• spherinder (3-ball × 1-ball), whose hypersurface is two 3-balls and a tube-like cell connected at the respective bounding spheres of the 3-balls
• duocylinder (2-ball × 2-ball)
• glome (4-ball), whose hypersurface is a 3-sphere without any connecting boundaries.
These constructions correspond to the five partitions of 4, the number of dimensions.
If the two ends of a spherinder are connected together, or equivalently if a sphere is dragged around a circle perpendicular to its 3-space, it traces out a spheritorus. If the two ends of an uncapped spherinder are rolled inward, the resulting shape is a torisphere.
Spherindrical coordinate system
One can define a "spherindrical" coordinate system (r, θ, φ, w), consisting of spherical coordinates with an extra coordinate w. This is analogous to how cylindrical coordinates are defined: r and φ being polar coordinates with an elevation coordinate z. Spherindrical coordinates can be converted to Cartesian coordinates using the formulas
${\begin{aligned}x&=r\cos \varphi \sin \theta \\y&=r\sin \varphi \sin \theta \\z&=r\cos \theta \\w&=w\end{aligned}}$
where r is the radius, θ is the zenith angle, φ is the azimuthal angle, and w is the height. Cartesian coordinates can be converted to spherindrical coordinates using the formulas
${\begin{aligned}r&={\sqrt {x^{2}+y^{2}+z^{2}}}\\\varphi &=\arctan {\frac {y}{x}}\\\theta &=\operatorname {arccot} {\frac {z}{\sqrt {x^{2}+y^{2}}}}\\w&=w\end{aligned}}$
The hypervolume element for spherindrical coordinates is $\mathrm {d} H=r^{2}\sin {\theta }\,\mathrm {d} r\,\mathrm {d} \theta \,\mathrm {d} \varphi \,\mathrm {d} w,$ which can be derived by computing the Jacobian.
Measurements
Hypervolume
Given a spherinder with a spherical base of radius r and a height h, the hypervolume of the spherinder is given by
$H={\frac {4}{3}}\pi r^{3}h$
Surface volume
The surface volume of a spherinder, like the surface area of a cylinder, is made up of three parts:
• the volume of the top base: $ {\frac {4}{3}}\pi r^{3}$
• the volume of the bottom base: $ {\frac {4}{3}}\pi r^{3}$
• the volume of the lateral 3D surface: $ 4\pi r^{2}h$, which is the surface area of the spherical base times the height
Therefore, the total surface volume is
$SV={\frac {8}{3}}\pi r^{3}+4\pi r^{2}h$
Proof
The above formulas for hypervolume and surface volume can be proven using integration. The hypervolume of an arbitrary 4D region is given by the quadruple integral
$H=\iiiint \limits _{D}\mathrm {d} H$
The hypervolume of the spherinder can be integrated over spherindrical coordinates.
$H_{\mathrm {spherinder} }=\iiiint \limits _{D}\mathrm {d} H=\int _{0}^{h}\int _{0}^{2\pi }\int _{0}^{\pi }\int _{0}^{R}r^{2}\sin {\theta }\,\mathrm {d} r\,\mathrm {d} \theta \,\mathrm {d} \varphi \,\mathrm {d} w={\frac {4}{3}}\pi R^{3}h$
Related 4-polytopes
The spherinder is related to the uniform prismatic polychora, which are cartesian product of a regular or semiregular polyhedron and a line segment. There are eighteen convex uniform prisms based on the Platonic and Archimedean solids (tetrahedral prism, truncated tetrahedral prism, cubic prism, cuboctahedral prism, octahedral prism, rhombicuboctahedral prism, truncated cubic prism, truncated octahedral prism, truncated cuboctahedral prism, snub cubic prism, dodecahedral prism, icosidodecahedral prism, icosahedral prism, truncated dodecahedral prism, rhombicosidodecahedral prism, truncated icosahedral prism, truncated icosidodecahedral prism, snub dodecahedral prism), plus an infinite family based on antiprisms, and another infinite family of uniform duoprisms, which are products of two regular polygons.
See also
• Clifford torus
References
• The Fourth Dimension Simply Explained, Henry P. Manning, Munn & Company, 1910, New York. Available from the University of Virginia library. Also accessible online: The Fourth Dimension Simply Explained—contains a description of duoprisms and duocylinders (double cylinders)
• The Visual Guide To Extra Dimensions: Visualizing The Fourth Dimension, Higher-Dimensional Polytopes, And Curved Hypersurfaces, Chris McMullen, 2008, ISBN 978-1438298924
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Capsule (geometry)
A capsule (from Latin capsula, "small box or chest"), or stadium of revolution, is a basic three-dimensional geometric shape consisting of a cylinder with hemispherical ends.[1] Another name for this shape is spherocylinder.[2][3][4][5]
It can also be referred to as an oval although the sides (either vertical or horizontal) are straight parallel.
Usages
The shape is used for some objects like containers for pressurised gases, windows of places like a jet, software buttons, building domes (like the U.S. Capitol, having the windows of the top hat that depict The Apotheosis of Washington inside designed with the appearance of the shape & placed in an omnidirectional pattern), mirrors, and pharmaceutical capsules.
In chemistry and physics, this shape is used as a basic model for non-spherical particles. It appears, in particular as a model for the molecules in liquid crystals[6][3][4] or for the particles in granular matter.[5][7][8]
Formulas
The volume $V$ of a capsule is calculated by adding the volume of a ball of radius $r$ (that accounts for the two hemispheres) to the volume of the cylindrical part. Hence, if the cylinder has height $h$,
$V={\frac {4}{3}}\pi r^{3}+(\pi r^{2}h)=\pi r^{2}\left({\frac {4}{3}}r+h\right)$.
The surface area of a capsule of radius $r$ whose cylinder part has height $h$ is $2\pi r(2r+h)$.
Generalization
A capsule can be equivalently described as the Minkowski sum of a ball of radius $r$ with a line segment of length $a$.[5] By this description, capsules can be straightforwardly generalized as Minkowski sums of a ball with a polyhedron. The resulting shape is called a spheropolyhedron.[7][8]
Related shapes
A capsule is the three-dimensional shape obtained by revolving the two-dimensional stadium around the line of symmetry that bisects the semicircles.
References
1. Sarkar, Dipankar; Halas, N. J. (1997). "General vector basis function solution of Maxwell's equations". Physical Review E. 56 (1, part B): 1102–1112. doi:10.1103/PhysRevE.56.1102. MR 1459098.
2. Kihara, Taro (1951). "The Second Virial Coefficient of Non-Spherical Molecules". Journal of the Physical Society of Japan. 6 (5): 289–296. doi:10.1143/JPSJ.6.289.
3. Frenkel, Daan (September 10, 1987). "Onsager's spherocylinders revisited". Journal of Physical Chemistry. 91 (19): 4912–4916. doi:10.1021/j100303a008. hdl:1874/8823. S2CID 96013495.
4. Dzubiella, Joachim; Schmidt, Matthias; Löwen, Hartmut (2000). "Topological defects in nematic droplets of hard spherocylinders". Physical Review E. 62 (4): 5081–5091. arXiv:cond-mat/9906388. Bibcode:2000PhRvE..62.5081D. doi:10.1103/PhysRevE.62.5081. PMID 11089056. S2CID 31381033.
5. Pournin, Lionel; Weber, Mats; Tsukahara, Michel; Ferrez, Jean-Albert; Ramaioli, Marco; Liebling, Thomas M. (2005). "Three-dimensional distinct element simulation of spherocylinder crystallization". Granular Matter. 7 (2–3): 119–126. doi:10.1007/s10035-004-0188-4.
6. Onsager, Lars (May 1949). "The effects of shape on the interaction of colloidal particles". Annals of the New York Academy of Sciences. 51 (4): 627–659. doi:10.1111/j.1749-6632.1949.tb27296.x. S2CID 84562683.
7. Pournin, Lionel; Liebling, Thomas M. (2005). "A generalization of Distinct Element Method to tridimensional particles with complex shapes". Powders and Grains 2005 Proceedings vol. II. A.A. Balkema, Rotterdam. pp. 1375–1378.
8. Pournin, Lionel; Liebling, Thomas M. (2009). "From spheres to spheropolyhedra: generalized Distinct Element Methodology and algorithm analysis". In Cook, William; Lovász, László; Vygen, Jens (eds.). Research Trends in Combinatorial Optimization. Springer, Berlin. pp. 347–363. doi:10.1007/978-3-540-76796-1_16. ISBN 978-3-540-76795-4.
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Coordinate system
In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space.[1][2] The order of the coordinates is significant, and they are sometimes identified by their position in an ordered tuple and sometimes by a letter, as in "the x-coordinate". The coordinates are taken to be real numbers in elementary mathematics, but may be complex numbers or elements of a more abstract system such as a commutative ring. The use of a coordinate system allows problems in geometry to be translated into problems about numbers and vice versa; this is the basis of analytic geometry.[3]
Common coordinate systems
Number line
Main article: Number line
The simplest example of a coordinate system is the identification of points on a line with real numbers using the number line. In this system, an arbitrary point O (the origin) is chosen on a given line. The coordinate of a point P is defined as the signed distance from O to P, where the signed distance is the distance taken as positive or negative depending on which side of the line P lies. Each point is given a unique coordinate and each real number is the coordinate of a unique point.[4]
Cartesian coordinate system
Main article: Cartesian coordinate system
The prototypical example of a coordinate system is the Cartesian coordinate system. In the plane, two perpendicular lines are chosen and the coordinates of a point are taken to be the signed distances to the lines.
In three dimensions, three mutually orthogonal planes are chosen and the three coordinates of a point are the signed distances to each of the planes.[5] This can be generalized to create n coordinates for any point in n-dimensional Euclidean space.
Depending on the direction and order of the coordinate axes, the three-dimensional system may be a right-handed or a left-handed system. This is one of many coordinate systems.
Polar coordinate system
Main article: Polar coordinate system
Another common coordinate system for the plane is the polar coordinate system.[6] A point is chosen as the pole and a ray from this point is taken as the polar axis. For a given angle θ, there is a single line through the pole whose angle with the polar axis is θ (measured counterclockwise from the axis to the line). Then there is a unique point on this line whose signed distance from the origin is r for given number r. For a given pair of coordinates (r, θ) there is a single point, but any point is represented by many pairs of coordinates. For example, (r, θ), (r, θ+2π) and (−r, θ+π) are all polar coordinates for the same point. The pole is represented by (0, θ) for any value of θ.
Cylindrical and spherical coordinate systems
Main articles: Cylindrical coordinate system and Spherical coordinate system
There are two common methods for extending the polar coordinate system to three dimensions. In the cylindrical coordinate system, a z-coordinate with the same meaning as in Cartesian coordinates is added to the r and θ polar coordinates giving a triple (r, θ, z).[7] Spherical coordinates take this a step further by converting the pair of cylindrical coordinates (r, z) to polar coordinates (ρ, φ) giving a triple (ρ, θ, φ).[8]
Homogeneous coordinate system
Main article: Homogeneous coordinates
A point in the plane may be represented in homogeneous coordinates by a triple (x, y, z) where x/z and y/z are the Cartesian coordinates of the point.[9] This introduces an "extra" coordinate since only two are needed to specify a point on the plane, but this system is useful in that it represents any point on the projective plane without the use of infinity. In general, a homogeneous coordinate system is one where only the ratios of the coordinates are significant and not the actual values.
Other commonly used systems
Some other common coordinate systems are the following:
• Curvilinear coordinates are a generalization of coordinate systems generally; the system is based on the intersection of curves.
• Orthogonal coordinates: coordinate surfaces meet at right angles
• Skew coordinates: coordinate surfaces are not orthogonal
• The log-polar coordinate system represents a point in the plane by the logarithm of the distance from the origin and an angle measured from a reference line intersecting the origin.
• Plücker coordinates are a way of representing lines in 3D Euclidean space using a six-tuple of numbers as homogeneous coordinates.
• Generalized coordinates are used in the Lagrangian treatment of mechanics.
• Canonical coordinates are used in the Hamiltonian treatment of mechanics.
• Barycentric coordinate system as used for ternary plots and more generally in the analysis of triangles.
• Trilinear coordinates are used in the context of triangles.
There are ways of describing curves without coordinates, using intrinsic equations that use invariant quantities such as curvature and arc length. These include:
• The Whewell equation relates arc length and the tangential angle.
• The Cesàro equation relates arc length and curvature.
Coordinates of geometric objects
Coordinates systems are often used to specify the position of a point, but they may also be used to specify the position of more complex figures such as lines, planes, circles or spheres. For example, Plücker coordinates are used to determine the position of a line in space.[10] When there is a need, the type of figure being described is used to distinguish the type of coordinate system, for example the term line coordinates is used for any coordinate system that specifies the position of a line.
It may occur that systems of coordinates for two different sets of geometric figures are equivalent in terms of their analysis. An example of this is the systems of homogeneous coordinates for points and lines in the projective plane. The two systems in a case like this are said to be dualistic. Dualistic systems have the property that results from one system can be carried over to the other since these results are only different interpretations of the same analytical result; this is known as the principle of duality.[11]
Transformations
See also: Active and passive transformation
Main article: List of common coordinate transformations
There are often many different possible coordinate systems for describing geometrical figures. The relationship between different systems is described by coordinate transformations, which give formulas for the coordinates in one system in terms of the coordinates in another system. For example, in the plane, if Cartesian coordinates (x, y) and polar coordinates (r, θ) have the same origin, and the polar axis is the positive x axis, then the coordinate transformation from polar to Cartesian coordinates is given by x = r cosθ and y = r sinθ.
With every bijection from the space to itself two coordinate transformations can be associated:
• Such that the new coordinates of the image of each point are the same as the old coordinates of the original point (the formulas for the mapping are the inverse of those for the coordinate transformation)
• Such that the old coordinates of the image of each point are the same as the new coordinates of the original point (the formulas for the mapping are the same as those for the coordinate transformation)
For example, in 1D, if the mapping is a translation of 3 to the right, the first moves the origin from 0 to 3, so that the coordinate of each point becomes 3 less, while the second moves the origin from 0 to −3, so that the coordinate of each point becomes 3 more.
Coordinate lines/curves and planes/surfaces
"Coordinate line" redirects here. Not to be confused with Line coordinates.
"Coordinate plane" redirects here. Not to be confused with Plane coordinates.
In two dimensions, if one of the coordinates in a point coordinate system is held constant and the other coordinate is allowed to vary, then the resulting curve is called a coordinate curve. If the coordinate curves are, in fact, straight lines, they may be called coordinate lines. In Cartesian coordinate systems, coordinates lines are mutually orthogonal, and are known as coordinate axes. For other coordinate systems the coordinates curves may be general curves. For example, the coordinate curves in polar coordinates obtained by holding r constant are the circles with center at the origin. A coordinate system for which some coordinate curves are not lines is called a curvilinear coordinate system.[12] This procedure does not always make sense, for example there are no coordinate curves in a homogeneous coordinate system.
In three-dimensional space, if one coordinate is held constant and the other two are allowed to vary, then the resulting surface is called a coordinate surface. For example, the coordinate surfaces obtained by holding ρ constant in the spherical coordinate system are the spheres with center at the origin. In three-dimensional space the intersection of two coordinate surfaces is a coordinate curve. In the Cartesian coordinate system we may speak of coordinate planes.
Similarly, coordinate hypersurfaces are the (n − 1)-dimensional spaces resulting from fixing a single coordinate of an n-dimensional coordinate system.[13]
Coordinate maps
Main article: Coordinate map
Further information: Manifold
The concept of a coordinate map, or coordinate chart is central to the theory of manifolds. A coordinate map is essentially a coordinate system for a subset of a given space with the property that each point has exactly one set of coordinates. More precisely, a coordinate map is a homeomorphism from an open subset of a space X to an open subset of Rn.[14] It is often not possible to provide one consistent coordinate system for an entire space. In this case, a collection of coordinate maps are put together to form an atlas covering the space. A space equipped with such an atlas is called a manifold and additional structure can be defined on a manifold if the structure is consistent where the coordinate maps overlap. For example, a differentiable manifold is a manifold where the change of coordinates from one coordinate map to another is always a differentiable function.
Orientation-based coordinates
In geometry and kinematics, coordinate systems are used to describe the (linear) position of points and the angular position of axes, planes, and rigid bodies.[15] In the latter case, the orientation of a second (typically referred to as "local") coordinate system, fixed to the node, is defined based on the first (typically referred to as "global" or "world" coordinate system). For instance, the orientation of a rigid body can be represented by an orientation matrix, which includes, in its three columns, the Cartesian coordinates of three points. These points are used to define the orientation of the axes of the local system; they are the tips of three unit vectors aligned with those axes.
Geographic systems
The Earth as a whole is one of the most common geometric spaces requiring the precise measurement of location, and thus coordinate systems. Starting with the Greeks of the Hellenistic period, a variety of coordinate systems have been developed based on the types above, including:
• Geographic coordinate system, the spherical coordinates of latitude and longitude
• Projected coordinate systems, including thousands of cartesian coordinate systems, each based on a map projection to create a planar surface of the world or a region.
• Geocentric coordinate system, a three-dimensional cartesian coordinate system that models the earth as an object, and are most commonly used for modeling the orbits of satellites, including the Global Positioning System and other satellite navigation systems.
See also
• Absolute angular momentum
• Alphanumeric grid
• Axes conventions in engineering
• Celestial coordinate system
• Coordinate-free
• Fractional coordinates
• Frame of reference
• Galilean transformation
• Grid reference
• Nomogram, graphical representations of different coordinate systems
• Reference system
• Rotation of axes
• Translation of axes
Relativistic coordinate systems
• Eddington–Finkelstein coordinates
• Gaussian polar coordinates
• Gullstrand–Painlevé coordinates
• Isotropic coordinates
• Kruskal–Szekeres coordinates
• Schwarzschild coordinates
References
Citations
1. Woods p. 1
2. Weisstein, Eric W. "Coordinate System". MathWorld.
3. Weisstein, Eric W. "Coordinates". MathWorld.
4. Stewart, James B.; Redlin, Lothar; Watson, Saleem (2008). College Algebra (5th ed.). Brooks Cole. pp. 13–19. ISBN 978-0-495-56521-5.
5. Moon P, Spencer DE (1988). "Rectangular Coordinates (x, y, z)". Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions (corrected 2nd, 3rd print ed.). New York: Springer-Verlag. pp. 9–11 (Table 1.01). ISBN 978-0-387-18430-2.
6. Finney, Ross; George Thomas; Franklin Demana; Bert Waits (June 1994). Calculus: Graphical, Numerical, Algebraic (Single Variable Version ed.). Addison-Wesley Publishing Co. ISBN 0-201-55478-X.
7. Margenau, Henry; Murphy, George M. (1956). The Mathematics of Physics and Chemistry. New York City: D. van Nostrand. p. 178. ISBN 978-0-88275-423-9. LCCN 55010911. OCLC 3017486.
8. Morse PM, Feshbach H (1953). Methods of Theoretical Physics, Part I. New York: McGraw-Hill. p. 658. ISBN 0-07-043316-X. LCCN 52011515.
9. Jones, Alfred Clement (1912). An Introduction to Algebraical Geometry. Clarendon.
10. Hodge, W.V.D.; D. Pedoe (1994) [1947]. Methods of Algebraic Geometry, Volume I (Book II). Cambridge University Press. ISBN 978-0-521-46900-5.
11. Woods p. 2
12. Tang, K. T. (2006). Mathematical Methods for Engineers and Scientists. Vol. 2. Springer. p. 13. ISBN 3-540-30268-9.
13. Liseikin, Vladimir D. (2007). A Computational Differential Geometry Approach to Grid Generation. Springer. p. 38. ISBN 978-3-540-34235-9.
14. Munkres, James R. (2000) Topology. Prentice Hall. ISBN 0-13-181629-2.
15. Hanspeter Schaub; John L. Junkins (2003). "Rigid body kinematics". Analytical Mechanics of Space Systems. American Institute of Aeronautics and Astronautics. p. 71. ISBN 1-56347-563-4.
Sources
• Voitsekhovskii, M.I.; Ivanov, A.B. (2001) [1994], "Coordinates", Encyclopedia of Mathematics, EMS Press
• Woods, Frederick S. (1922). Higher Geometry. Ginn and Co. pp. 1ff.
• Shigeyuki Morita; Teruko Nagase; Katsumi Nomizu (2001). Geometry of Differential Forms. AMS Bookstore. p. 12. ISBN 0-8218-1045-6.
External links
Look up coordinate system or coordinate in Wiktionary, the free dictionary.
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Spherical harmonics
In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations in many scientific fields.
Since the spherical harmonics form a complete set of orthogonal functions and thus an orthonormal basis, each function defined on the surface of a sphere can be written as a sum of these spherical harmonics. This is similar to periodic functions defined on a circle that can be expressed as a sum of circular functions (sines and cosines) via Fourier series. Like the sines and cosines in Fourier series, the spherical harmonics may be organized by (spatial) angular frequency, as seen in the rows of functions in the illustration on the right. Further, spherical harmonics are basis functions for irreducible representations of SO(3), the group of rotations in three dimensions, and thus play a central role in the group theoretic discussion of SO(3).
Spherical harmonics originate from solving Laplace's equation in the spherical domains. Functions that are solutions to Laplace's equation are called harmonics. Despite their name, spherical harmonics take their simplest form in Cartesian coordinates, where they can be defined as homogeneous polynomials of degree $\ell $ in $(x,y,z)$ that obey Laplace's equation. The connection with spherical coordinates arises immediately if one uses the homogeneity to extract a factor of radial dependence $r^{\ell }$ from the above-mentioned polynomial of degree $\ell $; the remaining factor can be regarded as a function of the spherical angular coordinates $\theta $ and $\varphi $ only, or equivalently of the orientational unit vector $\mathbf {r} $ specified by these angles. In this setting, they may be viewed as the angular portion of a set of solutions to Laplace's equation in three dimensions, and this viewpoint is often taken as an alternative definition. Notice, however, that spherical harmonics are not functions on the sphere which are harmonic with respect to the Laplace-Beltrami operator for the standard round metric on the sphere: the only harmonic functions in this sense on the sphere are the constants, since harmonic functions satisfy the Maximum principle. Spherical harmonics, as functions on the sphere, are eigenfunctions of the Laplace-Beltrami operator (see the section Higher dimensions below).
A specific set of spherical harmonics, denoted $Y_{\ell }^{m}(\theta ,\varphi )$ or $Y_{\ell }^{m}({\mathbf {r} })$, are known as Laplace's spherical harmonics, as they were first introduced by Pierre Simon de Laplace in 1782.[1] These functions form an orthogonal system, and are thus basic to the expansion of a general function on the sphere as alluded to above.
Spherical harmonics are important in many theoretical and practical applications, including the representation of multipole electrostatic and electromagnetic fields, electron configurations, gravitational fields, geoids, the magnetic fields of planetary bodies and stars, and the cosmic microwave background radiation. In 3D computer graphics, spherical harmonics play a role in a wide variety of topics including indirect lighting (ambient occlusion, global illumination, precomputed radiance transfer, etc.) and modelling of 3D shapes.
History
Spherical harmonics were first investigated in connection with the Newtonian potential of Newton's law of universal gravitation in three dimensions. In 1782, Pierre-Simon de Laplace had, in his Mécanique Céleste, determined that the gravitational potential $\mathbb {R} ^{3}\to \mathbb {R} $ at a point x associated with a set of point masses mi located at points xi was given by
$V(\mathbf {x} )=\sum _{i}{\frac {m_{i}}{|\mathbf {x} _{i}-\mathbf {x} |}}.$
Each term in the above summation is an individual Newtonian potential for a point mass. Just prior to that time, Adrien-Marie Legendre had investigated the expansion of the Newtonian potential in powers of r = |x| and r1 = |x1|. He discovered that if r ≤ r1 then
${\frac {1}{|\mathbf {x} _{1}-\mathbf {x} |}}=P_{0}(\cos \gamma ){\frac {1}{r_{1}}}+P_{1}(\cos \gamma ){\frac {r}{r_{1}^{2}}}+P_{2}(\cos \gamma ){\frac {r^{2}}{r_{1}^{3}}}+\cdots $
where γ is the angle between the vectors x and x1. The functions $P_{i}:[-1,1]\to \mathbb {R} $ are the Legendre polynomials, and they can be derived as a special case of spherical harmonics. Subsequently, in his 1782 memoir, Laplace investigated these coefficients using spherical coordinates to represent the angle γ between x1 and x. (See Applications of Legendre polynomials in physics for a more detailed analysis.)
In 1867, William Thomson (Lord Kelvin) and Peter Guthrie Tait introduced the solid spherical harmonics in their Treatise on Natural Philosophy, and also first introduced the name of "spherical harmonics" for these functions. The solid harmonics were homogeneous polynomial solutions $\mathbb {R} ^{3}\to \mathbb {R} $ of Laplace's equation
${\frac {\partial ^{2}u}{\partial x^{2}}}+{\frac {\partial ^{2}u}{\partial y^{2}}}+{\frac {\partial ^{2}u}{\partial z^{2}}}=0.$
By examining Laplace's equation in spherical coordinates, Thomson and Tait recovered Laplace's spherical harmonics. (See the section below, "Harmonic polynomial representation".) The term "Laplace's coefficients" was employed by William Whewell to describe the particular system of solutions introduced along these lines, whereas others reserved this designation for the zonal spherical harmonics that had properly been introduced by Laplace and Legendre.
The 19th century development of Fourier series made possible the solution of a wide variety of physical problems in rectangular domains, such as the solution of the heat equation and wave equation. This could be achieved by expansion of functions in series of trigonometric functions. Whereas the trigonometric functions in a Fourier series represent the fundamental modes of vibration in a string, the spherical harmonics represent the fundamental modes of vibration of a sphere in much the same way. Many aspects of the theory of Fourier series could be generalized by taking expansions in spherical harmonics rather than trigonometric functions. Moreover, analogous to how trigonometric functions can equivalently be written as complex exponentials, spherical harmonics also possessed an equivalent form as complex-valued functions. This was a boon for problems possessing spherical symmetry, such as those of celestial mechanics originally studied by Laplace and Legendre.
The prevalence of spherical harmonics already in physics set the stage for their later importance in the 20th century birth of quantum mechanics. The (complex-valued) spherical harmonics $S^{2}\to \mathbb {C} $ are eigenfunctions of the square of the orbital angular momentum operator
$-i\hbar \mathbf {r} \times \nabla ,$
and therefore they represent the different quantized configurations of atomic orbitals.
Laplace's spherical harmonics
Laplace's equation imposes that the Laplacian of a scalar field f is zero. (Here the scalar field is understood to be complex, i.e. to correspond to a (smooth) function $f:\mathbb {R} ^{3}\to \mathbb {C} $.) In spherical coordinates this is:[2]
$\nabla ^{2}f={\frac {1}{r^{2}}}{\frac {\partial }{\partial r}}\left(r^{2}{\frac {\partial f}{\partial r}}\right)+{\frac {1}{r^{2}\sin \theta }}{\frac {\partial }{\partial \theta }}\left(\sin \theta {\frac {\partial f}{\partial \theta }}\right)+{\frac {1}{r^{2}\sin ^{2}\theta }}{\frac {\partial ^{2}f}{\partial \varphi ^{2}}}=0.$
Consider the problem of finding solutions of the form f(r, θ, φ) = R(r) Y(θ, φ). By separation of variables, two differential equations result by imposing Laplace's equation:
${\frac {1}{R}}{\frac {d}{dr}}\left(r^{2}{\frac {dR}{dr}}\right)=\lambda ,\qquad {\frac {1}{Y}}{\frac {1}{\sin \theta }}{\frac {\partial }{\partial \theta }}\left(\sin \theta {\frac {\partial Y}{\partial \theta }}\right)+{\frac {1}{Y}}{\frac {1}{\sin ^{2}\theta }}{\frac {\partial ^{2}Y}{\partial \varphi ^{2}}}=-\lambda .$
The second equation can be simplified under the assumption that Y has the form Y(θ, φ) = Θ(θ) Φ(φ). Applying separation of variables again to the second equation gives way to the pair of differential equations
${\frac {1}{\Phi }}{\frac {d^{2}\Phi }{d\varphi ^{2}}}=-m^{2}$
$\lambda \sin ^{2}\theta +{\frac {\sin \theta }{\Theta }}{\frac {d}{d\theta }}\left(\sin \theta {\frac {d\Theta }{d\theta }}\right)=m^{2}$
for some number m. A priori, m is a complex constant, but because Φ must be a periodic function whose period evenly divides 2π, m is necessarily an integer and Φ is a linear combination of the complex exponentials e± imφ. The solution function Y(θ, φ) is regular at the poles of the sphere, where θ = 0, π. Imposing this regularity in the solution Θ of the second equation at the boundary points of the domain is a Sturm–Liouville problem that forces the parameter λ to be of the form λ = ℓ (ℓ + 1) for some non-negative integer with ℓ ≥ |m|; this is also explained below in terms of the orbital angular momentum. Furthermore, a change of variables t = cos θ transforms this equation into the Legendre equation, whose solution is a multiple of the associated Legendre polynomial Pm
ℓ
(cos θ)
. Finally, the equation for R has solutions of the form R(r) = A rℓ + B r−ℓ − 1; requiring the solution to be regular throughout R3 forces B = 0.[3]
Here the solution was assumed to have the special form Y(θ, φ) = Θ(θ) Φ(φ). For a given value of ℓ, there are 2ℓ + 1 independent solutions of this form, one for each integer m with −ℓ ≤ m ≤ ℓ. These angular solutions $Y_{\ell }^{m}:S^{2}\to \mathbb {C} $ are a product of trigonometric functions, here represented as a complex exponential, and associated Legendre polynomials:
$Y_{\ell }^{m}(\theta ,\varphi )=Ne^{im\varphi }P_{\ell }^{m}(\cos {\theta })$
which fulfill
$r^{2}\nabla ^{2}Y_{\ell }^{m}(\theta ,\varphi )=-\ell (\ell +1)Y_{\ell }^{m}(\theta ,\varphi ).$
Here $Y_{\ell }^{m}:S^{2}\to \mathbb {C} $ is called a spherical harmonic function of degree ℓ and order m, $P_{\ell }^{m}:[-1,1]\to \mathbb {R} $ is an associated Legendre polynomial, N is a normalization constant,[4] and θ and φ represent colatitude and longitude, respectively. In particular, the colatitude θ, or polar angle, ranges from 0 at the North Pole, to π/2 at the Equator, to π at the South Pole, and the longitude φ, or azimuth, may assume all values with 0 ≤ φ < 2π. For a fixed integer ℓ, every solution Y(θ, φ), $Y:S^{2}\to \mathbb {C} $, of the eigenvalue problem
$r^{2}\nabla ^{2}Y=-\ell (\ell +1)Y$
is a linear combination of $Y_{\ell }^{m}:S^{2}\to \mathbb {C} $. In fact, for any such solution, rℓ Y(θ, φ) is the expression in spherical coordinates of a homogeneous polynomial $\mathbb {R} ^{3}\to \mathbb {C} $ that is harmonic (see below), and so counting dimensions shows that there are 2ℓ + 1 linearly independent such polynomials.
The general solution $f:\mathbb {R} ^{3}\to \mathbb {C} $ to Laplace's equation $\Delta f=0$ in a ball centered at the origin is a linear combination of the spherical harmonic functions multiplied by the appropriate scale factor rℓ,
$f(r,\theta ,\varphi )=\sum _{\ell =0}^{\infty }\sum _{m=-\ell }^{\ell }f_{\ell }^{m}r^{\ell }Y_{\ell }^{m}(\theta ,\varphi ),$
where the $f_{\ell }^{m}\in \mathbb {C} $ are constants and the factors rℓ Yℓm are known as (regular) solid harmonics $\mathbb {R} ^{3}\to \mathbb {C} $. Such an expansion is valid in the ball
$r<R={\frac {1}{\limsup _{\ell \to \infty }|f_{\ell }^{m}|^{{1}/{\ell }}}}.$
For $r>R$, the solid harmonics with negative powers of $r$ (the irregular solid harmonics $\mathbb {R} ^{3}\setminus \{\mathbf {0} \}\to \mathbb {C} $) are chosen instead. In that case, one needs to expand the solution of known regions in Laurent series (about $r=\infty $), instead of the Taylor series (about $r=0$) used above, to match the terms and find series expansion coefficients $f_{\ell }^{m}\in \mathbb {C} $.
Orbital angular momentum
In quantum mechanics, Laplace's spherical harmonics are understood in terms of the orbital angular momentum[5]
$\mathbf {L} =-i\hbar (\mathbf {x} \times \mathbf {\nabla } )=L_{x}\mathbf {i} +L_{y}\mathbf {j} +L_{z}\mathbf {k} .$
The ħ is conventional in quantum mechanics; it is convenient to work in units in which ħ = 1. The spherical harmonics are eigenfunctions of the square of the orbital angular momentum
${\begin{aligned}\mathbf {L} ^{2}&=-r^{2}\nabla ^{2}+\left(r{\frac {\partial }{\partial r}}+1\right)r{\frac {\partial }{\partial r}}\\&=-{\frac {1}{\sin \theta }}{\frac {\partial }{\partial \theta }}\sin \theta {\frac {\partial }{\partial \theta }}-{\frac {1}{\sin ^{2}\theta }}{\frac {\partial ^{2}}{\partial \varphi ^{2}}}.\end{aligned}}$
Laplace's spherical harmonics are the joint eigenfunctions of the square of the orbital angular momentum and the generator of rotations about the azimuthal axis:
${\begin{aligned}L_{z}&=-i\left(x{\frac {\partial }{\partial y}}-y{\frac {\partial }{\partial x}}\right)\\&=-i{\frac {\partial }{\partial \varphi }}.\end{aligned}}$
These operators commute, and are densely defined self-adjoint operators on the weighted Hilbert space of functions f square-integrable with respect to the normal distribution as the weight function on R3:
${\frac {1}{(2\pi )^{3/2}}}\int _{\mathbb {R} ^{3}}|f(x)|^{2}e^{-|x|^{2}/2}\,dx<\infty .$
Furthermore, L2 is a positive operator.
If Y is a joint eigenfunction of L2 and Lz, then by definition
${\begin{aligned}\mathbf {L} ^{2}Y&=\lambda Y\\L_{z}Y&=mY\end{aligned}}$
for some real numbers m and λ. Here m must in fact be an integer, for Y must be periodic in the coordinate φ with period a number that evenly divides 2π. Furthermore, since
$\mathbf {L} ^{2}=L_{x}^{2}+L_{y}^{2}+L_{z}^{2}$
and each of Lx, Ly, Lz are self-adjoint, it follows that λ ≥ m2.
Denote this joint eigenspace by Eλ,m, and define the raising and lowering operators by
${\begin{aligned}L_{+}&=L_{x}+iL_{y}\\L_{-}&=L_{x}-iL_{y}\end{aligned}}$
Then L+ and L− commute with L2, and the Lie algebra generated by L+, L−, Lz is the special linear Lie algebra of order 2, ${\mathfrak {sl}}_{2}(\mathbb {C} )$, with commutation relations
$[L_{z},L_{+}]=L_{+},\quad [L_{z},L_{-}]=-L_{-},\quad [L_{+},L_{-}]=2L_{z}.$
Thus L+ : Eλ,m → Eλ,m+1 (it is a "raising operator") and L− : Eλ,m → Eλ,m−1 (it is a "lowering operator"). In particular, Lk
+
: Eλ,m → Eλ,m+k
must be zero for k sufficiently large, because the inequality λ ≥ m2 must hold in each of the nontrivial joint eigenspaces. Let Y ∈ Eλ,m be a nonzero joint eigenfunction, and let k be the least integer such that
$L_{+}^{k}Y=0.$
Then, since
$L_{-}L_{+}=\mathbf {L} ^{2}-L_{z}^{2}-L_{z}$
it follows that
$0=L_{-}L_{+}^{k}Y=(\lambda -(m+k)^{2}-(m+k))Y.$
Thus λ = ℓ(ℓ + 1) for the positive integer ℓ = m + k.
The foregoing has been all worked out in the spherical coordinate representation, $\langle \theta ,\varphi |lm\rangle =Y_{l}^{m}(\theta ,\varphi )$ but may be expressed more abstractly in the complete, orthonormal spherical ket basis.
Harmonic polynomial representation
See also: § Higher dimensions
The spherical harmonics can be expressed as the restriction to the unit sphere of certain polynomial functions $\mathbb {R} ^{3}\to \mathbb {C} $. Specifically, we say that a (complex-valued) polynomial function $p:\mathbb {R} ^{3}\to \mathbb {C} $ is homogeneous of degree $\ell $ if
$p(\lambda \mathbf {x} )=\lambda ^{\ell }p(\mathbf {x} )$
for all real numbers $\lambda \in \mathbb {R} $ and all $x\in \mathbb {R} ^{3}$. We say that $p$ is harmonic if
$\Delta p=0,$
where $\Delta $ is the Laplacian. Then for each $\ell $, we define
$\mathbf {A} _{\ell }=\left\{{\text{harmonic polynomials }}\mathbb {R} ^{3}\to \mathbb {C} {\text{ that are homogeneous of degree }}\ell \right\}.$
For example, when $\ell =1$, $\mathbf {A} _{1}$ is just the 3-dimensional space of all linear functions $\mathbb {R} ^{3}\to \mathbb {C} $, since any such function is automatically harmonic. Meanwhile, when $\ell =2$, we have a 5-dimensional space:
$\mathbf {A} _{2}=\operatorname {span} _{\mathbb {C} }(x_{1}x_{2},\,x_{1}x_{3},\,x_{2}x_{3},\,x_{1}^{2}-x_{2}^{2},\,x_{1}^{2}-x_{3}^{2}).$
For any $\ell $, the space $\mathbf {H} _{\ell }$ of spherical harmonics of degree $\ell $ is just the space of restrictions to the sphere $S^{2}$ of the elements of $\mathbf {A} _{\ell }$.[6] As suggested in the introduction, this perspective is presumably the origin of the term “spherical harmonic” (i.e., the restriction to the sphere of a harmonic function).
For example, for any $c\in \mathbb {C} $ the formula
$p(x_{1},x_{2},x_{3})=c(x_{1}+ix_{2})^{\ell }$
defines a homogeneous polynomial of degree $\ell $ with domain and codomain $\mathbb {R} ^{3}\to \mathbb {C} $, which happens to be independent of $x_{3}$. This polynomial is easily seen to be harmonic. If we write $p$ in spherical coordinates $(r,\theta ,\varphi )$ and then restrict to $r=1$, we obtain
$p(\theta ,\varphi )=c\sin(\theta )^{\ell }(\cos(\varphi )+i\sin(\varphi ))^{\ell },$
which can be rewritten as
$p(\theta ,\varphi )=c\left({\sqrt {1-\cos ^{2}(\theta )}}\right)^{\ell }e^{i\ell \varphi }.$
After using the formula for the associated Legendre polynomial $P_{\ell }^{\ell }$, we may recognize this as the formula for the spherical harmonic $Y_{\ell }^{\ell }(\theta ,\varphi ).$[7] (See the section below on special cases of the spherical harmonics.)
Conventions
Orthogonality and normalization
Several different normalizations are in common use for the Laplace spherical harmonic functions $S^{2}\to \mathbb {C} $. Throughout the section, we use the standard convention that for $m>0$ (see associated Legendre polynomials)
$P_{\ell }^{-m}=(-1)^{m}{\frac {(\ell -m)!}{(\ell +m)!}}P_{\ell }^{m}$
which is the natural normalization given by Rodrigues' formula.
In acoustics,[8] the Laplace spherical harmonics are generally defined as (this is the convention used in this article)
$Y_{\ell }^{m}(\theta ,\varphi )={\sqrt {{\frac {(2\ell +1)}{4\pi }}{\frac {(\ell -m)!}{(\ell +m)!}}}}\,P_{\ell }^{m}(\cos {\theta })\,e^{im\varphi }$
while in quantum mechanics:[9][10]
$Y_{\ell }^{m}(\theta ,\varphi )=(-1)^{m}{\sqrt {{\frac {(2\ell +1)}{4\pi }}{\frac {(\ell -m)!}{(\ell +m)!}}}}\,P_{\ell }^{m}(\cos {\theta })\,e^{im\varphi }$
where $P_{\ell }^{m}$ are associated Legendre polynomials without the Condon–Shortley phase (to avoid counting the phase twice).
In both definitions, the spherical harmonics are orthonormal
$\int _{\theta =0}^{\pi }\int _{\varphi =0}^{2\pi }Y_{\ell }^{m}\,Y_{\ell '}^{m'}{}^{*}\,d\Omega =\delta _{\ell \ell '}\,\delta _{mm'},$
where δij is the Kronecker delta and dΩ = sin(θ) dφ dθ. This normalization is used in quantum mechanics because it ensures that probability is normalized, i.e.,
$\int {|Y_{\ell }^{m}|^{2}d\Omega }=1.$
The disciplines of geodesy[11] and spectral analysis use
$Y_{\ell }^{m}(\theta ,\varphi )={\sqrt {{(2\ell +1)}{\frac {(\ell -m)!}{(\ell +m)!}}}}\,P_{\ell }^{m}(\cos {\theta })\,e^{im\varphi }$
which possess unit power
${\frac {1}{4\pi }}\int _{\theta =0}^{\pi }\int _{\varphi =0}^{2\pi }Y_{\ell }^{m}\,Y_{\ell '}^{m'}{}^{*}d\Omega =\delta _{\ell \ell '}\,\delta _{mm'}.$
The magnetics[11] community, in contrast, uses Schmidt semi-normalized harmonics
$Y_{\ell }^{m}(\theta ,\varphi )={\sqrt {\frac {(\ell -m)!}{(\ell +m)!}}}\,P_{\ell }^{m}(\cos {\theta })\,e^{im\varphi }$
which have the normalization
$\int _{\theta =0}^{\pi }\int _{\varphi =0}^{2\pi }Y_{\ell }^{m}\,Y_{\ell '}^{m'}{}^{*}d\Omega ={\frac {4\pi }{(2\ell +1)}}\delta _{\ell \ell '}\,\delta _{mm'}.$
In quantum mechanics this normalization is sometimes used as well, and is named Racah's normalization after Giulio Racah.
It can be shown that all of the above normalized spherical harmonic functions satisfy
$Y_{\ell }^{m}{}^{*}(\theta ,\varphi )=(-1)^{m}Y_{\ell }^{-m}(\theta ,\varphi ),$
where the superscript * denotes complex conjugation. Alternatively, this equation follows from the relation of the spherical harmonic functions with the Wigner D-matrix.
Condon–Shortley phase
One source of confusion with the definition of the spherical harmonic functions concerns a phase factor of $(-1)^{m}$, commonly referred to as the Condon–Shortley phase in the quantum mechanical literature. In the quantum mechanics community, it is common practice to either include this phase factor in the definition of the associated Legendre polynomials, or to append it to the definition of the spherical harmonic functions. There is no requirement to use the Condon–Shortley phase in the definition of the spherical harmonic functions, but including it can simplify some quantum mechanical operations, especially the application of raising and lowering operators. The geodesy[12] and magnetics communities never include the Condon–Shortley phase factor in their definitions of the spherical harmonic functions nor in the ones of the associated Legendre polynomials.[13]
Real form
A real basis of spherical harmonics $Y_{\ell m}:S^{2}\to \mathbb {R} $ can be defined in terms of their complex analogues $Y_{\ell }^{m}:S^{2}\to \mathbb {C} $ by setting
${\begin{aligned}Y_{\ell m}&={\begin{cases}{\dfrac {i}{\sqrt {2}}}\left(Y_{\ell }^{m}-(-1)^{m}\,Y_{\ell }^{-m}\right)&{\text{if}}\ m<0\\Y_{\ell }^{0}&{\text{if}}\ m=0\\{\dfrac {1}{\sqrt {2}}}\left(Y_{\ell }^{-m}+(-1)^{m}\,Y_{\ell }^{m}\right)&{\text{if}}\ m>0.\end{cases}}\\&={\begin{cases}{\dfrac {i}{\sqrt {2}}}\left(Y_{\ell }^{-|m|}-(-1)^{m}\,Y_{\ell }^{|m|}\right)&{\text{if}}\ m<0\\Y_{\ell }^{0}&{\text{if}}\ m=0\\{\dfrac {1}{\sqrt {2}}}\left(Y_{\ell }^{-|m|}+(-1)^{m}\,Y_{\ell }^{|m|}\right)&{\text{if}}\ m>0.\end{cases}}\\&={\begin{cases}{\sqrt {2}}\,(-1)^{m}\,\Im [{Y_{\ell }^{|m|}}]&{\text{if}}\ m<0\\Y_{\ell }^{0}&{\text{if}}\ m=0\\{\sqrt {2}}\,(-1)^{m}\,\Re [{Y_{\ell }^{m}}]&{\text{if}}\ m>0.\end{cases}}\end{aligned}}$
The Condon–Shortley phase convention is used here for consistency. The corresponding inverse equations defining the complex spherical harmonics $Y_{\ell }^{m}:S^{2}\to \mathbb {C} $ in terms of the real spherical harmonics $Y_{\ell m}:S^{2}\to \mathbb {R} $ are
$Y_{\ell }^{m}={\begin{cases}{\dfrac {1}{\sqrt {2}}}\left(Y_{\ell |m|}-iY_{\ell ,-|m|}\right)&{\text{if}}\ m<0\\[4pt]Y_{\ell 0}&{\text{if}}\ m=0\\[4pt]{\dfrac {(-1)^{m}}{\sqrt {2}}}\left(Y_{\ell |m|}+iY_{\ell ,-|m|}\right)&{\text{if}}\ m>0.\end{cases}}$
The real spherical harmonics $Y_{\ell m}:S^{2}\to \mathbb {R} $ are sometimes known as tesseral spherical harmonics.[14] These functions have the same orthonormality properties as the complex ones $Y_{\ell }^{m}:S^{2}\to \mathbb {C} $ above. The real spherical harmonics $Y_{\ell m}$ with m > 0 are said to be of cosine type, and those with m < 0 of sine type. The reason for this can be seen by writing the functions in terms of the Legendre polynomials as
$Y_{\ell m}={\begin{cases}\left(-1\right)^{m}{\sqrt {2}}{\sqrt {{\dfrac {2\ell +1}{4\pi }}{\dfrac {(\ell -|m|)!}{(\ell +|m|)!}}}}\;P_{\ell }^{|m|}(\cos \theta )\ \sin(|m|\varphi )&{\text{if }}m<0\\[4pt]{\sqrt {\dfrac {2\ell +1}{4\pi }}}\ P_{\ell }^{m}(\cos \theta )&{\text{if }}m=0\\[4pt]\left(-1\right)^{m}{\sqrt {2}}{\sqrt {{\dfrac {2\ell +1}{4\pi }}{\dfrac {(\ell -m)!}{(\ell +m)!}}}}\;P_{\ell }^{m}(\cos \theta )\ \cos(m\varphi )&{\text{if }}m>0\,.\end{cases}}$
The same sine and cosine factors can be also seen in the following subsection that deals with the Cartesian representation.
See here for a list of real spherical harmonics up to and including $\ell =4$, which can be seen to be consistent with the output of the equations above.
Use in quantum chemistry
As is known from the analytic solutions for the hydrogen atom, the eigenfunctions of the angular part of the wave function are spherical harmonics. However, the solutions of the non-relativistic Schrödinger equation without magnetic terms can be made real. This is why the real forms are extensively used in basis functions for quantum chemistry, as the programs don't then need to use complex algebra. Here, it is important to note that the real functions span the same space as the complex ones would.
For example, as can be seen from the table of spherical harmonics, the usual p functions ($\ell =1$) are complex and mix axis directions, but the real versions are essentially just x, y, and z.
Spherical harmonics in Cartesian form
The complex spherical harmonics $Y_{\ell }^{m}$ give rise to the solid harmonics by extending from $S^{2}$ to all of $\mathbb {R} ^{3}$ as a homogeneous function of degree $\ell $, i.e. setting
$R_{\ell }^{m}(v):=\|v\|^{\ell }Y_{\ell }^{m}\left({\frac {v}{\|v\|}}\right)$
It turns out that $R_{\ell }^{m}$ is basis of the space of harmonic and homogeneous polynomials of degree $\ell $. More specifically, it is the (unique up to normalization) Gelfand-Tsetlin-basis of this representation of the rotational group $SO(3)$ and an explicit formula for $R_{\ell }^{m}$ in cartesian coordinates can be derived from that fact.
The Herglotz generating function
If the quantum mechanical convention is adopted for the $Y_{\ell }^{m}:S^{2}\to \mathbb {C} $, then
$e^{v{\mathbf {a} }\cdot {\mathbf {r} }}=\sum _{\ell =0}^{\infty }\sum _{m=-\ell }^{\ell }{\sqrt {\frac {4\pi }{2\ell +1}}}{\frac {r^{\ell }v^{\ell }{\lambda ^{m}}}{\sqrt {(\ell +m)!(\ell -m)!}}}Y_{\ell }^{m}(\mathbf {r} /r).$
Here, $\mathbf {r} $ is the vector with components $(x,y,z)\in \mathbb {R} ^{3}$, $r=|\mathbf {r} |$, and
${\mathbf {a} }={\mathbf {\hat {z}} }-{\frac {\lambda }{2}}\left({\mathbf {\hat {x}} }+i{\mathbf {\hat {y}} }\right)+{\frac {1}{2\lambda }}\left({\mathbf {\hat {x}} }-i{\mathbf {\hat {y}} }\right).$
$\mathbf {a} $ is a vector with complex coordinates:
$\mathbf {a} =[{\frac {1}{2}}({\frac {1}{\lambda }}-\lambda ),-{\frac {i}{2}}({\frac {1}{\lambda }}+\lambda ),1].$
The essential property of $\mathbf {a} $ is that it is null:
$\mathbf {a} \cdot \mathbf {a} =0.$
It suffices to take $v$ and $\lambda $ as real parameters. In naming this generating function after Herglotz, we follow Courant & Hilbert 1962, §VII.7, who credit unpublished notes by him for its discovery.
Essentially all the properties of the spherical harmonics can be derived from this generating function.[15] An immediate benefit of this definition is that if the vector $\mathbf {r} $ is replaced by the quantum mechanical spin vector operator $\mathbf {J} $, such that ${\mathcal {Y}}_{\ell }^{m}({\mathbf {J} })$ is the operator analogue of the solid harmonic $r^{\ell }Y_{\ell }^{m}(\mathbf {r} /r)$,[16] one obtains a generating function for a standardized set of spherical tensor operators, ${\mathcal {Y}}_{\ell }^{m}({\mathbf {J} })$:
$e^{v{\mathbf {a} }\cdot {\mathbf {J} }}=\sum _{\ell =0}^{\infty }\sum _{m=-\ell }^{\ell }{\sqrt {\frac {4\pi }{2\ell +1}}}{\frac {v^{\ell }{\lambda ^{m}}}{\sqrt {(\ell +m)!(\ell -m)!}}}{\mathcal {Y}}_{\ell }^{m}({\mathbf {J} }).$
The parallelism of the two definitions ensures that the ${\mathcal {Y}}_{\ell }^{m}$'s transform under rotations (see below) in the same way as the $Y_{\ell }^{m}$'s, which in turn guarantees that they are spherical tensor operators, $T_{q}^{(k)}$, with $k={\ell }$ and $q=m$, obeying all the properties of such operators, such as the Clebsch-Gordan composition theorem, and the Wigner-Eckart theorem. They are, moreover, a standardized set with a fixed scale or normalization.
See also: Spherical basis
Separated Cartesian form
The Herglotzian definition yields polynomials which may, if one wishes, be further factorized into a polynomial of $z$ and another of $x$ and $y$, as follows (Condon–Shortley phase):
$r^{\ell }\,{\begin{pmatrix}Y_{\ell }^{m}\\Y_{\ell }^{-m}\end{pmatrix}}=\left[{\frac {2\ell +1}{4\pi }}\right]^{1/2}{\bar {\Pi }}_{\ell }^{m}(z){\begin{pmatrix}\left(-1\right)^{m}(A_{m}+iB_{m})\\(A_{m}-iB_{m})\end{pmatrix}},\qquad m>0.$
and for m = 0:
$r^{\ell }\,Y_{\ell }^{0}\equiv {\sqrt {\frac {2\ell +1}{4\pi }}}{\bar {\Pi }}_{\ell }^{0}.$
Here
$A_{m}(x,y)=\sum _{p=0}^{m}{\binom {m}{p}}x^{p}y^{m-p}\cos \left((m-p){\frac {\pi }{2}}\right),$
$B_{m}(x,y)=\sum _{p=0}^{m}{\binom {m}{p}}x^{p}y^{m-p}\sin \left((m-p){\frac {\pi }{2}}\right),$
and
${\bar {\Pi }}_{\ell }^{m}(z)=\left[{\frac {(\ell -m)!}{(\ell +m)!}}\right]^{1/2}\sum _{k=0}^{\left\lfloor (\ell -m)/2\right\rfloor }(-1)^{k}2^{-\ell }{\binom {\ell }{k}}{\binom {2\ell -2k}{\ell }}{\frac {(\ell -2k)!}{(\ell -2k-m)!}}\;r^{2k}\;z^{\ell -2k-m}.$
For $m=0$ this reduces to
${\bar {\Pi }}_{\ell }^{0}(z)=\sum _{k=0}^{\left\lfloor \ell /2\right\rfloor }(-1)^{k}2^{-\ell }{\binom {\ell }{k}}{\binom {2\ell -2k}{\ell }}\;r^{2k}\;z^{\ell -2k}.$
The factor ${\bar {\Pi }}_{\ell }^{m}(z)$ is essentially the associated Legendre polynomial $P_{\ell }^{m}(\cos \theta )$, and the factors $(A_{m}\pm iB_{m})$ are essentially $e^{\pm im\varphi }$.
Examples
Using the expressions for ${\bar {\Pi }}_{\ell }^{m}(z)$, $A_{m}(x,y)$, and $B_{m}(x,y)$ listed explicitly above we obtain:
$Y_{3}^{1}=-{\frac {1}{r^{3}}}\left[{\tfrac {7}{4\pi }}\cdot {\tfrac {3}{16}}\right]^{1/2}\left(5z^{2}-r^{2}\right)\left(x+iy\right)=-\left[{\tfrac {7}{4\pi }}\cdot {\tfrac {3}{16}}\right]^{1/2}\left(5\cos ^{2}\theta -1\right)\left(\sin \theta e^{i\varphi }\right)$
$Y_{4}^{-2}={\frac {1}{r^{4}}}\left[{\tfrac {9}{4\pi }}\cdot {\tfrac {5}{32}}\right]^{1/2}\left(7z^{2}-r^{2}\right)\left(x-iy\right)^{2}=\left[{\tfrac {9}{4\pi }}\cdot {\tfrac {5}{32}}\right]^{1/2}\left(7\cos ^{2}\theta -1\right)\left(\sin ^{2}\theta e^{-2i\varphi }\right)$
It may be verified that this agrees with the function listed here and here.
Real forms
Using the equations above to form the real spherical harmonics, it is seen that for $m>0$ only the $A_{m}$ terms (cosines) are included, and for $m<0$ only the $B_{m}$ terms (sines) are included:
$r^{\ell }\,{\begin{pmatrix}Y_{\ell m}\\Y_{\ell -m}\end{pmatrix}}={\sqrt {\frac {2\ell +1}{2\pi }}}{\bar {\Pi }}_{\ell }^{m}(z){\begin{pmatrix}A_{m}\\B_{m}\end{pmatrix}},\qquad m>0.$
and for m = 0:
$r^{\ell }\,Y_{\ell 0}\equiv {\sqrt {\frac {2\ell +1}{4\pi }}}{\bar {\Pi }}_{\ell }^{0}.$
Special cases and values
1. When $m=0$, the spherical harmonics $Y_{\ell }^{m}:S^{2}\to \mathbb {C} $ reduce to the ordinary Legendre polynomials:
$Y_{\ell }^{0}(\theta ,\varphi )={\sqrt {\frac {2\ell +1}{4\pi }}}P_{\ell }(\cos \theta ).$
2. When $m=\pm \ell $,
$Y_{\ell }^{\pm \ell }(\theta ,\varphi )={\frac {(\mp 1)^{\ell }}{2^{\ell }\ell !}}{\sqrt {\frac {(2\ell +1)!}{4\pi }}}\sin ^{\ell }\theta \,e^{\pm i\ell \varphi },$ !}}{\sqrt {\frac {(2\ell +1)!}{4\pi }}}\sin ^{\ell }\theta \,e^{\pm i\ell \varphi },}
or more simply in Cartesian coordinates,
$r^{\ell }Y_{\ell }^{\pm \ell }({\mathbf {r} })={\frac {(\mp 1)^{\ell }}{2^{\ell }\ell !}}{\sqrt {\frac {(2\ell +1)!}{4\pi }}}(x\pm iy)^{\ell }.$ !}}{\sqrt {\frac {(2\ell +1)!}{4\pi }}}(x\pm iy)^{\ell }.}
3. At the north pole, where $\theta =0$, and $\varphi $ is undefined, all spherical harmonics except those with $m=0$ vanish:
$Y_{\ell }^{m}(0,\varphi )=Y_{\ell }^{m}({\mathbf {z} })={\sqrt {\frac {2\ell +1}{4\pi }}}\delta _{m0}.$
Symmetry properties
The spherical harmonics have deep and consequential properties under the operations of spatial inversion (parity) and rotation.
Parity
The spherical harmonics have definite parity. That is, they are either even or odd with respect to inversion about the origin. Inversion is represented by the operator $P\Psi (\mathbf {r} )=\Psi (-\mathbf {r} )$. Then, as can be seen in many ways (perhaps most simply from the Herglotz generating function), with $\mathbf {r} $ being a unit vector,
$Y_{\ell }^{m}(-\mathbf {r} )=(-1)^{\ell }Y_{\ell }^{m}(\mathbf {r} ).$
In terms of the spherical angles, parity transforms a point with coordinates $\{\theta ,\varphi \}$ to $\{\pi -\theta ,\pi +\varphi \}$. The statement of the parity of spherical harmonics is then
$Y_{\ell }^{m}(\theta ,\varphi )\to Y_{\ell }^{m}(\pi -\theta ,\pi +\varphi )=(-1)^{\ell }Y_{\ell }^{m}(\theta ,\varphi )$
(This can be seen as follows: The associated Legendre polynomials gives (−1)ℓ+m and from the exponential function we have (−1)m, giving together for the spherical harmonics a parity of (−1)ℓ.)
Parity continues to hold for real spherical harmonics, and for spherical harmonics in higher dimensions: applying a point reflection to a spherical harmonic of degree ℓ changes the sign by a factor of (−1)ℓ.
Rotations
Consider a rotation ${\mathcal {R}}$ about the origin that sends the unit vector $\mathbf {r} $ to $\mathbf {r} '$. Under this operation, a spherical harmonic of degree $\ell $ and order $m$ transforms into a linear combination of spherical harmonics of the same degree. That is,
$Y_{\ell }^{m}({\mathbf {r} }')=\sum _{m'=-\ell }^{\ell }A_{mm'}Y_{\ell }^{m'}({\mathbf {r} }),$
where $A_{mm'}$ is a matrix of order $(2\ell +1)$ that depends on the rotation ${\mathcal {R}}$. However, this is not the standard way of expressing this property. In the standard way one writes,
$Y_{\ell }^{m}({\mathbf {r} }')=\sum _{m'=-\ell }^{\ell }[D_{mm'}^{(\ell )}({\mathcal {R}})]^{*}Y_{\ell }^{m'}({\mathbf {r} }),$
where $D_{mm'}^{(\ell )}({\mathcal {R}})^{*}$ is the complex conjugate of an element of the Wigner D-matrix. In particular when $\mathbf {r} '$ is a $\phi _{0}$ rotation of the azimuth we get the identity,
$Y_{\ell }^{m}({\mathbf {r} }')=Y_{\ell }^{m}({\mathbf {r} })e^{im\phi _{0}}.$
The rotational behavior of the spherical harmonics is perhaps their quintessential feature from the viewpoint of group theory. The $Y_{\ell }^{m}$'s of degree $\ell $ provide a basis set of functions for the irreducible representation of the group SO(3) of dimension $(2\ell +1)$. Many facts about spherical harmonics (such as the addition theorem) that are proved laboriously using the methods of analysis acquire simpler proofs and deeper significance using the methods of symmetry.
Spherical harmonics expansion
The Laplace spherical harmonics $Y_{\ell }^{m}:S^{2}\to \mathbb {C} $ form a complete set of orthonormal functions and thus form an orthonormal basis of the Hilbert space of square-integrable functions $L_{\mathbb {C} }^{2}(S^{2})$. On the unit sphere $S^{2}$, any square-integrable function $f:S^{2}\to \mathbb {C} $ can thus be expanded as a linear combination of these:
$f(\theta ,\varphi )=\sum _{\ell =0}^{\infty }\sum _{m=-\ell }^{\ell }f_{\ell }^{m}\,Y_{\ell }^{m}(\theta ,\varphi ).$
This expansion holds in the sense of mean-square convergence — convergence in L2 of the sphere — which is to say that
$\lim _{N\to \infty }\int _{0}^{2\pi }\int _{0}^{\pi }\left|f(\theta ,\varphi )-\sum _{\ell =0}^{N}\sum _{m=-\ell }^{\ell }f_{\ell }^{m}Y_{\ell }^{m}(\theta ,\varphi )\right|^{2}\sin \theta \,d\theta \,d\varphi =0.$
The expansion coefficients are the analogs of Fourier coefficients, and can be obtained by multiplying the above equation by the complex conjugate of a spherical harmonic, integrating over the solid angle Ω, and utilizing the above orthogonality relationships. This is justified rigorously by basic Hilbert space theory. For the case of orthonormalized harmonics, this gives:
$f_{\ell }^{m}=\int _{\Omega }f(\theta ,\varphi )\,Y_{\ell }^{m*}(\theta ,\varphi )\,d\Omega =\int _{0}^{2\pi }d\varphi \int _{0}^{\pi }\,d\theta \,\sin \theta f(\theta ,\varphi )Y_{\ell }^{m*}(\theta ,\varphi ).$
If the coefficients decay in ℓ sufficiently rapidly — for instance, exponentially — then the series also converges uniformly to f.
A square-integrable function $f:S^{2}\to \mathbb {R} $ can also be expanded in terms of the real harmonics $Y_{\ell m}:S^{2}\to \mathbb {R} $ above as a sum
$f(\theta ,\varphi )=\sum _{\ell =0}^{\infty }\sum _{m=-\ell }^{\ell }f_{\ell m}\,Y_{\ell m}(\theta ,\varphi ).$
The convergence of the series holds again in the same sense, namely the real spherical harmonics $Y_{\ell m}:S^{2}\to \mathbb {R} $ form a complete set of orthonormal functions and thus form an orthonormal basis of the Hilbert space of square-integrable functions $L_{\mathbb {R} }^{2}(S^{2})$. The benefit of the expansion in terms of the real harmonic functions $Y_{\ell m}$ is that for real functions $f:S^{2}\to \mathbb {R} $ the expansion coefficients $f_{\ell m}$ are guaranteed to be real, whereas their coefficients $f_{\ell }^{m}$ in their expansion in terms of the $Y_{\ell }^{m}$ (considering them as functions $f:S^{2}\to \mathbb {C} \supset \mathbb {R} $) do not have that property.
Spectrum analysis
Power spectrum in signal processing
The total power of a function f is defined in the signal processing literature as the integral of the function squared, divided by the area of its domain. Using the orthonormality properties of the real unit-power spherical harmonic functions, it is straightforward to verify that the total power of a function defined on the unit sphere is related to its spectral coefficients by a generalization of Parseval's theorem (here, the theorem is stated for Schmidt semi-normalized harmonics, the relationship is slightly different for orthonormal harmonics):
${\frac {1}{4\,\pi }}\int _{\Omega }|f(\Omega )|^{2}\,d\Omega =\sum _{\ell =0}^{\infty }S_{f\!f}(\ell ),$
where
$S_{f\!f}(\ell )={\frac {1}{2\ell +1}}\sum _{m=-\ell }^{\ell }|f_{\ell m}|^{2}$
is defined as the angular power spectrum (for Schmidt semi-normalized harmonics). In a similar manner, one can define the cross-power of two functions as
${\frac {1}{4\,\pi }}\int _{\Omega }f(\Omega )\,g^{\ast }(\Omega )\,d\Omega =\sum _{\ell =0}^{\infty }S_{fg}(\ell ),$
where
$S_{fg}(\ell )={\frac {1}{2\ell +1}}\sum _{m=-\ell }^{\ell }f_{\ell m}g_{\ell m}^{\ast }$
is defined as the cross-power spectrum. If the functions f and g have a zero mean (i.e., the spectral coefficients f00 and g00 are zero), then Sff(ℓ) and Sfg(ℓ) represent the contributions to the function's variance and covariance for degree ℓ, respectively. It is common that the (cross-)power spectrum is well approximated by a power law of the form
$S_{f\!f}(\ell )=C\,\ell ^{\beta }.$
When β = 0, the spectrum is "white" as each degree possesses equal power. When β < 0, the spectrum is termed "red" as there is more power at the low degrees with long wavelengths than higher degrees. Finally, when β > 0, the spectrum is termed "blue". The condition on the order of growth of Sff(ℓ) is related to the order of differentiability of f in the next section.
Differentiability properties
One can also understand the differentiability properties of the original function f in terms of the asymptotics of Sff(ℓ). In particular, if Sff(ℓ) decays faster than any rational function of ℓ as ℓ → ∞, then f is infinitely differentiable. If, furthermore, Sff(ℓ) decays exponentially, then f is actually real analytic on the sphere.
The general technique is to use the theory of Sobolev spaces. Statements relating the growth of the Sff(ℓ) to differentiability are then similar to analogous results on the growth of the coefficients of Fourier series. Specifically, if
$\sum _{\ell =0}^{\infty }(1+\ell ^{2})^{s}S_{ff}(\ell )<\infty ,$
then f is in the Sobolev space Hs(S2). In particular, the Sobolev embedding theorem implies that f is infinitely differentiable provided that
$S_{ff}(\ell )=O(\ell ^{-s})\quad {\rm {{as\ }\ell \to \infty }}$
for all s.
Algebraic properties
Addition theorem
A mathematical result of considerable interest and use is called the addition theorem for spherical harmonics. Given two vectors r and r′, with spherical coordinates $(r,\theta ,\varphi )$ and $(r',\theta ',\varphi ')$, respectively, the angle $\gamma $ between them is given by the relation
$\cos \gamma =\cos \theta '\cos \theta +\sin \theta \sin \theta '\cos(\varphi -\varphi ')$
in which the role of the trigonometric functions appearing on the right-hand side is played by the spherical harmonics and that of the left-hand side is played by the Legendre polynomials.
The addition theorem states[17]
$P_{\ell }(\mathbf {x} \cdot \mathbf {y} )={\frac {4\pi }{2\ell +1}}\sum _{m=-\ell }^{\ell }Y_{\ell m}(\mathbf {y} )\,Y_{\ell m}^{*}(\mathbf {x} )\quad \forall \,\ell \in \mathbb {N} _{0}\;\forall \,\mathbf {x} ,\mathbf {y} \in \mathbb {R} ^{3}\colon \;\|\mathbf {x} \|_{2}=\|\mathbf {y} \|_{2}=1\,,$
(1)
where Pℓ is the Legendre polynomial of degree ℓ. This expression is valid for both real and complex harmonics.[18] The result can be proven analytically, using the properties of the Poisson kernel in the unit ball, or geometrically by applying a rotation to the vector y so that it points along the z-axis, and then directly calculating the right-hand side.[19]
In particular, when x = y, this gives Unsöld's theorem[20]
$\sum _{m=-\ell }^{\ell }Y_{\ell m}^{*}(\mathbf {x} )\,Y_{\ell m}(\mathbf {x} )={\frac {2\ell +1}{4\pi }}$
which generalizes the identity cos2θ + sin2θ = 1 to two dimensions.
In the expansion (1), the left-hand side $P_{\ell }(\mathbf {x} \cdot \mathbf {y} )$ is a constant multiple of the degree ℓ zonal spherical harmonic. From this perspective, one has the following generalization to higher dimensions. Let Yj be an arbitrary orthonormal basis of the space Hℓ of degree ℓ spherical harmonics on the n-sphere. Then $Z_{\mathbf {x} }^{(\ell )}$, the degree ℓ zonal harmonic corresponding to the unit vector x, decomposes as[21]
$Z_{\mathbf {x} }^{(\ell )}({\mathbf {y} })=\sum _{j=1}^{\dim(\mathbf {H} _{\ell })}{\overline {Y_{j}({\mathbf {x} })}}\,Y_{j}({\mathbf {y} })$
(2)
Furthermore, the zonal harmonic $Z_{\mathbf {x} }^{(\ell )}({\mathbf {y} })$ is given as a constant multiple of the appropriate Gegenbauer polynomial:
$Z_{\mathbf {x} }^{(\ell )}({\mathbf {y} })=C_{\ell }^{((n-2)/2)}({\mathbf {x} }\cdot {\mathbf {y} })$
(3)
Combining (2) and (3) gives (1) in dimension n = 2 when x and y are represented in spherical coordinates. Finally, evaluating at x = y gives the functional identity
${\frac {\dim \mathbf {H} _{\ell }}{\omega _{n-1}}}=\sum _{j=1}^{\dim(\mathbf {H} _{\ell })}|Y_{j}({\mathbf {x} })|^{2}$
where ωn−1 is the volume of the (n−1)-sphere.
Contraction rule
Another useful identity expresses the product of two spherical harmonics as a sum over spherical harmonics[22]
$Y_{a,\alpha }\left(\theta ,\varphi \right)Y_{b,\beta }\left(\theta ,\varphi \right)={\sqrt {\frac {\left(2a+1\right)\left(2b+1\right)}{4\pi }}}\sum _{c=0}^{\infty }\sum _{\gamma =-c}^{c}\left(-1\right)^{\gamma }{\sqrt {2c+1}}{\begin{pmatrix}a&b&c\\\alpha &\beta &-\gamma \end{pmatrix}}{\begin{pmatrix}a&b&c\\0&0&0\end{pmatrix}}Y_{c,\gamma }\left(\theta ,\varphi \right).$
Many of the terms in this sum are trivially zero. The values of $c$ and $\gamma $ that result in non-zero terms in this sum are determined by the selection rules for the 3j-symbols.
Clebsch–Gordan coefficients
Main article: Clebsch–Gordan coefficients
The Clebsch–Gordan coefficients are the coefficients appearing in the expansion of the product of two spherical harmonics in terms of spherical harmonics themselves. A variety of techniques are available for doing essentially the same calculation, including the Wigner 3-jm symbol, the Racah coefficients, and the Slater integrals. Abstractly, the Clebsch–Gordan coefficients express the tensor product of two irreducible representations of the rotation group as a sum of irreducible representations: suitably normalized, the coefficients are then the multiplicities.
Visualization of the spherical harmonics
The Laplace spherical harmonics $Y_{\ell }^{m}$ can be visualized by considering their "nodal lines", that is, the set of points on the sphere where $\Re [Y_{\ell }^{m}]=0$, or alternatively where $\Im [Y_{\ell }^{m}]=0$. Nodal lines of $Y_{\ell }^{m}$ are composed of ℓ circles: there are |m| circles along longitudes and ℓ−|m| circles along latitudes. One can determine the number of nodal lines of each type by counting the number of zeros of $Y_{\ell }^{m}$ in the $\theta $ and $\varphi $ directions respectively. Considering $Y_{\ell }^{m}$ as a function of $\theta $, the real and imaginary components of the associated Legendre polynomials each possess ℓ−|m| zeros, each giving rise to a nodal 'line of latitude'. On the other hand, considering $Y_{\ell }^{m}$ as a function of $\varphi $, the trigonometric sin and cos functions possess 2|m| zeros, each of which gives rise to a nodal 'line of longitude'.
When the spherical harmonic order m is zero (upper-left in the figure), the spherical harmonic functions do not depend upon longitude, and are referred to as zonal. Such spherical harmonics are a special case of zonal spherical functions. When ℓ = |m| (bottom-right in the figure), there are no zero crossings in latitude, and the functions are referred to as sectoral. For the other cases, the functions checker the sphere, and they are referred to as tesseral.
More general spherical harmonics of degree ℓ are not necessarily those of the Laplace basis $Y_{\ell }^{m}$, and their nodal sets can be of a fairly general kind.[23]
List of spherical harmonics
Main article: Table of spherical harmonics
Analytic expressions for the first few orthonormalized Laplace spherical harmonics $Y_{\ell }^{m}:S^{2}\to \mathbb {C} $ that use the Condon–Shortley phase convention:
$Y_{0}^{0}(\theta ,\varphi )={\frac {1}{2}}{\sqrt {\frac {1}{\pi }}}$
${\begin{aligned}Y_{1}^{-1}(\theta ,\varphi )&={\frac {1}{2}}{\sqrt {\frac {3}{2\pi }}}\,\sin \theta \,e^{-i\varphi }\\Y_{1}^{0}(\theta ,\varphi )&={\frac {1}{2}}{\sqrt {\frac {3}{\pi }}}\,\cos \theta \\Y_{1}^{1}(\theta ,\varphi )&={\frac {-1}{2}}{\sqrt {\frac {3}{2\pi }}}\,\sin \theta \,e^{i\varphi }\end{aligned}}$
${\begin{aligned}Y_{2}^{-2}(\theta ,\varphi )&={\frac {1}{4}}{\sqrt {\frac {15}{2\pi }}}\,\sin ^{2}\theta \,e^{-2i\varphi }\\Y_{2}^{-1}(\theta ,\varphi )&={\frac {1}{2}}{\sqrt {\frac {15}{2\pi }}}\,\sin \theta \,\cos \theta \,e^{-i\varphi }\\Y_{2}^{0}(\theta ,\varphi )&={\frac {1}{4}}{\sqrt {\frac {5}{\pi }}}\,(3\cos ^{2}\theta -1)\\Y_{2}^{1}(\theta ,\varphi )&={\frac {-1}{2}}{\sqrt {\frac {15}{2\pi }}}\,\sin \theta \,\cos \theta \,e^{i\varphi }\\Y_{2}^{2}(\theta ,\varphi )&={\frac {1}{4}}{\sqrt {\frac {15}{2\pi }}}\,\sin ^{2}\theta \,e^{2i\varphi }\end{aligned}}$
Higher dimensions
The classical spherical harmonics are defined as complex-valued functions on the unit sphere $S^{2}$ inside three-dimensional Euclidean space $\mathbb {R} ^{3}$. Spherical harmonics can be generalized to higher-dimensional Euclidean space $\mathbb {R} ^{n}$ as follows, leading to functions $S^{n-1}\to \mathbb {C} $.[24] Let Pℓ denote the space of complex-valued homogeneous polynomials of degree ℓ in n real variables, here considered as functions $\mathbb {R} ^{n}\to \mathbb {C} $. That is, a polynomial p is in Pℓ provided that for any real $\lambda \in \mathbb {R} $, one has
$p(\lambda \mathbf {x} )=\lambda ^{\ell }p(\mathbf {x} ).$
Let Aℓ denote the subspace of Pℓ consisting of all harmonic polynomials:
$\mathbf {A} _{\ell }:=\{p\in \mathbf {P} _{\ell }\,\mid \,\Delta p=0\}\,.$
These are the (regular) solid spherical harmonics. Let Hℓ denote the space of functions on the unit sphere
$S^{n-1}:=\{\mathbf {x} \in \mathbb {R} ^{n}\,\mid \,\left|x\right|=1\}$
obtained by restriction from Aℓ
$\mathbf {H} _{\ell }:=\left\{f:S^{n-1}\to \mathbb {C} \,\mid \,{\text{ for some }}p\in \mathbf {A} _{\ell },\,f(\mathbf {x} )=p(\mathbf {x} ){\text{ for all }}\mathbf {x} \in S^{n-1}\right\}.$
The following properties hold:
• The sum of the spaces Hℓ is dense in the set $C(S^{n-1})$ of continuous functions on $S^{n-1}$ with respect to the uniform topology, by the Stone–Weierstrass theorem. As a result, the sum of these spaces is also dense in the space L2(Sn−1) of square-integrable functions on the sphere. Thus every square-integrable function on the sphere decomposes uniquely into a series of spherical harmonics, where the series converges in the L2 sense.
• For all f ∈ Hℓ, one has
$\Delta _{S^{n-1}}f=-\ell (\ell +n-2)f.$
where ΔSn−1 is the Laplace–Beltrami operator on Sn−1. This operator is the analog of the angular part of the Laplacian in three dimensions; to wit, the Laplacian in n dimensions decomposes as
$\nabla ^{2}=r^{1-n}{\frac {\partial }{\partial r}}r^{n-1}{\frac {\partial }{\partial r}}+r^{-2}\Delta _{S^{n-1}}={\frac {\partial ^{2}}{\partial r^{2}}}+{\frac {n-1}{r}}{\frac {\partial }{\partial r}}+r^{-2}\Delta _{S^{n-1}}$
• It follows from the Stokes theorem and the preceding property that the spaces Hℓ are orthogonal with respect to the inner product from L2(Sn−1). That is to say,
$\int _{S^{n-1}}f{\bar {g}}\,\mathrm {d} \Omega =0$
for f ∈ Hℓ and g ∈ Hk for k ≠ ℓ.
• Conversely, the spaces Hℓ are precisely the eigenspaces of ΔSn−1. In particular, an application of the spectral theorem to the Riesz potential $\Delta _{S^{n-1}}^{-1}$ gives another proof that the spaces Hℓ are pairwise orthogonal and complete in L2(Sn−1).
• Every homogeneous polynomial p ∈ Pℓ can be uniquely written in the form[25]
$p(x)=p_{\ell }(x)+|x|^{2}p_{\ell -2}+\cdots +{\begin{cases}|x|^{\ell }p_{0}&\ell {\rm {\ even}}\\|x|^{\ell -1}p_{1}(x)&\ell {\rm {\ odd}}\end{cases}}$
where pj ∈ Aj. In particular,
$\dim \mathbf {H} _{\ell }={\binom {n+\ell -1}{n-1}}-{\binom {n+\ell -3}{n-1}}.$
An orthogonal basis of spherical harmonics in higher dimensions can be constructed inductively by the method of separation of variables, by solving the Sturm-Liouville problem for the spherical Laplacian
$\Delta _{S^{n-1}}=\sin ^{2-n}\varphi {\frac {\partial }{\partial \varphi }}\sin ^{n-2}\varphi {\frac {\partial }{\partial \varphi }}+\sin ^{-2}\varphi \Delta _{S^{n-2}}$
where φ is the axial coordinate in a spherical coordinate system on Sn−1. The end result of such a procedure is[26]
$Y_{\ell _{1},\dots \ell _{n-1}}(\theta _{1},\dots \theta _{n-1})={\frac {1}{\sqrt {2\pi }}}e^{i\ell _{1}\theta _{1}}\prod _{j=2}^{n-1}{}_{j}{\bar {P}}_{\ell _{j}}^{\ell _{j-1}}(\theta _{j})$
where the indices satisfy |ℓ1| ≤ ℓ2 ≤ ⋯ ≤ ℓn−1 and the eigenvalue is −ℓn−1(ℓn−1 + n−2). The functions in the product are defined in terms of the Legendre function
${}_{j}{\bar {P}}_{L}^{\ell }(\theta )={\sqrt {{\frac {2L+j-1}{2}}{\frac {(L+\ell +j-2)!}{(L-\ell )!}}}}\sin ^{\frac {2-j}{2}}(\theta )P_{L+{\frac {j-2}{2}}}^{-\left(\ell +{\frac {j-2}{2}}\right)}(\cos \theta )\,.$
Connection with representation theory
The space Hℓ of spherical harmonics of degree ℓ is a representation of the symmetry group of rotations around a point (SO(3)) and its double-cover SU(2). Indeed, rotations act on the two-dimensional sphere, and thus also on Hℓ by function composition
$\psi \mapsto \psi \circ \rho ^{-1}$
for ψ a spherical harmonic and ρ a rotation. The representation Hℓ is an irreducible representation of SO(3).[27]
The elements of Hℓ arise as the restrictions to the sphere of elements of Aℓ: harmonic polynomials homogeneous of degree ℓ on three-dimensional Euclidean space R3. By polarization of ψ ∈ Aℓ, there are coefficients $\psi _{i_{1}\dots i_{\ell }}$ symmetric on the indices, uniquely determined by the requirement
$\psi (x_{1},\dots ,x_{n})=\sum _{i_{1}\dots i_{\ell }}\psi _{i_{1}\dots i_{\ell }}x_{i_{1}}\cdots x_{i_{\ell }}.$
The condition that ψ be harmonic is equivalent to the assertion that the tensor $\psi _{i_{1}\dots i_{\ell }}$ must be trace free on every pair of indices. Thus as an irreducible representation of SO(3), Hℓ is isomorphic to the space of traceless symmetric tensors of degree ℓ.
More generally, the analogous statements hold in higher dimensions: the space Hℓ of spherical harmonics on the n-sphere is the irreducible representation of SO(n+1) corresponding to the traceless symmetric ℓ-tensors. However, whereas every irreducible tensor representation of SO(2) and SO(3) is of this kind, the special orthogonal groups in higher dimensions have additional irreducible representations that do not arise in this manner.
The special orthogonal groups have additional spin representations that are not tensor representations, and are typically not spherical harmonics. An exception are the spin representation of SO(3): strictly speaking these are representations of the double cover SU(2) of SO(3). In turn, SU(2) is identified with the group of unit quaternions, and so coincides with the 3-sphere. The spaces of spherical harmonics on the 3-sphere are certain spin representations of SO(3), with respect to the action by quaternionic multiplication.
Connection with hemispherical harmonics
Spherical harmonics can be separated into two set of functions.[28] One is hemispherical functions (HSH), orthogonal and complete on hemisphere. Another is complementary hemispherical harmonics (CHSH).
Generalizations
The angle-preserving symmetries of the two-sphere are described by the group of Möbius transformations PSL(2,C). With respect to this group, the sphere is equivalent to the usual Riemann sphere. The group PSL(2,C) is isomorphic to the (proper) Lorentz group, and its action on the two-sphere agrees with the action of the Lorentz group on the celestial sphere in Minkowski space. The analog of the spherical harmonics for the Lorentz group is given by the hypergeometric series; furthermore, the spherical harmonics can be re-expressed in terms of the hypergeometric series, as SO(3) = PSU(2) is a subgroup of PSL(2,C).
More generally, hypergeometric series can be generalized to describe the symmetries of any symmetric space; in particular, hypergeometric series can be developed for any Lie group.[29][30][31][32]
See also
Wikimedia Commons has media related to Spherical harmonics.
• Cubic harmonic (often used instead of spherical harmonics in computations)
• Cylindrical harmonics
• Spherical basis
• Spinor spherical harmonics
• Spin-weighted spherical harmonics
• Sturm–Liouville theory
• Table of spherical harmonics
• Vector spherical harmonics
• Atomic orbital
Notes
1. A historical account of various approaches to spherical harmonics in three dimensions can be found in Chapter IV of MacRobert 1967. The term "Laplace spherical harmonics" is in common use; see Courant & Hilbert 1962 and Meijer & Bauer 2004.
2. The approach to spherical harmonics taken here is found in (Courant & Hilbert 1962, §V.8, §VII.5).
3. Physical applications often take the solution that vanishes at infinity, making A = 0. This does not affect the angular portion of the spherical harmonics.
4. Weisstein, Eric W. "Spherical Harmonic". mathworld.wolfram.com. Retrieved 2023-05-10.
5. Edmonds 1957, §2.5
6. Hall 2013 Section 17.6
7. Hall 2013 Lemma 17.16
8. Williams, Earl G. (1999). Fourier acoustics : sound radiation and nearfield acoustical holography. San Diego, Calif.: Academic Press. ISBN 0080506909. OCLC 181010993.
9. Messiah, Albert (1999). Quantum mechanics : two volumes bound as one (Two vol. bound as one, unabridged reprint ed.). Mineola, NY: Dover. ISBN 9780486409245.
10. Claude Cohen-Tannoudji; Bernard Diu; Franck Laloë (1996). Quantum mechanics. Translated by Susan Reid Hemley; et al. Wiley-Interscience: Wiley. ISBN 9780471569527.
11. Blakely, Richard (1995). Potential theory in gravity and magnetic applications. Cambridge England New York: Cambridge University Press. p. 113. ISBN 978-0521415088.
12. Heiskanen and Moritz, Physical Geodesy, 1967, eq. 1-62
13. Weisstein, Eric W. "Condon-Shortley Phase". mathworld.wolfram.com. Retrieved 2022-11-02.
14. Whittaker & Watson 1927, p. 392.
15. See, e.g., Appendix A of Garg, A., Classical Electrodynamics in a Nutshell (Princeton University Press, 2012).
16. Li, Feifei; Braun, Carol; Garg, Anupam (2013), "The Weyl-Wigner-Moyal Formalism for Spin", Europhysics Letters, 102 (6): 60006, arXiv:1210.4075, Bibcode:2013EL....10260006L, doi:10.1209/0295-5075/102/60006, S2CID 119610178
17. Edmonds, A. R. (1996). Angular Momentum In Quantum Mechanics. Princeton University Press. p. 63.
18. This is valid for any orthonormal basis of spherical harmonics of degree ℓ. For unit power harmonics it is necessary to remove the factor of 4π.
19. Whittaker & Watson 1927, p. 395
20. Unsöld 1927
21. Stein & Weiss 1971, §IV.2
22. Brink, D. M.; Satchler, G. R. Angular Momentum. Oxford University Press. p. 146.
23. Eremenko, Jakobson & Nadirashvili 2007
24. Solomentsev 2001; Stein & Weiss 1971, §Iv.2
25. Cf. Corollary 1.8 of Axler, Sheldon; Ramey, Wade (1995), Harmonic Polynomials and Dirichlet-Type Problems
26. Higuchi, Atsushi (1987). "Symmetric tensor spherical harmonics on the N-sphere and their application to the de Sitter group SO(N,1)". Journal of Mathematical Physics. 28 (7): 1553–1566. Bibcode:1987JMP....28.1553H. doi:10.1063/1.527513.
27. Hall 2013 Corollary 17.17
28. Zheng Y, Wei K, Liang B, Li Y, Chu X (2019-12-23). "Zernike like functions on spherical cap: principle and applications in optical surface fitting and graphics rendering". Optics Express. 27 (26): 37180–37195. Bibcode:2019OExpr..2737180Z. doi:10.1364/OE.27.037180. ISSN 1094-4087. PMID 31878503.
29. N. Vilenkin, Special Functions and the Theory of Group Representations, Am. Math. Soc. Transl., vol. 22, (1968).
30. J. D. Talman, Special Functions, A Group Theoretic Approach, (based on lectures by E.P. Wigner), W. A. Benjamin, New York (1968).
31. W. Miller, Symmetry and Separation of Variables, Addison-Wesley, Reading (1977).
32. A. Wawrzyńczyk, Group Representations and Special Functions, Polish Scientific Publishers. Warszawa (1984).
References
Cited references
• Courant, Richard; Hilbert, David (1962), Methods of Mathematical Physics, Volume I, Wiley-Interscience.
• Edmonds, A.R. (1957), Angular Momentum in Quantum Mechanics, Princeton University Press, ISBN 0-691-07912-9
• Eremenko, Alexandre; Jakobson, Dmitry; Nadirashvili, Nikolai (2007), "On nodal sets and nodal domains on S² and R²", Annales de l'Institut Fourier, 57 (7): 2345–2360, doi:10.5802/aif.2335, ISSN 0373-0956, MR 2394544
• Hall, Brian C. (2013), Quantum Theory for Mathematicians, Graduate Texts in Mathematics, vol. 267, Springer, ISBN 978-1461471158
• MacRobert, T.M. (1967), Spherical harmonics: An elementary treatise on harmonic functions, with applications, Pergamon Press.
• Meijer, Paul Herman Ernst; Bauer, Edmond (2004), Group theory: The application to quantum mechanics, Dover, ISBN 978-0-486-43798-9.
• Solomentsev, E.D. (2001) [1994], "Spherical harmonics", Encyclopedia of Mathematics, EMS Press.
• Stein, Elias; Weiss, Guido (1971), Introduction to Fourier Analysis on Euclidean Spaces, Princeton, N.J.: Princeton University Press, ISBN 978-0-691-08078-9.
• Unsöld, Albrecht (1927), "Beiträge zur Quantenmechanik der Atome", Annalen der Physik, 387 (3): 355–393, Bibcode:1927AnP...387..355U, doi:10.1002/andp.19273870304.
• Whittaker, E. T.; Watson, G. N. (1927), A Course of Modern Analysis, Cambridge University Press, p. 392.
General references
• E.W. Hobson, The Theory of Spherical and Ellipsoidal Harmonics, (1955) Chelsea Pub. Co., ISBN 978-0-8284-0104-3.
• C. Müller, Spherical Harmonics, (1966) Springer, Lecture Notes in Mathematics, Vol. 17, ISBN 978-3-540-03600-5.
• E. U. Condon and G. H. Shortley, The Theory of Atomic Spectra, (1970) Cambridge at the University Press, ISBN 0-521-09209-4, See chapter 3.
• J.D. Jackson, Classical Electrodynamics, ISBN 0-471-30932-X
• Albert Messiah, Quantum Mechanics, volume II. (2000) Dover. ISBN 0-486-40924-4.
• Press, WH; Teukolsky, SA; Vetterling, WT; Flannery, BP (2007), "Section 6.7. Spherical Harmonics", Numerical Recipes: The Art of Scientific Computing (3rd ed.), New York: Cambridge University Press, ISBN 978-0-521-88068-8
• D. A. Varshalovich, A. N. Moskalev, V. K. Khersonskii Quantum Theory of Angular Momentum,(1988) World Scientific Publishing Co., Singapore, ISBN 9971-5-0107-4
• Weisstein, Eric W. "Spherical harmonics". MathWorld.
• Maddock, John, Spherical harmonics in Boost.Math
External links
• Spherical Harmonics at MathWorld
• Spherical Harmonics 3D representation
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Wikipedia
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Spheroidal wave equation
In mathematics, the spheroidal wave equation is given by
$(1-t^{2}){\frac {d^{2}y}{dt^{2}}}-2(b+1)t\,{\frac {dy}{dt}}+(c-4qt^{2})\,y=0$
It is a generalization of the Mathieu differential equation.[1] If $y(t)$ is a solution to this equation and we define $S(t):=(1-t^{2})^{b/2}y(t)$, then $S(t)$ is a prolate spheroidal wave function in the sense that it satisfies the equation[2]
$(1-t^{2}){\frac {d^{2}S}{dt^{2}}}-2t\,{\frac {dS}{dt}}+(c-4q+b+b^{2}+4q(1-t^{2})-{\frac {b^{2}}{1-t^{2}}})\,S=0$
See also
• Wave equation
References
1. see Abramowitz and Stegun, page 722
2. see Bateman, page 442
Bibliography
• M. Abramowitz and I. Stegun, Handbook of Mathematical function (US Gov. Printing Office, Washington DC, 1964)
• H. Bateman, Partial Differential Equations of Mathematical Physics (Dover Publications, New York, 1944)
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Wikipedia
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Spheroidal wave function
Spheroidal wave functions are solutions of the Helmholtz equation that are found by writing the equation in spheroidal coordinates and applying the technique of separation of variables, just like the use of spherical coordinates lead to spherical harmonics. They are called oblate spheroidal wave functions if oblate spheroidal coordinates are used and prolate spheroidal wave functions if prolate spheroidal coordinates are used.[1] If instead of the Helmholtz equation, the Laplace equation is solved in spheroidal coordinates using the method of separation of variables, the spheroidal wave functions reduce to the spheroidal harmonics. With oblate spheroidal coordinates, the solutions are called oblate harmonics and with prolate spheroidal coordinates, prolate harmonics. Both type of spheroidal harmonics are expressible in terms of Legendre functions.
See also
• Oblate spheroidal coordinates, especially the section Oblate spheroidal harmonics, for a more extensive discussion.
• Oblate spheroidal wave function
References
Notes
1. Flammer, C. (1957). Spheroidal wave functions. Stanford University Press Stanford, Calif.
Bibliography
• C. Niven On the Conduction of Heat in Ellipsoids of Revolution. Philosophical transactions of the Royal Society of London, v. 171 p. 117 (1880)
• M. Abramowitz and I. Stegun, Handbook of Mathematical function (US Gov. Printing Office, Washington DC, 1964)
• Volkmer, H. (2010), "Spheroidal wave function", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, MR 2723248.
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Sphinx tiling
In geometry, the sphinx tiling is a tessellation of the plane using the "sphinx", a pentagonal hexiamond formed by gluing six equilateral triangles together. The resultant shape is named for its reminiscence to the Great Sphinx at Giza. A sphinx can be dissected into any square number of copies of itself,[1] some of them mirror images, and repeating this process leads to a non-periodic tiling of the plane. The sphinx is therefore a rep-tile (a self-replicating tessellation).[2] It is one of few known pentagonal rep-tiles and is the only known pentagonal rep-tile whose sub-copies are equal in size.[3]
Dissection of the sphinx into four sub-copies
Dissection of the sphinx into nine sub-copies
See also
• Mosaic
References
1. Niţică, Viorel (2003), "Rep-tiles revisited", MASS selecta, Providence, RI: American Mathematical Society, pp. 205–217, MR 2027179.
2. Godrèche, C. (1989), "The sphinx: a limit-periodic tiling of the plane", Journal of Physics A: Mathematical and General, 22 (24): L1163–L1166, doi:10.1088/0305-4470/22/24/006, MR 1030678
3. Martin, Andy (2003), "The sphinx task centre problem", in Pritchard, Chris (ed.), The Changing Shape of Geometry, MAA Spectrum, Cambridge University Press, pp. 371–378, ISBN 9780521531627
External links
• Mathematics Centre Sphinx Album ...
• Weisstein, Eric W. "Sphinx". MathWorld.
Tessellation
Periodic
• Pythagorean
• Rhombille
• Schwarz triangle
• Rectangle
• Domino
• Uniform tiling and honeycomb
• Coloring
• Convex
• Kisrhombille
• Wallpaper group
• Wythoff
Aperiodic
• Ammann–Beenker
• Aperiodic set of prototiles
• List
• Einstein problem
• Socolar–Taylor
• Gilbert
• Penrose
• Pentagonal
• Pinwheel
• Quaquaversal
• Rep-tile and Self-tiling
• Sphinx
• Socolar
• Truchet
Other
• Anisohedral and Isohedral
• Architectonic and catoptric
• Circle Limit III
• Computer graphics
• Honeycomb
• Isotoxal
• List
• Packing
• Problems
• Domino
• Wang
• Heesch's
• Squaring
• Dividing a square into similar rectangles
• Prototile
• Conway criterion
• Girih
• Regular Division of the Plane
• Regular grid
• Substitution
• Voronoi
• Voderberg
By vertex type
Spherical
• 2n
• 33.n
• V33.n
• 42.n
• V42.n
Regular
• 2∞
• 36
• 44
• 63
Semi-
regular
• 32.4.3.4
• V32.4.3.4
• 33.42
• 33.∞
• 34.6
• V34.6
• 3.4.6.4
• (3.6)2
• 3.122
• 42.∞
• 4.6.12
• 4.82
Hyper-
bolic
• 32.4.3.5
• 32.4.3.6
• 32.4.3.7
• 32.4.3.8
• 32.4.3.∞
• 32.5.3.5
• 32.5.3.6
• 32.6.3.6
• 32.6.3.8
• 32.7.3.7
• 32.8.3.8
• 33.4.3.4
• 32.∞.3.∞
• 34.7
• 34.8
• 34.∞
• 35.4
• 37
• 38
• 3∞
• (3.4)3
• (3.4)4
• 3.4.62.4
• 3.4.7.4
• 3.4.8.4
• 3.4.∞.4
• 3.6.4.6
• (3.7)2
• (3.8)2
• 3.142
• 3.162
• (3.∞)2
• 3.∞2
• 42.5.4
• 42.6.4
• 42.7.4
• 42.8.4
• 42.∞.4
• 45
• 46
• 47
• 48
• 4∞
• (4.5)2
• (4.6)2
• 4.6.12
• 4.6.14
• V4.6.14
• 4.6.16
• V4.6.16
• 4.6.∞
• (4.7)2
• (4.8)2
• 4.8.10
• V4.8.10
• 4.8.12
• 4.8.14
• 4.8.16
• 4.8.∞
• 4.102
• 4.10.12
• 4.122
• 4.12.16
• 4.142
• 4.162
• 4.∞2
• (4.∞)2
• 54
• 55
• 56
• 5∞
• 5.4.6.4
• (5.6)2
• 5.82
• 5.102
• 5.122
• (5.∞)2
• 64
• 65
• 66
• 68
• 6.4.8.4
• (6.8)2
• 6.82
• 6.102
• 6.122
• 6.162
• 73
• 74
• 77
• 7.62
• 7.82
• 7.142
• 83
• 84
• 86
• 88
• 8.62
• 8.122
• 8.162
• ∞3
• ∞4
• ∞5
• ∞∞
• ∞.62
• ∞.82
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Spider diagram
In mathematics, a unitary spider diagram adds existential points to an Euler or a Venn diagram. The points indicate the existence of an attribute described by the intersection of contours in the Euler diagram. These points may be joined forming a shape like a spider. Joined points represent an "or" condition, also known as a logical disjunction.
This article is about the extension of Euler diagrams. For diagrams showing relationships among concepts, see concept map and mind map. For radar charts, see spider chart.
A spider diagram is a boolean expression involving unitary spider diagrams and the logical symbols $\land ,\lor ,\lnot $. For example, it may consist of the conjunction of two spider diagrams, the disjunction of two spider diagrams, or the negation of a spider diagram.
Example
In the image shown, the following conjunctions are apparent from the Euler diagram.
$A\land B$
$B\land C$
$F\land E$
$G\land F$
In the universe of discourse defined by this Euler diagram, in addition to the conjunctions specified above, all possible sets from A through B and D through G are available separately. The set C is only available as a subset of B. Often, in complicated diagrams, singleton sets and/or conjunctions may be obscured by other set combinations.
The two spiders in the example correspond to the following logical expressions:
• Red spider: $(F\land E)\lor (G)\lor (D)$
• Blue spider: $(A)\lor (C\land B)\lor (F)$
References
• Howse, J. and Stapleton, G. and Taylor, H. Spider Diagrams London Mathematical Society Journal of Computation and Mathematics, (2005) v. 8, pp. 145–194. ISSN 1461-1570 Accessed on January 8, 2012 here
• Stapleton, G. and Howse, J. and Taylor, J. and Thompson, S. What can spider diagrams say? Proc. Diagrams, (2004) v. 168, pp. 169–219. Accessed on January 4, 2012 here
• Stapleton, G. and Jamnik, M. and Masthoff, J. On the Readability of Diagrammatic Proofs Proc. Automated Reasoning Workshop, 2009. PDF
External links
Wikimedia Commons has media related to Spider diagrams.
• Brighton and Kent University - Euler Diagrams
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Polygon-circle graph
In the mathematical discipline of graph theory, a polygon-circle graph is an intersection graph of a set of convex polygons all of whose vertices lie on a common circle. These graphs have also been called spider graphs. This class of graphs was first suggested by Michael Fellows in 1988, motivated by the fact that it is closed under edge contraction and induced subgraph operations.[1]
"Spider graph" redirects here. For the chart, see radar chart. For the cricket term, see Glossary of cricket terms.
A polygon-circle graph can be represented as an "alternating sequence". Such a sequence can be gained by perturbing the polygons representing the graph (if necessary) so that no two share a vertex, and then listing for each vertex (in circular order, starting at an arbitrary point) the polygon attached to that vertex.
Closure under induced minors
Contracting an edge of a polygon-circle graph results in another polygon-circle graph. A geometric representation of the new graph may be formed by replacing the polygons corresponding to the two endpoints of the contracted edge by their convex hull. Alternatively, in the alternating sequence representing the original graph, combining the subsequences representing the endpoints of the contracted edge into a single subsequence produces an alternating sequence representation of the contracted graph. Polygon circle graphs are also closed under induced subgraph or equivalently vertex deletion operations: to delete a vertex, remove its polygon from the geometric representation, or remove its subsequence of points from the alternating sequence.
Recognition
M. Koebe announced a polynomial time recognition algorithm;[2] however, his preliminary version had "serious errors"[3] and a final version was never published.[1] Martin Pergel later proved that the problem of recognizing these graphs is NP-complete.[4] It is also NP-complete to determine whether a given graph can be represented as a polygon-circle graph with at most k vertices per polygon, for any k ≥ 3.[1]
Related graph families
The polygon-circle graphs are a generalization of the circle graphs, which are intersection graphs of the chords of a circle, and the trapezoid graphs, intersection graphs of trapezoids that all have their vertices on the same two parallel lines. They also include the circular arc graphs.[1][5]
Polygon-circle graphs are not, in general, perfect graphs, but they are near-perfect, in the sense that their chromatic numbers can be bounded by an (exponential) function of their clique numbers.[6]
References
1. Kratochvíl, Jan; Pergel, Martin (2004), "Two results on intersection graphs of polygons", Graph Drawing: 11th International Symposium, GD 2003 Perugia, Italy, September 21-24, 2003, Revised Papers, Lecture Notes in Computer Science, vol. 2912, Berlin: Springer, pp. 59–70, doi:10.1007/978-3-540-24595-7_6, MR 2177583.
2. Koebe, Manfred (1992), "On a new class of intersection graphs", Fourth Czechoslovakian Symposium on Combinatorics, Graphs and Complexity (Prachatice, 1990), Ann. Discrete Math., vol. 51, North-Holland, Amsterdam, pp. 141–143, doi:10.1016/S0167-5060(08)70618-6, MR 1206256.
3. Spinrad, Jeremy P. (2003), Efficient graph representations, Fields Institute Monographs, vol. 19, American Mathematical Society, Providence, RI, p. 41, ISBN 0-8218-2815-0, MR 1971502.
4. Pergel, Martin (2007), "Recognition of polygon-circle graphs and graphs of interval filaments is NP-complete", Graph-Theoretic Concepts in Computer Science: 33rd International Workshop, WG 2007, Dornburg, Germany, June 21-23, 2007, Revised Papers, Lecture Notes in Computer Science, vol. 4769, Berlin: Springer, pp. 238–247, doi:10.1007/978-3-540-74839-7_23, MR 2428581.
5. Spider graphs, Information System on Graph Classes and their Inclusions, retrieved 2016-07-11.
6. Kostochka, Alexandr; Kratochvíl, Jan (1997), "Covering and coloring polygon-circle graphs", Discrete Mathematics, 163 (1–3): 299–305, doi:10.1016/S0012-365X(96)00344-5, MR 1428585.
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Spieker center
In geometry, the Spieker center is a special point associated with a plane triangle. It is defined as the center of mass of the perimeter of the triangle. The Spieker center of a triangle △ABC is the center of gravity of a homogeneous wire frame in the shape of △ABC.[1][2] The point is named in honor of the 19th-century German geometer Theodor Spieker.[3] The Spieker center is a triangle center and it is listed as the point X(10) in Clark Kimberling's Encyclopedia of Triangle Centers.
Location
The following result can be used to locate the Spieker center of any triangle.[1]
The Spieker center of triangle △ABC is the incenter of the medial triangle of △ABC.
That is, the Spieker center of △ABC is the center of the circle inscribed in the medial triangle of △ABC. This circle is known as the Spieker circle.
The Spieker center is also located at the intersection of the three cleavers of triangle △ABC. A cleaver of a triangle is a line segment that bisects the perimeter of the triangle and has one endpoint at the midpoint of one of the three sides. Each cleaver contains the center of mass of the boundary of △ABC, so the three cleavers meet at the Spieker center.
To see that the incenter of the medial triangle coincides with the intersection point of the cleavers, consider a homogeneous wireframe in the shape of triangle △ABC consisting of three wires in the form of line segments having lengths a, b, c. The wire frame has the same center of mass as a system of three particles of masses a, b, c placed at the midpoints D, E, F of the sides BC, CA, AB. The centre of mass of the particles at E and F is the point P which divides the segment EF in the ratio c : b. The line DP is the internal bisector of ∠D. The centre of mass of the three particle system thus lies on the internal bisector of ∠D. Similar arguments show that the center mass of the three particle system lies on the internal bisectors of ∠E and ∠F also. It follows that the center of mass of the wire frame is the point of concurrence of the internal bisectors of the angles of the triangle △DEF , which is the incenter of the medial triangle △DEF .
Properties
Let S be the Spieker center of triangle △ABC.
• The trilinear coordinates of S are
$bc(b+c):ca(c+a):ab(a+b).$[4]
• The barycentric coordinates of S are
$b+c:c+a:a+b.$[4]
• S is the radical center of the three excircles.[5]
• S is the cleavance center of triangle △ABC [1]
• S is collinear with the incenter (I), the centroid (G), and the Nagel point (N) of triangle △ABC. Moreover,[6]
$IS=SM,\quad IG=2\cdot GS,\quad MG=2\cdot IG.$
Thus on a suitably scaled and positioned number line, I = 0, G = 2, S = 3, and M = 6.
• S lies on the Kiepert hyperbola. S is the point of concurrence of the lines AX, BY, CZ where △XBC, △YCA, △ZAB are similar, isosceles and similarly situated triangles constructed on the sides of triangle △ABC as bases, having the common base angle[7]
$\theta =\tan ^{-1}\left[\tan \left({\frac {A}{2}}\right)\tan \left({\frac {B}{2}}\right)\tan \left({\frac {C}{2}}\right)\right].$
References
1. Honsberger, Ross (1995). Episodes in Nineteenth and Twentieth Century Euclidean Geometry. Mathematical Association of America. pp. 3–4.
2. Kimberling, Clark. "Spieker center". Retrieved 5 May 2012.
3. Spieker, Theodor (1888). Lehrbuch der ebenen Geometrie. Potsdam, Germany.{{cite book}}: CS1 maint: location missing publisher (link)
4. Kimberling, Clark. "Encyclopedia of Triangle Centers". Retrieved 5 May 2012.
5. Odenhal, Boris (2010), "Some triangle centers associated with the circles tangent to the excircles" (PDF), Forum Geometricorum, 10: 35–40
6. Bogomolny, A. "Nagel Line from Interactive Mathematics Miscellany and Puzzles". Retrieved 5 May 2012.
7. Weisstein, Eric W. "Kiepert Hyperbola". MathWorld.
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Spieker circle
In geometry, the incircle of the medial triangle of a triangle is the Spieker circle, named after 19th-century German geometer Theodor Spieker.[1] Its center, the Spieker center, in addition to being the incenter of the medial triangle, is the center of mass of the uniform-density boundary of triangle.[1] The Spieker center is also the point where all three cleavers of the triangle (perimeter bisectors with an endpoint at a side's midpoint) intersect each other.[1]
History
The Spieker circle and Spieker center are named after Theodor Spieker, a mathematician and professor from Potsdam, Germany. In 1862, he published Lehrbuch der ebenen geometrie mit übungsaufgaben für höhere lehranstalten, dealing with planar geometry. Due to this publication, influential in the lives of many famous scientists and mathematicians including Albert Einstein, Spieker became the mathematician for whom the Spieker circle and center were named.[1]
Construction
To find the Spieker circle of a triangle, the medial triangle must first be constructed from the midpoints of each side of the original triangle.[1] The circle is then constructed in such a way that each side of the medial triangle is tangent to the circle within the medial triangle, creating the incircle.[1] This circle center is named the Spieker center.
Nagel points and lines
Spieker circles also have relations to Nagel points. The incenter of the triangle and the Nagel point form a line within the Spieker circle. The middle of this line segment is the Spieker center.[1] The Nagel line is formed by the incenter of the triangle, the Nagel point, and the centroid of the triangle.[1] The Spieker center will always lie on this line.[1]
Nine-point circle and Euler line
Spieker circles were first found to be very similar to nine-point circles by Julian Coolidge. At this time, it was not yet identified as the Spieker circle, but is referred to as the "P circle" throughout the book.[2] The nine-point circle with the Euler line and the Spieker circle with the Nagel line are analogous to each other, but are not duals, only having dual-like similarities.[1] One similarity between the nine-point circle and the Spieker circle deals with their construction. The nine-point circle is the circumscribed circle of the medial triangle, while the Spieker circle is the inscribed circle of the medial triangle.[2] With relation to their associated lines, the incenter for the Nagel line relates to the circumcenter for the Euler line.[1] Another analogous point is the Nagel point and the othocenter, with the Nagel point associated with the Spieker circle and the orthocenter associated with the nine-point circle.[1] Each circle meets the sides of the medial triangle where the lines from the orthocenter, or the Nagel point, to the vertices of the original triangle meet the sides of the medial triangle.[2]
Spieker conic
The nine-point circle with the Euler line was generalized into the nine-point conic.[1] Through a similar process, due to the analogous properties of the two circles, the Spieker circle was also able to be generalized into the Spieker conic.[1] The Spieker conic is still found within the medial triangle and touches each side of the medial triangle, however it does not meet those sides of the triangle at the same points. If lines are constructed from each vertex of the medial triangle to the Nagel point, then the midpoint of each of those lines can be found.[3] Also, the midpoints of each side of the medial triangle are found and connected to the midpoint of the opposite line through the Nagel point.[3] Each of these lines share a common midpoint, S.[3] With each of these lines reflected through S, the result is 6 points within the medial triangle. Draw a conic through any 5 of these reflected points and the conic will touch the final point.[1] This was proven by de Villiers in 2006.[1]
Spieker radical circle
The Spieker radical circle is the circle, centered at the Spieker center, which is orthogonal to the three excircles of the medial triangle.[4][5]
References
1. de Villiers, Michael (June 2006). "A generalisation of the Spieker circle and Nagel line". Pythagoras. 63: 30–37.
2. Coolidge, Julian L. (1916). A treatise on the circle and the sphere. Oxford University Press. pp. 53–57.
3. de Villiers, M. (2007). "Spieker Conic and generalization of Nagle line". Dynamic Mathematics Learning.
4. Weisstein, Eric W. "Excircles Radical Circle". MathWorld- A Wolfram Web Resource.
5. Weisstein, Eric W. "Radical Circle". MathWorld- A Wolfram Web Resource.
• Johnson, Roger A. (1929). Modern Geometry. Boston: Houghton Mifflin. Dover reprint, 1960.
• Kimberling, Clark (1998). "Triangle centers and central triangles". Congressus Numerantium. 129: i–xxv, 1–295.
External links
• Spieker Conic and generalization of Nagel line at Dynamic Geometry Sketches Generalizes Spieker circle and associated Nagel line.
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Spin group
In mathematics the spin group Spin(n)[1][2] is a Lie group whose underlying manifold is the double cover of the special orthogonal group SO(n) = SO(n, R), such that there exists a short exact sequence of Lie groups (when n ≠ 2)
$1\to \mathrm {Z} _{2}\to \operatorname {Spin} (n)\to \operatorname {SO} (n)\to 1.$
Algebraic structure → Group theory
Group theory
Basic notions
• Subgroup
• Normal subgroup
• Quotient group
• (Semi-)direct product
Group homomorphisms
• kernel
• image
• direct sum
• wreath product
• simple
• finite
• infinite
• continuous
• multiplicative
• additive
• cyclic
• abelian
• dihedral
• nilpotent
• solvable
• action
• Glossary of group theory
• List of group theory topics
Finite groups
• Cyclic group Zn
• Symmetric group Sn
• Alternating group An
• Dihedral group Dn
• Quaternion group Q
• Cauchy's theorem
• Lagrange's theorem
• Sylow theorems
• Hall's theorem
• p-group
• Elementary abelian group
• Frobenius group
• Schur multiplier
Classification of finite simple groups
• cyclic
• alternating
• Lie type
• sporadic
• Discrete groups
• Lattices
• Integers ($\mathbb {Z} $)
• Free group
Modular groups
• PSL(2, $\mathbb {Z} $)
• SL(2, $\mathbb {Z} $)
• Arithmetic group
• Lattice
• Hyperbolic group
Topological and Lie groups
• Solenoid
• Circle
• General linear GL(n)
• Special linear SL(n)
• Orthogonal O(n)
• Euclidean E(n)
• Special orthogonal SO(n)
• Unitary U(n)
• Special unitary SU(n)
• Symplectic Sp(n)
• G2
• F4
• E6
• E7
• E8
• Lorentz
• Poincaré
• Conformal
• Diffeomorphism
• Loop
Infinite dimensional Lie group
• O(∞)
• SU(∞)
• Sp(∞)
Algebraic groups
• Linear algebraic group
• Reductive group
• Abelian variety
• Elliptic curve
The group multiplication law on the double cover is given by lifting the multiplication on $\operatorname {SO} (n)$.
As a Lie group, Spin(n) therefore shares its dimension, n(n − 1)/2, and its Lie algebra with the special orthogonal group.
For n > 2, Spin(n) is simply connected and so coincides with the universal cover of SO(n).
The non-trivial element of the kernel is denoted −1, which should not be confused with the orthogonal transform of reflection through the origin, generally denoted −I.
Spin(n) can be constructed as a subgroup of the invertible elements in the Clifford algebra Cl(n). A distinct article discusses the spin representations.
Motivation and physical interpretation
The spin group is used in physics to describe the symmetries of (electrically neutral, uncharged) fermions. Its complexification, Spinc, is used to describe electrically charged fermions, most notably the electron. Strictly speaking, the spin group describes a fermion in a zero-dimensional space; but of course, space is not zero-dimensional, and so the spin group is used to define spin structures on (pseudo-)Riemannian manifolds: the spin group is the structure group of a spinor bundle. The affine connection on a spinor bundle is the spin connection; the spin connection is useful as it can simplify and bring elegance to many intricate calculations in general relativity. The spin connection in turn enables the Dirac equation to be written in curved spacetime (effectively in the tetrad coordinates), which in turn provides a footing for quantum gravity, as well as a formalization of Hawking radiation (where one of a pair of entangled, virtual fermions falls past the event horizon, and the other does not). In short, the spin group is a vital cornerstone, centrally important for understanding advanced concepts in modern theoretical physics. In mathematics, the spin group is interesting in its own right: not only for these reasons, but for many more.
Construction
Construction of the Spin group often starts with the construction of a Clifford algebra over a real vector space V with a definite quadratic form q.[3] The Clifford algebra is the quotient of the tensor algebra TV of V by a two-sided ideal. The tensor algebra (over the reals) may be written as
$\mathrm {T} V=\mathbb {R} \oplus V\oplus (V\otimes V)\oplus \cdots $
The Clifford algebra Cl(V) is then the quotient algebra
$\operatorname {Cl} (V)=\mathrm {T} V/\left(v\otimes v-q(v)\right),$
where $q(v)$ is the quadratic form applied to a vector $v\in V$. The resulting space is finite dimensional, naturally graded (as a vector space), and can be written as
$\operatorname {Cl} (V)=\operatorname {Cl} ^{0}\oplus \operatorname {Cl} ^{1}\oplus \operatorname {Cl} ^{2}\oplus \cdots \oplus \operatorname {Cl} ^{n}$
where $n$ is the dimension of $V$, $\operatorname {Cl} ^{0}=\mathbf {R} $ and $\operatorname {Cl} ^{1}=V$. The spin algebra ${\mathfrak {spin}}$ is defined as
$\operatorname {Cl} ^{2}={\mathfrak {spin}}(V)={\mathfrak {spin}}(n),$
where the last is a short-hand for V being a real vector space of real dimension n. It is a Lie algebra; it has a natural action on V, and in this way can be shown to be isomorphic to the Lie algebra ${\mathfrak {so}}(n)$ of the special orthogonal group.
The pin group $\operatorname {Pin} (V)$ is a subgroup of $\operatorname {Cl} (V)$'s Clifford group of all elements of the form
$v_{1}v_{2}\cdots v_{k},$
where each $v_{i}\in V$ is of unit length: $q(v_{i})=1.$
The spin group is then defined as
$\operatorname {Spin} (V)=\operatorname {Pin} (V)\cap \operatorname {Cl} ^{\text{even}},$
where $\operatorname {Cl} ^{\text{even}}=\operatorname {Cl} ^{0}\oplus \operatorname {Cl} ^{2}\oplus \operatorname {Cl} ^{4}\oplus \cdots $ is the subspace generated by elements that are the product of an even number of vectors. That is, Spin(V) consists of all elements of Pin(V), given above, with the restriction to k being an even number. The restriction to the even subspace is key to the formation of two-component (Weyl) spinors, constructed below.
If the set $\{e_{i}\}$ are an orthonormal basis of the (real) vector space V, then the quotient above endows the space with a natural anti-commuting structure:
$e_{i}e_{j}=-e_{j}e_{i}$ for $i\neq j,$
which follows by considering $v\otimes v$ for $v=e_{i}+e_{j}$. This anti-commutation turns out to be of importance in physics, as it captures the spirit of the Pauli exclusion principle for fermions. A precise formulation is out of scope, here, but it involves the creation of a spinor bundle on Minkowski spacetime; the resulting spinor fields can be seen to be anti-commuting as a by-product of the Clifford algebra construction. This anti-commutation property is also key to the formulation of supersymmetry. The Clifford algebra and the spin group have many interesting and curious properties, some of which are listed below.
Geometric construction
The spin groups can be constructed less explicitly but without appealing to Clifford algebras. As a manifold, $\operatorname {Spin} (n)$ is the double cover of $\operatorname {SO} (n)$. Its multiplication law can be defined by lifting as follows. Call the covering map $p:\operatorname {Spin} (n)\rightarrow \operatorname {SO} (n)$. Then $p^{-1}(\{e\})$ is a set with two elements, and one can be chosen without loss of generality to be the identity. Call this ${\tilde {e}}$. Then to define multiplication in $\operatorname {Spin} (n)$, for $a,b\in \operatorname {Spin} (n)$ choose paths $\gamma _{a},\gamma _{b}$ satisfying $\gamma _{a}(0)=\gamma _{b}(0)={\tilde {e}}$, and $\gamma _{a}(1)=a,\gamma _{b}(1)=b$. These define a path $\gamma $ in $\operatorname {SO} (n)$ defined $\gamma (t)=p(\gamma _{a}(t))\cdot p(\gamma _{b}(t))$ satisfying $\gamma (0)=e$. Since $\operatorname {Spin} (n)$ is a double cover, there is a unique lift ${\tilde {\gamma }}$ of $\gamma $ with ${\tilde {\gamma }}(0)={\tilde {e}}$. Then define the product as $a\cdot b={\tilde {\gamma }}(1)$.
It can then be shown that this definition is independent of the paths $\gamma _{a},\gamma _{b}$, that the multiplication is continuous, and the group axioms are satisfied with inversion being continuous, making $\operatorname {Spin} (n)$ a Lie group.
Double covering
For a quadratic space V, a double covering of SO(V) by Spin(V) can be given explicitly, as follows. Let $\{e_{i}\}$ be an orthonormal basis for V. Define an antiautomorphism $t:\operatorname {Cl} (V)\to \operatorname {Cl} (V)$ by
$\left(e_{i}e_{j}\cdots e_{k}\right)^{t}=e_{k}\cdots e_{j}e_{i}.$
This can be extended to all elements of $a,b\in \operatorname {Cl} (V)$ by linearity. It is an antihomomorphism since
$(ab)^{t}=b^{t}a^{t}.$
Observe that Pin(V) can then be defined as all elements $a\in \operatorname {Cl} (V)$ for which
$aa^{t}=1.$
Now define the automorphism $\alpha \colon \operatorname {Cl} (V)\to \operatorname {Cl} (V)$ which on degree 1 elements is given by
$\alpha (v)=-v,\quad v\in V,$
and let $a^{*}$ denote $\alpha (a)^{t}$, which is an antiautomorphism of Cl(V). With this notation, an explicit double covering is the homomorphism $\operatorname {Pin} (V)\to \operatorname {O} (V)$ given by
$\rho (a)v=ava^{*},$
where $v\in V$. When a has degree 1 (i.e. $a\in V$), $\rho (a)$ corresponds a reflection across the hyperplane orthogonal to a; this follows from the anti-commuting property of the Clifford algebra.
This gives a double covering of both O(V) by Pin(V) and of SO(V) by Spin(V) because $a$ gives the same transformation as $-a$.
Spinor space
It is worth reviewing how spinor space and Weyl spinors are constructed, given this formalism. Given a real vector space V of dimension n = 2m an even number, its complexification is $V\otimes \mathbf {C} $. It can be written as the direct sum of a subspace $W$ of spinors and a subspace ${\overline {W}}$ of anti-spinors:
$V\otimes \mathbf {C} =W\oplus {\overline {W}}$
The space $W$ is spanned by the spinors $\eta _{k}=\left(e_{2k-1}-ie_{2k}\right)/{\sqrt {2}}$ for $1\leq k\leq m$ and the complex conjugate spinors span ${\overline {W}}$. It is straightforward to see that the spinors anti-commute, and that the product of a spinor and anti-spinor is a scalar.
The spinor space is defined as the exterior algebra $\textstyle {\bigwedge }W$. The (complexified) Clifford algebra acts naturally on this space; the (complexified) spin group corresponds to the length-preserving endomorphisms. There is a natural grading on the exterior algebra: the product of an odd number of copies of $W$ correspond to the physics notion of fermions; the even subspace corresponds to the bosons. The representations of the action of the spin group on the spinor space can be built in a relatively straightforward fashion.[3]
Complex case
Main article: Spin structure § SpinC structures
The SpinC group is defined by the exact sequence
$1\to \mathrm {Z} _{2}\to \operatorname {Spin} ^{\mathbf {C} }(n)\to \operatorname {SO} (n)\times \operatorname {U} (1)\to 1.$
It is a multiplicative subgroup of the complexification $\operatorname {Cl} (V)\otimes \mathbf {C} $ of the Clifford algebra, and specifically, it is the subgroup generated by Spin(V) and the unit circle in C. Alternately, it is the quotient
$\operatorname {Spin} ^{\mathbf {C} }(V)=\left(\operatorname {Spin} (V)\times S^{1}\right)/\sim $
where the equivalence $\sim $ identifies (a, u) with (−a, −u).
This has important applications in 4-manifold theory and Seiberg–Witten theory. In physics, the Spin group is appropriate for describing uncharged fermions, while the SpinC group is used to describe electrically charged fermions. In this case, the U(1) symmetry is specifically the gauge group of electromagnetism.
Exceptional isomorphisms
In low dimensions, there are isomorphisms among the classical Lie groups called exceptional isomorphisms. For instance, there are isomorphisms between low-dimensional spin groups and certain classical Lie groups, owing to low-dimensional isomorphisms between the root systems (and corresponding isomorphisms of Dynkin diagrams) of the different families of simple Lie algebras. Writing R for the reals, C for the complex numbers, H for the quaternions and the general understanding that Cl(n) is a short-hand for Cl(Rn) and that Spin(n) is a short-hand for Spin(Rn) and so on, one then has that[3]
Cleven(1) = R the real numbers
Pin(1) = {+i, −i, +1, −1}
Spin(1) = O(1) = {+1, −1} the orthogonal group of dimension zero.
--
Cleven(2) = C the complex numbers
Spin(2) = U(1) = SO(2), which acts on z in R2 by double phase rotation z ↦ u2z. Corresponds to the abelian $D_{1}$. dim = 1
--
Cleven(3) = H the quaternions
Spin(3) = Sp(1) = SU(2), corresponding to $B_{1}\cong C_{1}\cong A_{1}$. dim = 3
--
Cleven(4) = H ⊕ H
Spin(4) = SU(2) × SU(2), corresponding to $D_{2}\cong A_{1}\times A_{1}$. dim = 6
--
Cleven(5)= M(2, H) the two-by-two matrices with quaternionic coefficients
Spin(5) = Sp(2), corresponding to $B_{2}\cong C_{2}$. dim = 10
--
Cleven(6)= M(4, C) the four-by-four matrices with complex coefficients
Spin(6) = SU(4), corresponding to $D_{3}\cong A_{3}$. dim = 15
There are certain vestiges of these isomorphisms left over for n = 7, 8 (see Spin(8) for more details). For higher n, these isomorphisms disappear entirely.
Indefinite signature
In indefinite signature, the spin group Spin(p, q) is constructed through Clifford algebras in a similar way to standard spin groups. It is a double cover of SO0(p, q), the connected component of the identity of the indefinite orthogonal group SO(p, q). For p + q > 2, Spin(p, q) is connected; for (p, q) = (1, 1) there are two connected components.[4]: 193 As in definite signature, there are some accidental isomorphisms in low dimensions:
Spin(1, 1) = GL(1, R)
Spin(2, 1) = SL(2, R)
Spin(3, 1) = SL(2, C)
Spin(2, 2) = SL(2, R) × SL(2, R)
Spin(4, 1) = Sp(1, 1)
Spin(3, 2) = Sp(4, R)
Spin(5, 1) = SL(2, H)
Spin(4, 2) = SU(2, 2)
Spin(3, 3) = SL(4, R)
Spin(6, 2) = SU(2, 2, H)
Note that Spin(p, q) = Spin(q, p).
Topological considerations
Connected and simply connected Lie groups are classified by their Lie algebra. So if G is a connected Lie group with a simple Lie algebra, with G′ the universal cover of G, there is an inclusion
$\pi _{1}(G)\subset \operatorname {Z} (G'),$
with Z(G′) the center of G′. This inclusion and the Lie algebra ${\mathfrak {g}}$ of G determine G entirely (note that it is not the case that ${\mathfrak {g}}$ and π1(G) determine G entirely; for instance SL(2, R) and PSL(2, R) have the same Lie algebra and same fundamental group Z, but are not isomorphic).
The definite signature Spin(n) are all simply connected for n > 2, so they are the universal coverings of SO(n).
In indefinite signature, Spin(p, q) is not necessarily connected, and in general the identity component, Spin0(p, q), is not simply connected, thus it is not a universal cover. The fundamental group is most easily understood by considering the maximal compact subgroup of SO(p, q), which is SO(p) × SO(q), and noting that rather than being the product of the 2-fold covers (hence a 4-fold cover), Spin(p, q) is the "diagonal" 2-fold cover – it is a 2-fold quotient of the 4-fold cover. Explicitly, the maximal compact connected subgroup of Spin(p, q) is
Spin(p) × Spin(q)/{(1, 1), (−1, −1)}.
This allows us to calculate the fundamental groups of SO(p, q), taking p ≥ q:
$\pi _{1}({\mbox{SO}}(p,q))={\begin{cases}0&(p,q)=(1,1){\mbox{ or }}(1,0)\\\mathbb {Z} _{2}&p>2,q=0,1\\\mathbb {Z} &(p,q)=(2,0){\mbox{ or }}(2,1)\\\mathbb {Z} \times \mathbb {Z} &(p,q)=(2,2)\\\mathbb {Z} &p>2,q=2\\\mathbb {Z} _{2}&p,q>2\\\end{cases}}$
Thus once p, q > 2 the fundamental group is Z2, as it is a 2-fold quotient of a product of two universal covers.
The maps on fundamental groups are given as follows. For p, q > 2, this implies that the map π1(Spin(p, q)) → π1(SO(p, q)) is given by 1 ∈ Z2 going to (1, 1) ∈ Z2 × Z2. For p = 2, q > 2, this map is given by 1 ∈ Z → (1,1) ∈ Z × Z2. And finally, for p = q = 2, (1, 0) ∈ Z × Z is sent to (1,1) ∈ Z × Z and (0, 1) is sent to (1, −1).
Fundamental groups of SO(n)
The fundamental groups $\pi _{1}(\operatorname {SO} (n))$ can be more directly derived using results in homotopy theory. In particular we can find $\pi _{1}(\operatorname {SO} (n))$ for $n>3$ as the three smallest have familiar underlying manifolds: $SO(1)$ is the point manifold, $SO(2)\cong S^{1}$, and $SO(3)\cong \mathbb {RP} ^{3}$ (shown using the axis-angle representation).
The proof uses known results in algebraic topology.[5]
Proof
First consider the action of $\operatorname {SO} (n)$ on $\mathbb {R} ^{n}$, in particular on the vector $v=(1,0,\cdots ,0)$. The orbit of this vector is ${\text{Orbit}}_{{\text{SO}}(n)}(v)=S^{n-1}$, while the stabilizer is ${\text{Stab}}_{{\text{SO}}(n)}(v)={\text{SO}}(n-1)$. Thus from the orbit-stabilizer theorem one obtains an isomorphism
${\text{SO}}(n)/{\text{SO}}(n-1)\cong S^{n-1}.$
Geometrically, this provides a fibration
${\text{SO}}(n-1)\rightarrow {\text{SO}}(n)\rightarrow S^{n-1}.$
Then Theorem 4.41 in Hatcher tells us that there is a long exact sequence of homotopy groups
$\cdots \rightarrow \pi _{k}({\text{SO}}(n-1))\rightarrow \pi _{k}({\text{SO}}(n))\rightarrow \pi _{k}(S^{n-1})\rightarrow \pi _{k-1}({\text{SO}}(n-1))\rightarrow \cdots $
and we concentrate on a section at the end of the sequence:
$\pi _{2}(S^{n-1})\rightarrow \pi _{1}({\text{SO}}(n-1))\rightarrow \pi _{1}({\text{SO}}(n))\rightarrow \pi _{1}(S^{n-1}).$
Corollary 4.9 in Hatcher states $\pi _{k}(S^{n})=0$ for $k<n$. So for $n>3$, the exact sequence becomes
$0\rightarrow \pi _{1}({\text{SO}}(n-1))\rightarrow \pi _{1}({\text{SO}}(n))\rightarrow 0,$
hence $\pi _{1}({\text{SO}}(n))$ and $\pi _{1}({\text{SO}}(n-1))$ are isomorphic as long as $n>3$, so for $n>3$, we have $\pi _{1}({\text{SO}}(n))\cong \pi _{1}({\text{SO}}(3))$.
And since ${\text{SO}}(3)\cong \mathbb {RP} ^{3}\cong S^{3}/\{\pm 1\}$, we get $\pi _{1}({\text{SO}}(3))\cong \mathbb {Z} _{2}$.
The same argument can be used to show $\pi ({\text{SO}}(1,n)^{\uparrow })\cong \pi ({\text{SO}}(n))$, by considering a fibration
${\text{SO}}(n)\rightarrow {\text{SO}}(1,n)^{\uparrow }\rightarrow H^{n},$
where $H^{n}$ is the upper sheet of a two-sheeted hyperboloid, which is contractible, and ${\text{SO}}(1,n)^{\uparrow }$ is the identity component of the proper Lorentz group (the proper orthochronous Lorentz group).
Center
The center of the spin groups, for n ≥ 3, (complex and real) are given as follows:[4]: 208
${\begin{aligned}\operatorname {Z} (\operatorname {Spin} (n,\mathbf {C} ))&={\begin{cases}\mathrm {Z} _{2}&n=2k+1\\\mathrm {Z} _{4}&n=4k+2\\\mathrm {Z} _{2}\oplus \mathrm {Z} _{2}&n=4k\\\end{cases}}\\\operatorname {Z} (\operatorname {Spin} (p,q))&={\begin{cases}\mathrm {Z} _{2}&p{\text{ or }}q{\text{ odd}}\\\mathrm {Z} _{4}&n=4k+2,{\text{ and }}p,q{\text{ even}}\\\mathrm {Z} _{2}\oplus \mathrm {Z} _{2}&n=4k,{\text{ and }}p,q{\text{ even}}\\\end{cases}}\end{aligned}}$
Quotient groups
Quotient groups can be obtained from a spin group by quotienting out by a subgroup of the center, with the spin group then being a covering group of the resulting quotient, and both groups having the same Lie algebra.
Quotienting out by the entire center yields the minimal such group, the projective special orthogonal group, which is centerless, while quotienting out by {±1} yields the special orthogonal group – if the center equals {±1} (namely in odd dimension), these two quotient groups agree. If the spin group is simply connected (as Spin(n) is for n > 2), then Spin is the maximal group in the sequence, and one has a sequence of three groups,
Spin(n) → SO(n) → PSO(n),
splitting by parity yields:
Spin(2n) → SO(2n) → PSO(2n),
Spin(2n+1) → SO(2n+1) = PSO(2n+1),
which are the three compact real forms (or two, if SO = PSO) of the compact Lie algebra ${\mathfrak {so}}(n,\mathbf {R} ).$
The homotopy groups of the cover and the quotient are related by the long exact sequence of a fibration, with discrete fiber (the fiber being the kernel) – thus all homotopy groups for k > 1 are equal, but π0 and π1 may differ.
For n > 2, Spin(n) is simply connected (π0 = π1 = Z1 is trivial), so SO(n) is connected and has fundamental group Z2 while PSO(n) is connected and has fundamental group equal to the center of Spin(n).
In indefinite signature the covers and homotopy groups are more complicated – Spin(p, q) is not simply connected, and quotienting also affects connected components. The analysis is simpler if one considers the maximal (connected) compact SO(p) × SO(q) ⊂ SO(p, q) and the component group of Spin(p, q).
Whitehead tower
The spin group appears in a Whitehead tower anchored by the orthogonal group:
$\ldots \rightarrow {\text{Fivebrane}}(n)\rightarrow {\text{String}}(n)\rightarrow {\text{Spin}}(n)\rightarrow {\text{SO}}(n)\rightarrow {\text{O}}(n)$
The tower is obtained by successively removing (killing) homotopy groups of increasing order. This is done by constructing short exact sequences starting with an Eilenberg–MacLane space for the homotopy group to be removed. Killing the π3 homotopy group in Spin(n), one obtains the infinite-dimensional string group String(n).
Discrete subgroups
Discrete subgroups of the spin group can be understood by relating them to discrete subgroups of the special orthogonal group (rotational point groups).
Given the double cover Spin(n) → SO(n), by the lattice theorem, there is a Galois connection between subgroups of Spin(n) and subgroups of SO(n) (rotational point groups): the image of a subgroup of Spin(n) is a rotational point group, and the preimage of a point group is a subgroup of Spin(n), and the closure operator on subgroups of Spin(n) is multiplication by {±1}. These may be called "binary point groups"; most familiar is the 3-dimensional case, known as binary polyhedral groups.
Concretely, every binary point group is either the preimage of a point group (hence denoted 2G, for the point group G), or is an index 2 subgroup of the preimage of a point group which maps (isomorphically) onto the point group; in the latter case the full binary group is abstractly $\mathrm {C} _{2}\times G$ (since {±1} is central). As an example of these latter, given a cyclic group of odd order $\mathrm {Z} _{2k+1}$ in SO(n), its preimage is a cyclic group of twice the order, $\mathrm {C} _{4k+2}\cong \mathrm {Z} _{2k+1}\times \mathrm {Z} _{2},$ and the subgroup Z2k+1 < Spin(n) maps isomorphically to Z2k+1 < SO(n).
Of particular note are two series:
• higher binary tetrahedral groups, corresponding to the 2-fold cover of symmetries of the n-simplex; this group can also be considered as the double cover of the symmetric group, 2⋅An → An, with the alternating group being the (rotational) symmetry group of the n-simplex.
• higher binary octahedral groups, corresponding to the 2-fold covers of the hyperoctahedral group (symmetries of the hypercube, or equivalently of its dual, the cross-polytope).
For point groups that reverse orientation, the situation is more complicated, as there are two pin groups, so there are two possible binary groups corresponding to a given point group.
See also
• Clifford algebra
• Clifford analysis
• Spinor
• Spinor bundle
• Spin structure
• Table of Lie groups
• Anyon
• Orientation entanglement
Related groups
• Pin group Pin(n) – two-fold cover of orthogonal group, O(n)
• Metaplectic group Mp(2n) – two-fold cover of symplectic group, Sp(2n)
• String group String(n) – the next group in the Whitehead tower
References
1. Lawson, H. Blaine; Michelsohn, Marie-Louise (1989). Spin Geometry. Princeton University Press. ISBN 978-0-691-08542-5. page 14
2. Friedrich, Thomas (2000), Dirac Operators in Riemannian Geometry, American Mathematical Society, ISBN 978-0-8218-2055-1 page 15
3. Jürgen Jost, Riemannian Geometry and Geometric Analysis, (2002) Springer Verlag ISBN 3-540-42627-2 (See Chapter 1.)
4. Varadarajan, V. S. (2004). Supersymmetry for mathematicians : an introduction. Providence, R.I.: American Mathematical Society. ISBN 0821835742. OCLC 55487352.
5. Hatcher, Allen (2002). Algebraic topology (PDF). Cambridge: Cambridge University Press. ISBN 9780521795401. Retrieved 24 February 2023.
External links
• The essential dimension of spin groups is OEIS:A280191.
• Grothendieck's "torsion index" is OEIS:A096336.
Further reading
• Karoubi, Max (2008). K-Theory. Springer. pp. 210–214. ISBN 978-3-540-79889-7.
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Wikipedia
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Spin(7)-manifold
In mathematics, a Spin(7)-manifold is an eight-dimensional Riemannian manifold whose holonomy group is contained in Spin(7). Spin(7)-manifolds are Ricci-flat and admit a parallel spinor. They also admit a parallel 4-form, known as the Cayley form, which is a calibrating form for a special class of submanifolds called Cayley cycles.
History
The fact that Spin(7) might possibly arise as the holonomy group of certain Riemannian 8-manifolds was first suggested by the 1955 classification theorem of Marcel Berger, and this possibility remained consistent with the simplified proof of Berger's theorem given by Jim Simons in 1962. Although not a single example of such a manifold had yet been discovered, Edmond Bonan then showed in 1966 that, if such a manifold did in fact exist, it would carry a parallel 4-form, and that it would necessarily be Ricci-flat. The first local examples of 8-manifolds with holonomy Spin(7) were finally constructed around 1984 by Robert Bryant, and his full proof of their existence appeared in Annals of Mathematics in 1987.[1] Next, complete (but still noncompact) 8-manifolds with holonomy Spin(7) were explicitly constructed by Bryant and Salamon in 1989. The first examples of compact Spin(7)-manifolds were then constructed by Dominic Joyce in 1996.
See also
• G2 manifold
• Calabi–Yau manifold
References
1. Bryant, Robert L. (1987) "Metrics with exceptional holonomy," Annals of Mathematics (2)126, 525–576.
• E. Bonan (1966), "Sur les variétés riemanniennes à groupe d'holonomie G2 ou Spin(7)", C. R. Acad. Sci. Paris, 262: 127–129.
• Bryant, R.L.; Salamon, S.M. (1989), "On the construction of some complete metrics with exceptional holonomy", Duke Mathematical Journal, 58 (3): 829–850, doi:10.1215/s0012-7094-89-05839-0.
• Dominic Joyce (2000). Compact Manifolds with Special Holonomy. Oxford University Press. ISBN 0-19-850601-5.
• Karigiannis, Spiro (2009), "Flows of G2 and Spin(7) structures", Mathematical Institute, University of Oxford, 9 (4): 389–463.
String theory
Background
• Strings
• Cosmic strings
• History of string theory
• First superstring revolution
• Second superstring revolution
• String theory landscape
Theory
• Nambu–Goto action
• Polyakov action
• Bosonic string theory
• Superstring theory
• Type I string
• Type II string
• Type IIA string
• Type IIB string
• Heterotic string
• N=2 superstring
• F-theory
• String field theory
• Matrix string theory
• Non-critical string theory
• Non-linear sigma model
• Tachyon condensation
• RNS formalism
• GS formalism
String duality
• T-duality
• S-duality
• U-duality
• Montonen–Olive duality
Particles and fields
• Graviton
• Dilaton
• Tachyon
• Ramond–Ramond field
• Kalb–Ramond field
• Magnetic monopole
• Dual graviton
• Dual photon
Branes
• D-brane
• NS5-brane
• M2-brane
• M5-brane
• S-brane
• Black brane
• Black holes
• Black string
• Brane cosmology
• Quiver diagram
• Hanany–Witten transition
Conformal field theory
• Virasoro algebra
• Mirror symmetry
• Conformal anomaly
• Conformal algebra
• Superconformal algebra
• Vertex operator algebra
• Loop algebra
• Kac–Moody algebra
• Wess–Zumino–Witten model
Gauge theory
• Anomalies
• Instantons
• Chern–Simons form
• Bogomol'nyi–Prasad–Sommerfield bound
• Exceptional Lie groups (G2, F4, E6, E7, E8)
• ADE classification
• Dirac string
• p-form electrodynamics
Geometry
• Worldsheet
• Kaluza–Klein theory
• Compactification
• Why 10 dimensions?
• Kähler manifold
• Ricci-flat manifold
• Calabi–Yau manifold
• Hyperkähler manifold
• K3 surface
• G2 manifold
• Spin(7)-manifold
• Generalized complex manifold
• Orbifold
• Conifold
• Orientifold
• Moduli space
• Hořava–Witten theory
• K-theory (physics)
• Twisted K-theory
Supersymmetry
• Supergravity
• Superspace
• Lie superalgebra
• Lie supergroup
Holography
• Holographic principle
• AdS/CFT correspondence
M-theory
• Matrix theory
• Introduction to M-theory
String theorists
• Aganagić
• Arkani-Hamed
• Atiyah
• Banks
• Berenstein
• Bousso
• Cleaver
• Curtright
• Dijkgraaf
• Distler
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• Gates
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• Greene
• Gross
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• Hořava
• Gibbons
• Kachru
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• Zamolodchikov
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• Zumino
• Zwiebach
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Wikipedia
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SO(8)
In mathematics, SO(8) is the special orthogonal group acting on eight-dimensional Euclidean space. It could be either a real or complex simple Lie group of rank 4 and dimension 28.
Algebraic structure → Group theory
Group theory
Basic notions
• Subgroup
• Normal subgroup
• Quotient group
• (Semi-)direct product
Group homomorphisms
• kernel
• image
• direct sum
• wreath product
• simple
• finite
• infinite
• continuous
• multiplicative
• additive
• cyclic
• abelian
• dihedral
• nilpotent
• solvable
• action
• Glossary of group theory
• List of group theory topics
Finite groups
• Cyclic group Zn
• Symmetric group Sn
• Alternating group An
• Dihedral group Dn
• Quaternion group Q
• Cauchy's theorem
• Lagrange's theorem
• Sylow theorems
• Hall's theorem
• p-group
• Elementary abelian group
• Frobenius group
• Schur multiplier
Classification of finite simple groups
• cyclic
• alternating
• Lie type
• sporadic
• Discrete groups
• Lattices
• Integers ($\mathbb {Z} $)
• Free group
Modular groups
• PSL(2, $\mathbb {Z} $)
• SL(2, $\mathbb {Z} $)
• Arithmetic group
• Lattice
• Hyperbolic group
Topological and Lie groups
• Solenoid
• Circle
• General linear GL(n)
• Special linear SL(n)
• Orthogonal O(n)
• Euclidean E(n)
• Special orthogonal SO(n)
• Unitary U(n)
• Special unitary SU(n)
• Symplectic Sp(n)
• G2
• F4
• E6
• E7
• E8
• Lorentz
• Poincaré
• Conformal
• Diffeomorphism
• Loop
Infinite dimensional Lie group
• O(∞)
• SU(∞)
• Sp(∞)
Algebraic groups
• Linear algebraic group
• Reductive group
• Abelian variety
• Elliptic curve
Spin(8)
Like all special orthogonal groups of $n>2$, SO(8) is not simply connected, having a fundamental group isomorphic to Z2. The universal cover of SO(8) is the spin group Spin(8).
Center
The center of SO(8) is Z2, the diagonal matrices {±I} (as for all SO(2n) with 2n ≥ 4), while the center of Spin(8) is Z2×Z2 (as for all Spin(4n), 4n ≥ 4).
Triality
Main article: Triality
SO(8) is unique among the simple Lie groups in that its Dynkin diagram, (D4 under the Dynkin classification), possesses a three-fold symmetry. This gives rise to peculiar feature of Spin(8) known as triality. Related to this is the fact that the two spinor representations, as well as the fundamental vector representation, of Spin(8) are all eight-dimensional (for all other spin groups the spinor representation is either smaller or larger than the vector representation). The triality automorphism of Spin(8) lives in the outer automorphism group of Spin(8) which is isomorphic to the symmetric group S3 that permutes these three representations. The automorphism group acts on the center Z2 x Z2 (which also has automorphism group isomorphic to S3 which may also be considered as the general linear group over the finite field with two elements, S3 ≅GL(2,2)). When one quotients Spin(8) by one central Z2, breaking this symmetry and obtaining SO(8), the remaining outer automorphism group is only Z2. The triality symmetry acts again on the further quotient SO(8)/Z2.
Sometimes Spin(8) appears naturally in an "enlarged" form, as the automorphism group of Spin(8), which breaks up as a semidirect product: Aut(Spin(8)) ≅ PSO (8) ⋊ S3.
Unit octonions
Elements of SO(8) can be described with unit octonions, analogously to how elements of SO(2) can be described with unit complex numbers and elements of SO(4) can be described with unit quaternions. However the relationship is more complicated, partly due to the non-associativity of the octonions. A general element in SO(8) can be described as the product of 7 left-multiplications, 7 right-multiplications and also 7 bimultiplications by unit octonions (a bimultiplication being the composition of a left-multiplication and a right-multiplication by the same octonion and is unambiguously defined due to octonions obeying the Moufang identities).
It can be shown that an element of SO(8) can be constructed with bimultiplications, by first showing that pairs of reflections through the origin in 8-dimensional space correspond to pairs of bimultiplications by unit octonions. The triality automorphism of Spin(8) described below provides similar constructions with left multiplications and right multiplications.[1]
Octonions and triality
If $x,y,z\in \mathbb {O} $ and $(xy)z=1$, it can be shown that this is equivalent to $x(yz)=1$, meaning that $xyz=1$ without ambiguity. A triple of maps $(\alpha ,\beta ,\gamma )$ that preserve this identity, so that $x^{\alpha }y^{\beta }z^{\gamma }=1$ is called an isotopy. If the three maps of an isotopy are in $\operatorname {SO(8)} $, the isotopy is called an orthogonal isotopy. If $\gamma \in \operatorname {SO(8)} $, then following the above $\gamma $ can be described as the product of bimultiplications of unit octonions, say $\gamma =B_{u_{1}}...B_{u_{n}}$. Let $\alpha ,\beta \in \operatorname {SO(8)} $ be the corresponding products of left and right multiplications by the conjugates (i.e., the multiplicative inverses) of the same unit octonions, so $\alpha =L_{\overline {u_{1}}}...L_{\overline {u_{n}}}$, $\beta =R_{\overline {u_{1}}}...R_{\overline {u_{n}}}$. A simple calculation shows that $(\alpha ,\beta ,\gamma )$ is an isotopy. As a result of the non-associativity of the octonions, the only other orthogonal isotopy for $\gamma $ is $(-\alpha ,-\beta ,\gamma )$. As the set of orthogonal isotopies produce a 2-to-1 cover of $\operatorname {SO} (8)$, they must in fact be $\operatorname {Spin} (8)$.
Multiplicative inverses of octonions are two-sided, which means that $xyz=1$ is equivalent to $yzx=1$. This means that a given isotopy $(\alpha ,\beta ,\gamma )$ can be permuted cyclically to give two further isotopies $(\beta ,\gamma ,\alpha )$ and $(\gamma ,\alpha ,\beta )$. This produces an order 3 outer automorphism of $\operatorname {Spin} (8)$. This "triality" automorphism is exceptional among spin groups. There is no triality automorphism of $\operatorname {SO} (8)$, as for a given $\gamma $ the corresponding maps $\alpha ,\beta $ are only uniquely determined up to sign.[1]
Root system
$(\pm 1,\pm 1,0,0)$
$(\pm 1,0,\pm 1,0)$
$(\pm 1,0,0,\pm 1)$
$(0,\pm 1,\pm 1,0)$
$(0,\pm 1,0,\pm 1)$
$(0,0,\pm 1,\pm 1)$
Weyl group
Its Weyl/Coxeter group has 4! × 8 = 192 elements.
Cartan matrix
${\begin{pmatrix}2&-1&-1&-1\\-1&2&0&0\\-1&0&2&0\\-1&0&0&2\end{pmatrix}}$
See also
• Octonions
• Clifford algebra
• G2
References
1. John H. Conway; Derek A. Smith (23 January 2003). On Quaternions and Octonions. Taylor & Francis. ISBN 978-1-56881-134-5.
• Adams, J.F. (1996), Lectures on exceptional Lie groups, Chicago Lectures in Mathematics, University of Chicago Press, ISBN 0-226-00526-7
• Chevalley, Claude (1997), The algebraic theory of spinors and Clifford algebras, Collected works, vol. 2, Springer-Verlag, ISBN 3-540-57063-2 (originally published in 1954 by Columbia University Press)
• Porteous, Ian R. (1995), Clifford algebras and the classical groups, Cambridge Studies in Advanced Mathematics, vol. 50, Cambridge University Press, ISBN 0-521-55177-3
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Spin chain
A spin chain is a type of model in statistical physics. Spin chains were originally formulated to model magnetic systems, which typically consist of particles with magnetic spin located at fixed sites on a lattice. A prototypical example is the quantum Heisenberg model. Interactions between the sites are modelled by operators which act on two different sites, often neighboring sites.
They can be seen as a quantum version of statistical lattice models, such as the Ising model, in the sense that the parameter describing the spin at each site is promoted from a variable taking values in a discrete set (typically $\{+1,-1\}$, representing 'spin up' and 'spin down') to a variable taking values in a vector space (typically the spin-1/2 or two-dimensional representation of ${\mathfrak {su}}(2)$).
History
The prototypical example of a spin chain is the Heisenberg model, described by Werner Heisenberg in 1928.[1] This models a one-dimensional lattice of fixed particles with spin 1/2. A simple version (the antiferromagnetic XXX model) was solved, that is, the spectrum of the Hamiltonian of the Heisenberg model was determined, by Hans Bethe using the Bethe ansatz.[2] Now the term Bethe ansatz is used generally to refer to many ansatzes used to solve exactly solvable problems in spin chain theory such as for the other variations of the Heisenberg model (XXZ, XYZ), and even in statistical lattice theory, such as for the six-vertex model.
Another spin chain with physical applications is the Hubbard model, introduced by John Hubbard in 1963.[3] This model was shown to be exactly solvable by Elliott Lieb and Fa-Yueh Wu in 1968.[4]
Another example of (a class of) spin chains is the Gaudin model, described and solved by Michel Gaudin in 1976[5]
Mathematical description
The lattice is described by a graph $G$ with vertex set $V$ and edge set $E$.
The model has an associated Lie algebra ${\mathfrak {sl}}_{2}:={\mathfrak {sl}}(2,\mathbb {C} )$. More generally, this Lie algebra can be taken to be any complex, finite-dimensional semi-simple Lie algebra ${\mathfrak {g}}$. More generally still it can be taken to be an arbitrary Lie algebra.
Each vertex $v\in V$ has an associated representation of the Lie algebra ${\mathfrak {g}}$, labelled $V_{v}$. This is a quantum generalization of statistical lattice models, where each vertex has an associated 'spin variable'.
The Hilbert space ${\mathcal {H}}$ for the whole system, which could be called the configuration space, is the tensor product of the representation spaces at each vertex:
${\mathcal {H}}=\bigotimes _{v\in V}V_{v}.$
A Hamiltonian is then an operator on the Hilbert space. In the theory of spin chains, there are possibly many Hamiltonians which mutually commute. This allows the operators to be simultaneously diagonalized.
There is a notion of exact solvability for spin chains, often stated as determining the spectrum of the model. In precise terms, this means determining the simultaneous eigenvectors of the Hilbert space for the Hamiltonians of the system as well as the eigenvalues of each eigenvector with respect to each Hamiltonian.
Examples
Spin 1/2 XXX model in detail
The prototypical example, and a particular example of the Heisenberg spin chain, is known as the spin 1/2 Heisenberg XXX model.[6]
The graph $G$ is the periodic 1-dimensional lattice with $N$-sites. Explicitly, this is given by $V=\{1,\cdots ,N\}$, and the elements of $E$ being $\{n,n+1\}$ with $N+1$ identified with $1$.
The associated Lie algebra is ${\mathfrak {sl}}_{2}$.
At site $n$ there is an associated Hilbert space $h_{n}$ which is isomorphic to the two dimensional representation of ${\mathfrak {sl}}_{2}$ (and therefore further isomorphic to $\mathbb {C} ^{2}$). The Hilbert space of system configurations is ${\mathcal {H}}=\bigotimes _{n=1}^{N}h_{n}$, of dimension $2^{N}$.
Given an operator $A$ on the two-dimensional representation $h$ of ${\mathfrak {sl}}_{2}$, denote by $A^{(n)}$ the operator on ${\mathcal {H}}$ which acts as $A$ on $h_{n}$ and as identity on the other $h_{m}$ with $m\neq n$. Explicitly, it can be written
$A^{(n)}=1\otimes \cdots \otimes \underbrace {A} _{n}\otimes \cdots \otimes 1,$
where the 1 denotes identity.
The Hamiltonian is essentially, up to an affine transformation, $H=\sum _{n=1}^{N}\sigma _{i}^{(n)}\sigma _{i}^{(n+1)}$ with implied summation over index $i$, and where $\sigma _{i}$ are the Pauli matrices. The Hamiltonian has ${\mathfrak {sl}}_{2}$ symmetry under the action of the three total spin operators $\sigma _{i}=\sum _{n=1}^{N}\sigma _{i}^{(n)}$.
The central problem is then to determine the spectrum (eigenvalues and eigenvectors in ${\mathcal {H}}$) of the Hamiltonian. This is solved by the method of an Algebraic Bethe ansatz, discovered by Hans Bethe and further explored by Ludwig Faddeev.
List of spin chains
• Quantum Heisenberg model
• Inozemtsev model
• Haldane–Shastry model
• Quantum Gaudin model
See also
• Lattice model (physics)
• Exactly solvable
References
1. Heisenberg, Werner (September 1928). "Zur Theorie des Ferromagnetismus". Zeitschrift für Physik. 49 (9–10): 619–636. Bibcode:1928ZPhy...49..619H. doi:10.1007/BF01328601. S2CID 122524239. Retrieved 4 October 2022.
2. Bethe, H. (March 1931). "Zur Theorie der Metalle. I. Eigenwerte und Eigenfunktionen der linearen Atomkette". Zeitschrift für Physik. 71 (3–4): 205–226. doi:10.1007/BF01341708. S2CID 124225487.
3. Hubbard, John (26 November 1963). "Electron correlations in narrow energy bands". Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences. 276 (1365): 238–257. doi:10.1098/rspa.1963.0204.
4. Lieb, Elliott H.; Wu, F. Y. (17 June 1968). "Absence of Mott Transition in an Exact Solution of the Short-Range, One-Band Model in One Dimension". Physical Review Letters. 20 (25): 1445–1448. doi:10.1103/PhysRevLett.20.1445.
5. Gaudin, Michel (1976). "Diagonalisation d'une classe d'hamiltoniens de spin". Journal de Physique. 37 (10): 1087–1098. doi:10.1051/jphys:0197600370100108700. Retrieved 26 September 2022.
6. Faddeev, Ludwig (1996). "How Algebraic Bethe Ansatz works for integrable model". arXiv:hep-th/9605187.
External links
• Spin chain in nLab
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Wikipedia
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Spin connection
In differential geometry and mathematical physics, a spin connection is a connection on a spinor bundle. It is induced, in a canonical manner, from the affine connection. It can also be regarded as the gauge field generated by local Lorentz transformations. In some canonical formulations of general relativity, a spin connection is defined on spatial slices and can also be regarded as the gauge field generated by local rotations.
The spin connection occurs in two common forms: the Levi-Civita spin connection, when it is derived from the Levi-Civita connection, and the affine spin connection, when it is obtained from the affine connection. The difference between the two of these is that the Levi-Civita connection is by definition the unique torsion-free connection, whereas the affine connection (and so the affine spin connection) may contain torsion.
Definition
Let $e_{\mu }^{\;\,a}$ be the local Lorentz frame fields or vierbein (also known as a tetrad), which is a set of orthonormal space time vector fields that diagonalize the metric tensor
$g_{\mu \nu }=e_{\mu }^{\;\,a}e_{\nu }^{\;\,b}\eta _{ab},$
where $g_{\mu \nu }$ is the spacetime metric and $\eta _{ab}$ is the Minkowski metric. Here, Latin letters denote the local Lorentz frame indices; Greek indices denote general coordinate indices. This simply expresses that $g_{\mu \nu }$, when written in terms of the basis $e_{\mu }^{\;\,a}$, is locally flat. The Greek vierbein indices can be raised or lowered by the metric, i.e. $g^{\mu \nu }$ or $g_{\mu \nu }$. The Latin or "Lorentzian" vierbein indices can be raised or lowered by $\eta ^{ab}$ or $\eta _{ab}$ respectively. For example, $e^{\mu a}=g^{\mu \nu }e_{\nu }^{\;\,a}$ and $e_{\nu a}=\eta _{ab}e_{\nu }^{\;\,b}$
The torsion-free spin connection is given by
$\omega _{\mu }^{\ ab}=e_{\nu }^{\ a}\Gamma _{\ \sigma \mu }^{\nu }e^{\sigma b}+e_{\nu }^{\ a}\partial _{\mu }e^{\nu b}=e_{\nu }^{\ a}\Gamma _{\ \sigma \mu }^{\nu }e^{\sigma b}-e^{\nu b}\partial _{\mu }e_{\nu }^{\ a},$
where $\Gamma _{\mu \nu }^{\sigma }$ are the Christoffel symbols. This definition should be taken as defining the torsion-free spin connection, since, by convention, the Christoffel symbols are derived from the Levi-Civita connection, which is the unique metric compatible, torsion-free connection on a Riemannian Manifold. In general, there is no restriction: the spin connection may also contain torsion.
Note that $\omega _{\mu }^{\ ab}=e_{\nu }^{\ a}\partial _{;\mu }e^{\nu b}=e_{\nu }^{\ a}(\partial _{\mu }e^{\nu b}+\Gamma _{\ \sigma \mu }^{\nu }e^{\sigma b})$ using the gravitational covariant derivative $\partial _{;\mu }e^{\nu b}$ of the contravariant vector $e^{\nu b}$. The spin connection may be written purely in terms of the vierbein field as[1]
$\omega _{\mu }^{\ ab}={\tfrac {1}{2}}e^{\nu a}(\partial _{\mu }e_{\nu }^{\ b}-\partial _{\nu }e_{\mu }^{\ b})-{\tfrac {1}{2}}e^{\nu b}(\partial _{\mu }e_{\nu }^{\ a}-\partial _{\nu }e_{\mu }^{\ a})-{\tfrac {1}{2}}e^{\rho a}e^{\sigma b}(\partial _{\rho }e_{\sigma c}-\partial _{\sigma }e_{\rho c})e_{\mu }^{\ c},$
which by definition is anti-symmetric in its internal indices $a,b$.
The spin connection $\omega _{\mu }^{\ ab}$ defines a covariant derivative $D_{\mu }$ on generalized tensors. For example, its action on $V_{\nu }^{\ a}$ is
$D_{\mu }V_{\nu }^{\ a}=\partial _{\mu }V_{\nu }^{\ a}+{{\omega _{\mu }}^{a}}_{b}V_{\nu }^{\ b}-\Gamma _{\ \nu \mu }^{\sigma }V_{\sigma }^{\ a}$
Cartan's structure equations
In the Cartan formalism, the spin connection is used to define both torsion and curvature. These are easiest to read by working with differential forms, as this hides some of the profusion of indexes. The equations presented here are effectively a restatement of those that can be found in the article on the connection form and the curvature form. The primary difference is that these retain the indexes on the vierbein, instead of completely hiding them. More narrowly, the Cartan formalism is to be interpreted in its historical setting, as a generalization of the idea of an affine connection to a homogeneous space; it is not yet as general as the idea of a principal connection on a fiber bundle. It serves as a suitable half-way point between the narrower setting in Riemannian geometry and the fully abstract fiber bundle setting, thus emphasizing the similarity to gauge theory. Note that Cartan's structure equations, as expressed here, have a direct analog: the Maurer–Cartan equations for Lie groups (that is, they are the same equations, but in a different setting and notation).
Writing the vierbeins as differential forms
$e^{a}=e_{\mu }^{\;\,a}dx^{\mu }$
for the orthonormal coordinates on the cotangent bundle, the affine spin connection one-form is
$\omega ^{ab}=\omega _{\mu }^{\;\;ab}dx^{\mu }$
The torsion 2-form is given by
$\Theta ^{a}=de^{a}+\omega _{\;b}^{a}\wedge e^{b}$
while the curvature 2-form is
$R_{\;\,b}^{a}=d\omega _{\;\,b}^{a}+\omega _{\;c}^{a}\wedge \omega _{\;\,b}^{c}={\tfrac {1}{2}}R_{\;\,bcd}^{a}e^{c}\wedge e^{d}$
These two equations, taken together are called Cartan's structure equations.[2] Consistency requires that the Bianchi identities be obeyed. The first Bianchi identity is obtained by taking the exterior derivative of the torsion:
$d\Theta ^{a}+\omega _{\;b}^{a}\wedge \Theta ^{b}=R_{\;\,b}^{a}\wedge e^{b}$
while the second by differentiating the curvature:
$dR_{\;\,b}^{a}+\omega _{\;c}^{a}\wedge R_{\;\,b}^{c}-R_{\;\,c}^{a}\wedge \omega _{\;b}^{c}=0.$
The covariant derivative for a generic differential form $V_{\;\,b}^{a}$ of degree p is defined by
$DV_{\;\,b}^{a}=dV_{\;\,b}^{a}+\omega _{\;c}^{a}\wedge V_{\;\,b}^{c}-(-1)^{p}V_{\;\,c}^{a}\wedge \omega _{\;b}^{c}.$
Bianchi's second identity then becomes
$DR_{\;\,b}^{a}=0.$
The difference between a connection with torsion, and the unique torsionless connection is given by the contorsion tensor. Connections with torsion are commonly found in theories of teleparallelism, Einstein–Cartan theory, gauge theory gravity and supergravity.
Derivation
Metricity
It is easy to deduce by raising and lowering indices as needed that the frame fields defined by $g_{\mu \nu }={e_{\mu }}^{a}{e_{\nu }}^{b}\eta _{ab}$ will also satisfy ${e_{\mu }}^{a}{e^{\mu }}_{b}=\delta _{b}^{a}$ and ${e_{\mu }}^{b}{e^{\nu }}_{b}=\delta _{\mu }^{\nu }$. We expect that $D_{\mu }$ will also annihilate the Minkowski metric $\eta _{ab}$,
$D_{\mu }\eta _{ab}=\partial _{\mu }\eta _{ab}-{\omega _{\mu a}}^{c}\eta _{cb}-{\omega _{\mu b}}^{c}\eta _{ac}=0.$
This implies that the connection is anti-symmetric in its internal indices, ${\omega _{\mu }}^{ab}=-{\omega _{\mu }}^{ba}.$ This is also deduced by taking the gravitational covariant derivative $\partial _{;\beta }({e_{\mu }}^{a}{e^{\mu }}_{b})=0$ which implies that $\partial _{;\beta }{e_{\mu }}^{a}{e^{\mu }}_{b}=-{e_{\mu }}^{a}\partial _{;\beta }{e^{\mu }}_{b}$ thus ultimately, ${\omega _{\beta }}^{ab}=-{\omega _{\beta }}^{ba}$. This is sometimes called the metricity condition;[2] it is analogous to the more commonly stated metricity condition that $g_{\mu \nu ;\alpha }=0.$ ;\alpha }=0.} Note that this condition holds only for the Levi-Civita spin connection, and not for the affine spin connection in general.
By substituting the formula for the Christoffel symbols ${\Gamma ^{\nu }}_{\sigma \mu }={\tfrac {1}{2}}g^{\nu \delta }\left(\partial _{\sigma }g_{\delta \mu }+\partial _{\mu }g_{\sigma \delta }-\partial _{\delta }g_{\sigma \mu }\right)$ written in terms of the ${e_{\mu }}^{a}$, the spin connection can be written entirely in terms of the ${e_{\mu }}^{a}$,
${\omega _{\mu }}^{ab}=e^{\nu [a}({{e_{\nu }}^{b]}}_{,\mu }-{{e_{\mu }}^{b]}}_{,\nu }+e^{\sigma |b]}{e_{\mu }}^{c}e_{\nu c,\sigma })$
where antisymmetrization of indices has an implicit factor of 1/2.
By the metric compatibility
This formula can be derived another way. To directly solve the compatibility condition for the spin connection ${\omega _{\mu }}^{ab}$, one can use the same trick that was used to solve $\nabla _{\rho }g_{\alpha \beta }=0$ for the Christoffel symbols ${\Gamma ^{\gamma }}_{\alpha \beta }$. First contract the compatibility condition to give
${e^{\alpha }}_{b}{e^{\beta }}_{c}(\partial _{[\alpha }e_{\beta ]a}+{\omega _{[\alpha a}}^{d}\;e_{\beta ]d})=0.$
Then, do a cyclic permutation of the free indices $a,b,$ and $c$, and add and subtract the three resulting equations:
$\Omega _{bca}+\Omega _{abc}-\Omega _{cab}+2{e^{\alpha }}_{b}\omega _{\alpha ac}=0$
where we have used the definition $\Omega _{bca}:={e^{\alpha }}_{b}{e^{\beta }}_{c}\partial _{[\alpha }e_{\beta ]a}$. The solution for the spin connection is
$\omega _{\alpha ca}={\tfrac {1}{2}}{e_{\alpha }}^{b}(\Omega _{bca}+\Omega _{abc}-\Omega _{cab}).$
From this we obtain the same formula as before.
Applications
The spin connection arises in the Dirac equation when expressed in the language of curved spacetime, see Dirac equation in curved spacetime. Specifically there are problems coupling gravity to spinor fields: there are no finite-dimensional spinor representations of the general covariance group. However, there are of course spinorial representations of the Lorentz group. This fact is utilized by employing tetrad fields describing a flat tangent space at every point of spacetime. The Dirac matrices $\gamma ^{a}$ are contracted onto vierbiens,
$\gamma ^{a}{e^{\mu }}_{a}(x)=\gamma ^{\mu }(x).$
We wish to construct a generally covariant Dirac equation. Under a flat tangent space Lorentz transformation the spinor transforms as
$\psi \mapsto e^{i\epsilon ^{ab}(x)\sigma _{ab}}\psi $
We have introduced local Lorentz transformations on flat tangent space generated by the $\sigma _{ab}$'s, such that $\epsilon _{ab}$ is a function of space-time. This means that the partial derivative of a spinor is no longer a genuine tensor. As usual, one introduces a connection field ${\omega _{\mu }}^{ab}$ that allows us to gauge the Lorentz group. The covariant derivative defined with the spin connection is,
$\nabla _{\mu }\psi =\left(\partial _{\mu }-{\tfrac {i}{4}}{\omega _{\mu }}^{ab}\sigma _{ab}\right)\psi =\left(\partial _{\mu }-{\tfrac {i}{4}}e^{\nu a}\partial _{;\mu }{e_{\nu }}^{b}\sigma _{ab}\right)\psi ,$
and is a genuine tensor and Dirac's equation is rewritten as
$(i\gamma ^{\mu }\nabla _{\mu }-m)\psi =0.$
The generally covariant fermion action couples fermions to gravity when added to the first order tetradic Palatini action,
${\mathcal {L}}=-{1 \over 2\kappa ^{2}}e\,{e^{\mu }}_{a}{e^{\nu }}_{b}{\Omega _{\mu \nu }}^{ab}[\omega ]+e{\overline {\psi }}(i\gamma ^{\mu }\nabla _{\mu }-m)\psi $
where $ e:=\det {e_{\mu }}^{a}={\sqrt {-g}}$ and ${\Omega _{\mu \nu }}^{ab}$ is the curvature of the spin connection.
The tetradic Palatini formulation of general relativity which is a first order formulation of the Einstein–Hilbert action where the tetrad and the spin connection are the basic independent variables. In the 3+1 version of Palatini formulation, the information about the spatial metric, $q_{ab}(x)$, is encoded in the triad $e_{a}^{i}$ (three-dimensional, spatial version of the tetrad). Here we extend the metric compatibility condition $D_{a}q_{bc}=0$ to $e_{a}^{i}$, that is, $D_{a}e_{b}^{i}=0$ and we obtain a formula similar to the one given above but for the spatial spin connection $\Gamma _{a}^{ij}$.
The spatial spin connection appears in the definition of Ashtekar–Barbero variables which allows 3+1 general relativity to be rewritten as a special type of $\mathrm {SU} (2)$ Yang–Mills gauge theory. One defines $\Gamma _{a}^{i}=\epsilon ^{ijk}\Gamma _{a}^{jk}$. The Ashtekar–Barbero connection variable is then defined as $A_{a}^{i}=\Gamma _{a}^{i}+\beta c_{a}^{i}$ where $c_{a}^{i}=c_{ab}e^{bi}$ and $c_{ab}$ is the extrinsic curvature and $\beta $ is the Immirzi parameter. With $A_{a}^{i}$ as the configuration variable, the conjugate momentum is the densitized triad $E_{a}^{i}=\left|\det(e)\right|e_{a}^{i}$. With 3+1 general relativity rewritten as a special type of $\mathrm {SU} (2)$ Yang–Mills gauge theory, it allows the importation of non-perturbative techniques used in Quantum chromodynamics to canonical quantum general relativity.
See also
• Ashtekar variables
• Dirac operator
• Cartan connection
• Levi-Civita connection
• Ricci calculus
• Supergravity
• Torsion tensor
• Contorsion tensor
• Dirac equation in curved spacetime
References
1. M.B. Green, J.H. Schwarz, E. Witten, "Superstring theory", Vol. 2.
2. Tohru Eguchi, Peter B. Gilkey and Andrew J. Hanson, "Gravitation, Gauge Theories and Differential Geometry", Physics Reports 66 (1980) pp 213-393.
• Hehl, F.W.; von der Heyde, P.; Kerlick, G.D.; Nester, J.M. (1976), "General relativity with spin and torsion: Foundations and prospects", Rev. Mod. Phys. 48, 393.
• Kibble, T.W.B. (1961), "Lorentz invariance and the gravitational field", J. Math. Phys. 2, 212.
• Poplawski, N.J. (2009), "Spacetime and fields", arXiv:0911.0334
• Sciama, D.W. (1964), "The physical structure of general relativity", Rev. Mod. Phys. 36, 463.
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Spin geometry
In mathematics, spin geometry is the area of differential geometry and topology where objects like spin manifolds and Dirac operators, and the various associated index theorems have come to play a fundamental role both in mathematics and in mathematical physics.
An important generalisation is the theory of symplectic Dirac operators in symplectic spin geometry and symplectic topology, which have become important fields of mathematical research.
See also
• Symplectic topology
• Spinor
• Spinor bundle
• Spin manifold
Books
• Lawson, H. Blaine; Michelsohn, Marie-Louise (1989). Spin Geometry. Princeton University Press. ISBN 978-0-691-08542-5.
• Friedrich, Thomas (2000), Dirac Operators in Riemannian Geometry, American Mathematical Society, ISBN 978-0-8218-2055-1
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Spin representation
In mathematics, the spin representations are particular projective representations of the orthogonal or special orthogonal groups in arbitrary dimension and signature (i.e., including indefinite orthogonal groups). More precisely, they are two equivalent representations of the spin groups, which are double covers of the special orthogonal groups. They are usually studied over the real or complex numbers, but they can be defined over other fields.
Elements of a spin representation are called spinors. They play an important role in the physical description of fermions such as the electron.
The spin representations may be constructed in several ways, but typically the construction involves (perhaps only implicitly) the choice of a maximal isotropic subspace in the vector representation of the group. Over the real numbers, this usually requires using a complexification of the vector representation. For this reason, it is convenient to define the spin representations over the complex numbers first, and derive real representations by introducing real structures.
The properties of the spin representations depend, in a subtle way, on the dimension and signature of the orthogonal group. In particular, spin representations often admit invariant bilinear forms, which can be used to embed the spin groups into classical Lie groups. In low dimensions, these embeddings are surjective and determine special isomorphisms between the spin groups and more familiar Lie groups; this elucidates the properties of spinors in these dimensions.
Set-up
Let V be a finite-dimensional real or complex vector space with a nondegenerate quadratic form Q. The (real or complex) linear maps preserving Q form the orthogonal group O(V, Q). The identity component of the group is called the special orthogonal group SO(V, Q). (For V real with an indefinite quadratic form, this terminology is not standard: the special orthogonal group is usually defined to be a subgroup with two components in this case.) Up to group isomorphism, SO(V, Q) has a unique connected double cover, the spin group Spin(V, Q). There is thus a group homomorphism h: Spin(V, Q) → SO(V, Q) whose kernel has two elements denoted {1, −1}, where 1 is the identity element. Thus, the group elements g and −g of Spin(V, Q) are equivalent after the homomorphism to SO(V, Q); that is, h(g) = h(−g) for any g in Spin(V, Q).
The groups O(V, Q), SO(V, Q) and Spin(V, Q) are all Lie groups, and for fixed (V, Q) they have the same Lie algebra, so(V, Q). If V is real, then V is a real vector subspace of its complexification VC = V ⊗R C, and the quadratic form Q extends naturally to a quadratic form QC on VC. This embeds SO(V, Q) as a subgroup of SO(VC, QC), and hence we may realise Spin(V, Q) as a subgroup of Spin(VC, QC). Furthermore, so(VC, QC) is the complexification of so(V, Q).
In the complex case, quadratic forms are determined uniquely up to isomorphism by the dimension n of V. Concretely, we may assume V = Cn and
$Q(z_{1},\ldots ,z_{n})=z_{1}^{2}+z_{2}^{2}+\cdots +z_{n}^{2}.$
The corresponding Lie groups are denoted O(n, C), SO(n, C), Spin(n, C) and their Lie algebra as so(n, C).
In the real case, quadratic forms are determined up to isomorphism by a pair of nonnegative integers (p, q) where n = p + q is the dimension of V, and p − q is the signature. Concretely, we may assume V = Rn and
$Q(x_{1},\ldots ,x_{n})=x_{1}^{2}+x_{2}^{2}+\cdots +x_{p}^{2}-(x_{p+1}^{2}+\cdots +x_{p+q}^{2}).$
The corresponding Lie groups and Lie algebra are denoted O(p, q), SO(p, q), Spin(p, q) and so(p, q). We write Rp,q in place of Rn to make the signature explicit.
The spin representations are, in a sense, the simplest representations of Spin(n, C) and Spin(p, q) that do not come from representations of SO(n, C) and SO(p, q). A spin representation is, therefore, a real or complex vector space S together with a group homomorphism ρ from Spin(n, C) or Spin(p, q) to the general linear group GL(S) such that the element −1 is not in the kernel of ρ.
If S is such a representation, then according to the relation between Lie groups and Lie algebras, it induces a Lie algebra representation, i.e., a Lie algebra homomorphism from so(n, C) or so(p, q) to the Lie algebra gl(S) of endomorphisms of S with the commutator bracket.
Spin representations can be analysed according to the following strategy: if S is a real spin representation of Spin(p, q), then its complexification is a complex spin representation of Spin(p, q); as a representation of so(p, q), it therefore extends to a complex representation of so(n, C). Proceeding in reverse, we therefore first construct complex spin representations of Spin(n, C) and so(n, C), then restrict them to complex spin representations of so(p, q) and Spin(p, q), then finally analyse possible reductions to real spin representations.
Complex spin representations
Let V = Cn with the standard quadratic form Q so that
${\mathfrak {so}}(V,Q)={\mathfrak {so}}(n,\mathbb {C} ).$
The symmetric bilinear form on V associated to Q by polarization is denoted ⟨.,.⟩.
Isotropic subspaces and root systems
A standard construction of the spin representations of so(n, C) begins with a choice of a pair (W, W∗) of maximal totally isotropic subspaces (with respect to Q) of V with W ∩ W∗ = 0. Let us make such a choice. If n = 2m or n = 2m + 1, then W and W∗ both have dimension m. If n = 2m, then V = W ⊕ W∗, whereas if n = 2m + 1, then V = W ⊕ U ⊕ W∗, where U is the 1-dimensional orthogonal complement to W ⊕ W∗. The bilinear form ⟨.,.⟩ associated to Q induces a pairing between W and W∗, which must be nondegenerate, because W and W∗ are totally isotropic subspaces and Q is nondegenerate. Hence W and W∗ are dual vector spaces.
More concretely, let a1, … am be a basis for W. Then there is a unique basis α1, ... αm of W∗ such that
$\langle \alpha _{i},a_{j}\rangle =\delta _{ij}.$
If A is an m × m matrix, then A induces an endomorphism of W with respect to this basis and the transpose AT induces a transformation of W∗ with
$\langle Aw,w^{*}\rangle =\langle w,A^{\mathrm {T} }w^{*}\rangle $
for all w in W and w∗ in W∗. It follows that the endomorphism ρA of V, equal to A on W, −AT on W∗ and zero on U (if n is odd), is skew,
$\langle \rho _{A}u,v\rangle =-\langle u,\rho _{A}v\rangle $
for all u, v in V, and hence (see classical group) an element of so(n, C) ⊂ End(V).
Using the diagonal matrices in this construction defines a Cartan subalgebra h of so(n, C): the rank of so(n, C) is m, and the diagonal n × n matrices determine an m-dimensional abelian subalgebra.
Let ε1, … εm be the basis of h∗ such that, for a diagonal matrix A, εk(ρA) is the kth diagonal entry of A. Clearly this is a basis for h∗. Since the bilinear form identifies so(n, C) with $\wedge ^{2}V$, explicitly,
$x\wedge y\mapsto \varphi _{x\wedge y},\quad \varphi _{x\wedge y}(v)=2(\langle y,v\rangle x-\langle x,v\rangle y),\quad x\wedge y\in \wedge ^{2}V,\quad x,y,v\in V,\quad \varphi _{x\wedge y}\in {\mathfrak {so}}(n,\mathbb {C} ),$[1]
it is now easy to construct the root system associated to h. The root spaces (simultaneous eigenspaces for the action of h) are spanned by the following elements:
$a_{i}\wedge a_{j},\;i\neq j,$ with root (simultaneous eigenvalue) $\varepsilon _{i}+\varepsilon _{j}$
$a_{i}\wedge \alpha _{j}$ (which is in h if i = j) with root $\varepsilon _{i}-\varepsilon _{j}$
$\alpha _{i}\wedge \alpha _{j},\;i\neq j,$ with root $-\varepsilon _{i}-\varepsilon _{j},$
and, if n is odd, and u is a nonzero element of U,
$a_{i}\wedge u,$ with root $\varepsilon _{i}$
$\alpha _{i}\wedge u,$ with root $-\varepsilon _{i}.$
Thus, with respect to the basis ε1, … εm, the roots are the vectors in h∗ that are permutations of
$(\pm 1,\pm 1,0,0,\dots ,0)$
together with the permutations of
$(\pm 1,0,0,\dots ,0)$
if n = 2m + 1 is odd.
A system of positive roots is given by εi + εj (i ≠ j), εi − εj (i < j) and (for n odd) εi. The corresponding simple roots are
$\varepsilon _{1}-\varepsilon _{2},\varepsilon _{2}-\varepsilon _{3},\ldots ,\varepsilon _{m-1}-\varepsilon _{m},\left\{{\begin{matrix}\varepsilon _{m-1}+\varepsilon _{m}&n=2m\\\varepsilon _{m}&n=2m+1.\end{matrix}}\right.$
The positive roots are nonnegative integer linear combinations of the simple roots.
Spin representations and their weights
One construction of the spin representations of so(n, C) uses the exterior algebra(s)
$S=\wedge ^{\bullet }W$ and/or $S'=\wedge ^{\bullet }W^{*}.$
There is an action of V on S such that for any element v = w + w∗ in W ⊕ W∗ and any ψ in S the action is given by:
$v\cdot \psi =2^{\frac {1}{2}}(w\wedge \psi +\iota (w^{*})\psi ),$
where the second term is a contraction (interior multiplication) defined using the bilinear form, which pairs W and W∗. This action respects the Clifford relations v2 = Q(v)1, and so induces a homomorphism from the Clifford algebra ClnC of V to End(S). A similar action can be defined on S′, so that both S and S′ are Clifford modules.
The Lie algebra so(n, C) is isomorphic to the complexified Lie algebra spinnC in ClnC via the mapping induced by the covering Spin(n) → SO(n)[2]
$v\wedge w\mapsto {\tfrac {1}{4}}[v,w].$
It follows that both S and S′ are representations of so(n, C). They are actually equivalent representations, so we focus on S.
The explicit description shows that the elements αi ∧ ai of the Cartan subalgebra h act on S by
$(\alpha _{i}\wedge a_{i})\cdot \psi ={\tfrac {1}{4}}(2^{\tfrac {1}{2}})^{2}(\iota (\alpha _{i})(a_{i}\wedge \psi )-a_{i}\wedge (\iota (\alpha _{i})\psi ))={\tfrac {1}{2}}\psi -a_{i}\wedge (\iota (\alpha _{i})\psi ).$
A basis for S is given by elements of the form
$a_{i_{1}}\wedge a_{i_{2}}\wedge \cdots \wedge a_{i_{k}}$
for 0 ≤ k ≤ m and i1 < ... < ik. These clearly span weight spaces for the action of h: αi ∧ ai has eigenvalue −1/2 on the given basis vector if i = ij for some j, and has eigenvalue 1/2 otherwise.
It follows that the weights of S are all possible combinations of
${\bigl (}\pm {\tfrac {1}{2}},\pm {\tfrac {1}{2}},\ldots \pm {\tfrac {1}{2}}{\bigr )}$
and each weight space is one-dimensional. Elements of S are called Dirac spinors.
When n is even, S is not an irreducible representation: $S_{+}=\wedge ^{\mathrm {even} }W$ and $S_{-}=\wedge ^{\mathrm {odd} }W$ are invariant subspaces. The weights divide into those with an even number of minus signs, and those with an odd number of minus signs. Both S+ and S− are irreducible representations of dimension 2m−1 whose elements are called Weyl spinors. They are also known as chiral spin representations or half-spin representations. With respect to the positive root system above, the highest weights of S+ and S− are
${\bigl (}{\tfrac {1}{2}},{\tfrac {1}{2}},\ldots {\tfrac {1}{2}},{\tfrac {1}{2}}{\bigr )}$ and ${\bigl (}{\tfrac {1}{2}},{\tfrac {1}{2}},\ldots {\tfrac {1}{2}},-{\tfrac {1}{2}}{\bigr )}$
respectively. The Clifford action identifies ClnC with End(S) and the even subalgebra is identified with the endomorphisms preserving S+ and S−. The other Clifford module S′ is isomorphic to S in this case.
When n is odd, S is an irreducible representation of so(n,C) of dimension 2m: the Clifford action of a unit vector u ∈ U is given by
$u\cdot \psi =\left\{{\begin{matrix}\psi &{\hbox{if }}\psi \in \wedge ^{\mathrm {even} }W\\-\psi &{\hbox{if }}\psi \in \wedge ^{\mathrm {odd} }W\end{matrix}}\right.$
and so elements of so(n,C) of the form u∧w or u∧w∗ do not preserve the even and odd parts of the exterior algebra of W. The highest weight of S is
${\bigl (}{\tfrac {1}{2}},{\tfrac {1}{2}},\ldots {\tfrac {1}{2}}{\bigr )}.$
The Clifford action is not faithful on S: ClnC can be identified with End(S) ⊕ End(S′), where u acts with the opposite sign on S′. More precisely, the two representations are related by the parity involution α of ClnC (also known as the principal automorphism), which is the identity on the even subalgebra, and minus the identity on the odd part of ClnC. In other words, there is a linear isomorphism from S to S′, which identifies the action of A in ClnC on S with the action of α(A) on S′.
Bilinear forms
if λ is a weight of S, so is −λ. It follows that S is isomorphic to the dual representation S∗.
When n = 2m + 1 is odd, the isomorphism B: S → S∗ is unique up to scale by Schur's lemma, since S is irreducible, and it defines a nondegenerate invariant bilinear form β on S via
$\beta (\varphi ,\psi )=B(\varphi )(\psi ).$
Here invariance means that
$\beta (\xi \cdot \varphi ,\psi )+\beta (\varphi ,\xi \cdot \psi )=0$
for all ξ in so(n,C) and φ, ψ in S — in other words the action of ξ is skew with respect to β. In fact, more is true: S∗ is a representation of the opposite Clifford algebra, and therefore, since ClnC only has two nontrivial simple modules S and S′, related by the parity involution α, there is an antiautomorphism τ of ClnC such that
$\quad \beta (A\cdot \varphi ,\psi )=\beta (\varphi ,\tau (A)\cdot \psi )\qquad (1)$
for any A in ClnC. In fact τ is reversion (the antiautomorphism induced by the identity on V) for m even, and conjugation (the antiautomorphism induced by minus the identity on V) for m odd. These two antiautomorphisms are related by parity involution α, which is the automorphism induced by minus the identity on V. Both satisfy τ(ξ) = −ξ for ξ in so(n,C).
When n = 2m, the situation depends more sensitively upon the parity of m. For m even, a weight λ has an even number of minus signs if and only if −λ does; it follows that there are separate isomorphisms B±: S± → S±∗ of each half-spin representation with its dual, each determined uniquely up to scale. These may be combined into an isomorphism B: S → S∗. For m odd, λ is a weight of S+ if and only if −λ is a weight of S−; thus there is an isomorphism from S+ to S−∗, again unique up to scale, and its transpose provides an isomorphism from S− to S+∗. These may again be combined into an isomorphism B: S → S∗.
For both m even and m odd, the freedom in the choice of B may be restricted to an overall scale by insisting that the bilinear form β corresponding to B satisfies (1), where τ is a fixed antiautomorphism (either reversion or conjugation).
Symmetry and the tensor square
The symmetry properties of β: S ⊗ S → C can be determined using Clifford algebras or representation theory. In fact much more can be said: the tensor square S ⊗ S must decompose into a direct sum of k-forms on V for various k, because its weights are all elements in h∗ whose components belong to {−1,0,1}. Now equivariant linear maps S ⊗ S → ∧kV∗ correspond bijectively to invariant maps ∧kV ⊗ S ⊗ S → C and nonzero such maps can be constructed via the inclusion of ∧kV into the Clifford algebra. Furthermore, if β(φ,ψ) = ε β(ψ,φ) and τ has sign εk on ∧kV then
$\beta (A\cdot \varphi ,\psi )=\varepsilon \varepsilon _{k}\beta (A\cdot \psi ,\varphi )$
for A in ∧kV.
If n = 2m+1 is odd then it follows from Schur's Lemma that
$S\otimes S\cong \bigoplus _{j=0}^{m}\wedge ^{2j}V^{*}$
(both sides have dimension 22m and the representations on the right are inequivalent). Because the symmetries are governed by an involution τ that is either conjugation or reversion, the symmetry of the ∧2jV∗ component alternates with j. Elementary combinatorics gives
$\sum _{j=0}^{m}(-1)^{j}\dim \wedge ^{2j}\mathbb {C} ^{2m+1}=(-1)^{{\frac {1}{2}}m(m+1)}2^{m}=(-1)^{{\frac {1}{2}}m(m+1)}(\dim \mathrm {S} ^{2}S-\dim \wedge ^{2}S)$
and the sign determines which representations occur in S2S and which occur in ∧2S.[3] In particular
$\beta (\phi ,\psi )=(-1)^{{\frac {1}{2}}m(m+1)}\beta (\psi ,\phi ),$ and
$\beta (v\cdot \phi ,\psi )=(-1)^{m}(-1)^{{\frac {1}{2}}m(m+1)}\beta (v\cdot \psi ,\phi )=(-1)^{m}\beta (\phi ,v\cdot \psi )$
for v ∈ V (which is isomorphic to ∧2mV), confirming that τ is reversion for m even, and conjugation for m odd.
If n = 2m is even, then the analysis is more involved, but the result is a more refined decomposition: S2S±, ∧2S± and S+ ⊗ S− can each be decomposed as a direct sum of k-forms (where for k = m there is a further decomposition into selfdual and antiselfdual m-forms).
The main outcome is a realisation of so(n,C) as a subalgebra of a classical Lie algebra on S, depending upon n modulo 8, according to the following table:
n mod 8 0 1 2 3 4 5 6 7
Spinor algebra ${\mathfrak {so}}(S_{+})\oplus {\mathfrak {so}}(S_{-})$ ${\mathfrak {so}}(S)$ ${\mathfrak {gl}}(S_{\pm })$ ${\mathfrak {sp}}(S)$ ${\mathfrak {sp}}(S_{+})\oplus {\mathfrak {sp}}(S_{-})$ ${\mathfrak {sp}}(S)$ ${\mathfrak {gl}}(S_{\pm })$ ${\mathfrak {so}}(S)$
For n ≤ 6, these embeddings are isomorphisms (onto sl rather than gl for n = 6):
${\mathfrak {so}}(2,\mathbb {C} )\cong {\mathfrak {gl}}(1,\mathbb {C} )\qquad (=\mathbb {C} )$
${\mathfrak {so}}(3,\mathbb {C} )\cong {\mathfrak {sp}}(2,\mathbb {C} )\qquad (={\mathfrak {sl}}(2,\mathbb {C} ))$
${\mathfrak {so}}(4,\mathbb {C} )\cong {\mathfrak {sp}}(2,\mathbb {C} )\oplus {\mathfrak {sp}}(2,\mathbb {C} )$
${\mathfrak {so}}(5,\mathbb {C} )\cong {\mathfrak {sp}}(4,\mathbb {C} )$
${\mathfrak {so}}(6,\mathbb {C} )\cong {\mathfrak {sl}}(4,\mathbb {C} ).$
Real representations
The complex spin representations of so(n,C) yield real representations S of so(p,q) by restricting the action to the real subalgebras. However, there are additional "reality" structures that are invariant under the action of the real Lie algebras. These come in three types.
1. There is an invariant complex antilinear map r: S → S with r2 = idS. The fixed point set of r is then a real vector subspace SR of S with SR ⊗ C = S. This is called a real structure.
2. There is an invariant complex antilinear map j: S → S with j2 = −idS. It follows that the triple i, j and k:=ij make S into a quaternionic vector space SH. This is called a quaternionic structure.
3. There is an invariant complex antilinear map b: S → S∗ that is invertible. This defines a pseudohermitian bilinear form on S and is called a hermitian structure.
The type of structure invariant under so(p,q) depends only on the signature p − q modulo 8, and is given by the following table.
p−q mod 8 0 1 2 3 4 5 6 7
Structure R + R R C H H + H H C R
Here R, C and H denote real, hermitian and quaternionic structures respectively, and R + R and H + H indicate that the half-spin representations both admit real or quaternionic structures respectively.
Description and tables
To complete the description of real representation, we must describe how these structures interact with the invariant bilinear forms. Since n = p + q ≅ p − q mod 2, there are two cases: the dimension and signature are both even, and the dimension and signature are both odd.
The odd case is simpler, there is only one complex spin representation S, and hermitian structures do not occur. Apart from the trivial case n = 1, S is always even-dimensional, say dim S = 2N. The real forms of so(2N,C) are so(K,L) with K + L = 2N and so∗(N,H), while the real forms of sp(2N,C) are sp(2N,R) and sp(K,L) with K + L = N. The presence of a Clifford action of V on S forces K = L in both cases unless pq = 0, in which case KL=0, which is denoted simply so(2N) or sp(N). Hence the odd spin representations may be summarized in the following table.
n mod 8 1, 7 3, 5
p-q mod 8 so(2N,C) sp(2N,C)
1, 7 R so(N,N) or so(2N) sp(2N,R)
3, 5 H so∗(N,H) sp(N/2,N/2)† or sp(N)
(†) N is even for n > 3 and for n = 3, this is sp(1).
The even-dimensional case is similar. For n > 2, the complex half-spin representations are even-dimensional. We have additionally to deal with hermitian structures and the real forms of sl(2N, C), which are sl(2N, R), su(K, L) with K + L = 2N, and sl(N, H). The resulting even spin representations are summarized as follows.
n mod 8 0 2, 6 4
p-q mod 8 so(2N,C)+so(2N,C) sl(2N,C) sp(2N,C)+sp(2N,C)
0 R+R so(N,N)+so(N,N)∗ sl(2N,R) sp(2N,R)+sp(2N,R)
2, 6 C so(2N,C) su(N,N) sp(2N,C)
4 H+H so∗(N,H)+so∗(N,H) sl(N,H) sp(N/2,N/2)+sp(N/2,N/2)†
(*) For pq = 0, we have instead so(2N) + so(2N)
(†) N is even for n > 4 and for pq = 0 (which includes n = 4 with N = 1), we have instead sp(N) + sp(N)
The low-dimensional isomorphisms in the complex case have the following real forms.
Euclidean signature Minkowskian signature Other signatures
${\mathfrak {so}}(2)\cong {\mathfrak {u}}(1)$ ${\mathfrak {so}}(1,1)\cong \mathbb {R} $
${\mathfrak {so}}(3)\cong {\mathfrak {sp}}(1)$ ${\mathfrak {so}}(2,1)\cong {\mathfrak {sl}}(2,\mathbb {R} )$
${\mathfrak {so}}(4)\cong {\mathfrak {sp}}(1)\oplus {\mathfrak {sp}}(1)$ ${\mathfrak {so}}(3,1)\cong {\mathfrak {sl}}(2,\mathbb {C} )$ ${\mathfrak {so}}(2,2)\cong {\mathfrak {sl}}(2,\mathbb {R} )\oplus {\mathfrak {sl}}(2,\mathbb {R} )$
${\mathfrak {so}}(5)\cong {\mathfrak {sp}}(2)$ ${\mathfrak {so}}(4,1)\cong {\mathfrak {sp}}(1,1)$ ${\mathfrak {so}}(3,2)\cong {\mathfrak {sp}}(4,\mathbb {R} )$
${\mathfrak {so}}(6)\cong {\mathfrak {su}}(4)$ ${\mathfrak {so}}(5,1)\cong {\mathfrak {sl}}(2,\mathbb {H} )$ ${\mathfrak {so}}(4,2)\cong {\mathfrak {su}}(2,2)$ ${\mathfrak {so}}(3,3)\cong {\mathfrak {sl}}(4,\mathbb {R} )$
The only special isomorphisms of real Lie algebras missing from this table are ${\mathfrak {so}}^{*}(3,\mathbb {H} )\cong {\mathfrak {su}}(3,1)$ and ${\mathfrak {so}}^{*}(4,\mathbb {H} )\cong {\mathfrak {so}}(6,2).$
Notes
1. Fulton & Harris 1991 Chapter 20, p.303. The factor 2 is not important, it is there to agree with the Clifford algebra construction.
2. since if $\alpha :q\to (v\to q.v.q^{-1})$ is the covering, then $d\alpha :q\to (v\to q.v-v.q)$, so $d\alpha (v.w)=2\varphi _{v,w}$ and since $v.w+w.v$ is a scalar, we get $d\alpha (1/4[v,w])=\varphi _{v,w}$
3. This sign can also be determined from the observation that if φ is a highest weight vector for S then φ⊗φ is a highest weight vector for ∧mV ≅ ∧m+1V, so this summand must occur in S2S.
References
• Brauer, Richard; Weyl, Hermann (1935), "Spinors in n dimensions", American Journal of Mathematics, American Journal of Mathematics, Vol. 57, No. 2, 57 (2): 425–449, doi:10.2307/2371218, JSTOR 2371218.
• Cartan, Élie (1966), The theory of spinors, Paris, Hermann (reprinted 1981, Dover Publications), ISBN 978-0-486-64070-9.
• Chevalley, Claude (1954), The algebraic theory of spinors and Clifford algebras, Columbia University Press (reprinted 1996, Springer), ISBN 978-3-540-57063-9.
• Deligne, Pierre (1999), "Notes on spinors", in P. Deligne; P. Etingof; D. S. Freed; L. C. Jeffrey; D. Kazhdan; J. W. Morgan; D. R. Morrison; E. Witten (eds.), Quantum Fields and Strings: A Course for Mathematicians, Providence: American Mathematical Society, pp. 99–135. See also the programme website for a preliminary version.
• Fulton, William; Harris, Joe (1991), Representation theory. A first course, Graduate Texts in Mathematics, Readings in Mathematics, vol. 129, New York: Springer-Verlag, ISBN 0-387-97495-4, MR 1153249.
• Harvey, F. Reese (1990), Spinors and Calibrations, Academic Press, ISBN 978-0-12-329650-4.
• Lawson, H. Blaine; Michelsohn, Marie-Louise (1989), Spin Geometry, Princeton University Press, ISBN 0-691-08542-0.
• Weyl, Hermann (1946), The Classical Groups: Their Invariants and Representations (2nd ed.), Princeton University Press (reprinted 1997), ISBN 978-0-691-05756-9.
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Wikipedia
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Spin structure
In differential geometry, a spin structure on an orientable Riemannian manifold (M, g) allows one to define associated spinor bundles, giving rise to the notion of a spinor in differential geometry.
Spin structures have wide applications to mathematical physics, in particular to quantum field theory where they are an essential ingredient in the definition of any theory with uncharged fermions. They are also of purely mathematical interest in differential geometry, algebraic topology, and K theory. They form the foundation for spin geometry.
Overview
In geometry and in field theory, mathematicians ask whether or not a given oriented Riemannian manifold (M,g) admits spinors. One method for dealing with this problem is to require that M has a spin structure.[1][2][3] This is not always possible since there is potentially a topological obstruction to the existence of spin structures. Spin structures will exist if and only if the second Stiefel–Whitney class w2(M) ∈ H2(M, Z2) of M vanishes. Furthermore, if w2(M) = 0, then the set of the isomorphism classes of spin structures on M is acted upon freely and transitively by H1(M, Z2) . As the manifold M is assumed to be oriented, the first Stiefel–Whitney class w1(M) ∈ H1(M, Z2) of M vanishes too. (The Stiefel–Whitney classes wi(M) ∈ Hi(M, Z2) of a manifold M are defined to be the Stiefel–Whitney classes of its tangent bundle TM.)
The bundle of spinors πS: S → M over M is then the complex vector bundle associated with the corresponding principal bundle πP: P → M of spin frames over M and the spin representation of its structure group Spin(n) on the space of spinors Δn. The bundle S is called the spinor bundle for a given spin structure on M.
A precise definition of spin structure on manifold was possible only after the notion of fiber bundle had been introduced; André Haefliger (1956) found the topological obstruction to the existence of a spin structure on an orientable Riemannian manifold and Max Karoubi (1968) extended this result to the non-orientable pseudo-Riemannian case.[4][5]
Spin structures on Riemannian manifolds
Definition
A spin structure on an orientable Riemannian manifold $(M,g)$ with an oriented vector bundle $E$ is an equivariant lift of the orthonormal frame bundle $P_{\operatorname {SO} }(E)\rightarrow M$ with respect to the double covering $\rho :\operatorname {Spin} (n)\rightarrow \operatorname {SO} (n)$ :\operatorname {Spin} (n)\rightarrow \operatorname {SO} (n)} . In other words, a pair $(P_{\operatorname {Spin} },\phi )$ is a spin structure on the SO(n)-principal bundle $\pi :P_{\operatorname {SO} }(E)\rightarrow M$ when
a) $\pi _{P}:P_{\operatorname {Spin} }\rightarrow M$ is a principal Spin(n)-bundle over $M$, and
b) $\phi :P_{\operatorname {Spin} }\rightarrow P_{\operatorname {SO} }(E)$ is an equivariant 2-fold covering map such that
$\pi \circ \phi =\pi _{P}\quad $and$\quad \phi (pq)=\phi (p)\rho (q)\quad $for all $p\in P_{\operatorname {Spin} }$ and $q\in \operatorname {Spin} (n)$.
Two spin structures $(P_{1},\phi _{1})$ and $(P_{2},\phi _{2})$ on the same oriented Riemannian manifold are called "equivalent" if there exists a Spin(n)-equivariant map $f:P_{1}\rightarrow P_{2}$ such that
$\phi _{2}\circ f=\phi _{1}\quad $ and $\quad f(pq)=f(p)q\quad $ for all $p\in P_{1}$ and $q\in \operatorname {Spin} (n)$.
In this case $\phi _{1}$ and $\phi _{2}$ are two equivalent double coverings.
The definition of spin structure on $(M,g)$ as a spin structure on the principal bundle $P_{\operatorname {SO} }(E)\rightarrow M$ is due to André Haefliger (1956).
Obstruction
Haefliger[1] found necessary and sufficient conditions for the existence of a spin structure on an oriented Riemannian manifold (M,g). The obstruction to having a spin structure is a certain element [k] of H2(M, Z2) . For a spin structure the class [k] is the second Stiefel–Whitney class w2(M) ∈ H2(M, Z2) of M. Hence, a spin structure exists if and only if the second Stiefel–Whitney class w2(M) ∈ H2(M, Z2) of M vanishes.
Spin structures on vector bundles
Let M be a paracompact topological manifold and E an oriented vector bundle on M of dimension n equipped with a fibre metric. This means that at each point of M, the fibre of E is an inner product space. A spinor bundle of E is a prescription for consistently associating a spin representation to every point of M. There are topological obstructions to being able to do it, and consequently, a given bundle E may not admit any spinor bundle. In case it does, one says that the bundle E is spin.
This may be made rigorous through the language of principal bundles. The collection of oriented orthonormal frames of a vector bundle form a frame bundle PSO(E), which is a principal bundle under the action of the special orthogonal group SO(n). A spin structure for PSO(E) is a lift of PSO(E) to a principal bundle PSpin(E) under the action of the spin group Spin(n), by which we mean that there exists a bundle map $\phi $ : PSpin(E) → PSO(E) such that
$\phi (pg)=\phi (p)\rho (g)$, for all p ∈ PSpin(E) and g ∈ Spin(n),
where ρ : Spin(n) → SO(n) is the mapping of groups presenting the spin group as a double-cover of SO(n).
In the special case in which E is the tangent bundle TM over the base manifold M, if a spin structure exists then one says that M is a spin manifold. Equivalently M is spin if the SO(n) principal bundle of orthonormal bases of the tangent fibers of M is a Z2 quotient of a principal spin bundle.
If the manifold has a cell decomposition or a triangulation, a spin structure can equivalently be thought of as a homotopy-class of trivialization of the tangent bundle over the 1-skeleton that extends over the 2-skeleton. If the dimension is lower than 3, one first takes a Whitney sum with a trivial line bundle.
Obstruction and classification
For an orientable vector bundle $\pi _{E}:E\to M$ a spin structure exists on $E$ if and only if the second Stiefel–Whitney class $w_{2}(E)$ vanishes. This is a result of Armand Borel and Friedrich Hirzebruch.[6] Furthermore, in the case $E\to M$ is spin, the number of spin structures are in bijection with $H^{1}(M,\mathbb {Z} /2)$. These results can be easily proven[7]pg 110-111 using a spectral sequence argument for the associated principal $\operatorname {SO} (n)$-bundle $P_{E}\to M$. Notice this gives a fibration
$\operatorname {SO} (n)\to P_{E}\to M$
hence the Serre spectral sequence can be applied. From general theory of spectral sequences, there is an exact sequence
$0\to E_{3}^{0,1}\to E_{2}^{0,1}\xrightarrow {d_{2}} E_{2}^{2,0}\to E_{3}^{2,0}\to 0$
where
${\begin{aligned}E_{2}^{0,1}&=H^{0}(M,H^{1}(\operatorname {SO} (n),\mathbb {Z} /2))=H^{1}(\operatorname {SO} (n),\mathbb {Z} /2)\\E_{2}^{2,0}&=H^{2}(M,H^{0}(\operatorname {SO} (n),\mathbb {Z} /2))=H^{2}(M,\mathbb {Z} /2)\end{aligned}}$
In addition, $E_{\infty }^{0,1}=E_{3}^{0,1}$ and $E_{\infty }^{0,1}=H^{1}(P_{E},\mathbb {Z} /2)/F^{1}(H^{1}(P_{E},\mathbb {Z} /2))$ for some filtration on $H^{1}(P_{E},\mathbb {Z} /2)$, hence we get a map
$H^{1}(P_{E},\mathbb {Z} /2)\to E_{3}^{0,1}$
giving an exact sequence
$H^{1}(P_{E},\mathbb {Z} /2)\to H^{1}(\operatorname {SO} (n),\mathbb {Z} /2)\to H^{2}(M,\mathbb {Z} /2)$
Now, a spin structure is exactly a double covering of $P_{E}$ fitting into a commutative diagram
${\begin{matrix}\operatorname {Spin} (n)&\to &{\tilde {P}}_{E}&\to &M\\\downarrow &&\downarrow &&\downarrow \\\operatorname {SO} (n)&\to &P_{E}&\to &M\end{matrix}}$
where the two left vertical maps are the double covering maps. Now, double coverings of $P_{E}$ are in bijection with index $2$ subgroups of $\pi _{1}(P_{E})$, which is in bijection with the set of group morphisms ${\text{Hom}}(\pi _{1}(E),\mathbb {Z} /2)$. But, from Hurewicz theorem and change of coefficients, this is exactly the cohomology group $H^{1}(P_{E},\mathbb {Z} /2)$. Applying the same argument to $\operatorname {SO} (n)$, the non-trivial covering $\operatorname {Spin} (n)\to \operatorname {SO} (n)$ corresponds to $1\in H^{1}(\operatorname {SO} (n),\mathbb {Z} /2)=\mathbb {Z} /2$, and the map to $H^{2}(M,\mathbb {Z} /2)$ is precisely the $w_{2}$ of the second Stiefel–Whitney class, hence $w_{2}(1)=w_{2}(E)$. If it vanishes, then the inverse image of $1$ under the map
$H^{1}(P_{E},\mathbb {Z} /2)\to H^{1}(\operatorname {SO} (n),\mathbb {Z} /2)$
is the set of double coverings giving spin structures. Now, this subset of $H^{1}(P_{E},\mathbb {Z} /2)$ can be identified with $H^{1}(M,\mathbb {Z} /2)$, showing this latter cohomology group classifies the various spin structures on the vector bundle $E\to M$. This can be done by looking at the long exact sequence of homotopy groups of the fibration
$\pi _{1}(\operatorname {SO} (n))\to \pi _{1}(P_{E})\to \pi _{1}(M)\to 1$
and applying ${\text{Hom}}(-,\mathbb {Z} /2)$, giving the sequence of cohomology groups
$0\to H^{1}(M,\mathbb {Z} /2)\to H^{1}(P_{E},\mathbb {Z} /2)\to H^{1}(\operatorname {SO} (n),\mathbb {Z} /2)$
Because $H^{1}(M,\mathbb {Z} /2)$ is the kernel, and the inverse image of $1\in H^{1}(\operatorname {SO} (n),\mathbb {Z} /2)$ is in bijection with the kernel, we have the desired result.
Remarks on classification
When spin structures exist, the inequivalent spin structures on a manifold have a one-to-one correspondence (not canonical) with the elements of H1(M,Z2), which by the universal coefficient theorem is isomorphic to H1(M,Z2). More precisely, the space of the isomorphism classes of spin structures is an affine space over H1(M,Z2).
Intuitively, for each nontrivial cycle on M a spin structure corresponds to a binary choice of whether a section of the SO(N) bundle switches sheets when one encircles the loop. If w2[8] vanishes then these choices may be extended over the two-skeleton, then (by obstruction theory) they may automatically be extended over all of M. In particle physics this corresponds to a choice of periodic or antiperiodic boundary conditions for fermions going around each loop. Note that on a complex manifold $X$ the second Stiefel-Whitney class can be computed as the first chern class ${\text{mod }}2$.
Examples
1. A genus g Riemann surface admits 22g inequivalent spin structures; see theta characteristic.
2. If H2(M,Z2) vanishes, M is spin. For example, Sn is spin for all $n\neq 2$. (Note that S2 is also spin, but for different reasons; see below.)
3. The complex projective plane CP2 is not spin.
4. More generally, all even-dimensional complex projective spaces CP2n are not spin.
5. All odd-dimensional complex projective spaces CP2n+1 are spin.
6. All compact, orientable manifolds of dimension 3 or less are spin.
7. All Calabi–Yau manifolds are spin.
Properties
• The  genus of a spin manifold is an integer, and is an even integer if in addition the dimension is 4 mod 8.
In general the  genus is a rational invariant, defined for any manifold, but it is not in general an integer.
This was originally proven by Hirzebruch and Borel, and can be proven by the Atiyah–Singer index theorem, by realizing the  genus as the index of a Dirac operator – a Dirac operator is a square root of a second order operator, and exists due to the spin structure being a "square root". This was a motivating example for the index theorem.
SpinC structures
A spinC structure is analogous to a spin structure on an oriented Riemannian manifold,[9] but uses the SpinC group, which is defined instead by the exact sequence
$1\to \mathbb {Z} _{2}\to \operatorname {Spin} ^{\mathbf {C} }(n)\to \operatorname {SO} (n)\times \operatorname {U} (1)\to 1.$
To motivate this, suppose that κ : Spin(n) → U(N) is a complex spinor representation. The center of U(N) consists of the diagonal elements coming from the inclusion i : U(1) → U(N), i.e., the scalar multiples of the identity. Thus there is a homomorphism
$\kappa \times i\colon {\mathrm {Spin} }(n)\times {\mathrm {U} }(1)\to {\mathrm {U} }(N).$
This will always have the element (−1,−1) in the kernel. Taking the quotient modulo this element gives the group SpinC(n). This is the twisted product
${\mathrm {Spin} }^{\mathbb {C} }(n)={\mathrm {Spin} }(n)\times _{\mathbb {Z} _{2}}{\mathrm {U} }(1)\,,$
where U(1) = SO(2) = S1. In other words, the group SpinC(n) is a central extension of SO(n) by S1.
Viewed another way, SpinC(n) is the quotient group obtained from Spin(n) × Spin(2) with respect to the normal Z2 which is generated by the pair of covering transformations for the bundles Spin(n) → SO(n) and Spin(2) → SO(2) respectively. This makes the SpinC group both a bundle over the circle with fibre Spin(n), and a bundle over SO(n) with fibre a circle.[10][11]
The fundamental group π1(SpinC(n)) is isomorphic to Z if n ≠ 2, and to Z ⊕ Z if n = 2.
If the manifold has a cell decomposition or a triangulation, a spinC structure can be equivalently thought of as a homotopy class of complex structure over the 2-skeleton that extends over the 3-skeleton. Similarly to the case of spin structures, one takes a Whitney sum with a trivial line bundle if the manifold is odd-dimensional.
Yet another definition is that a spinC structure on a manifold N is a complex line bundle L over N together with a spin structure on TN ⊕ L.
Obstruction
A spinC structure exists when the bundle is orientable and the second Stiefel–Whitney class of the bundle E is in the image of the map H2(M, Z) → H2(M, Z/2Z) (in other words, the third integral Stiefel–Whitney class vanishes). In this case one says that E is spinC. Intuitively, the lift gives the Chern class of the square of the U(1) part of any obtained spinC bundle. By a theorem of Hopf and Hirzebruch, closed orientable 4-manifolds always admit a spinC structure.
Classification
When a manifold carries a spinC structure at all, the set of spinC structures forms an affine space. Moreover, the set of spinC structures has a free transitive action of H2(M, Z). Thus, spinC-structures correspond to elements of H2(M, Z) although not in a natural way.
Geometric picture
This has the following geometric interpretation, which is due to Edward Witten. When the spinC structure is nonzero this square root bundle has a non-integral Chern class, which means that it fails the triple overlap condition. In particular, the product of transition functions on a three-way intersection is not always equal to one, as is required for a principal bundle. Instead it is sometimes −1.
This failure occurs at precisely the same intersections as an identical failure in the triple products of transition functions of the obstructed spin bundle. Therefore, the triple products of transition functions of the full spinc bundle, which are the products of the triple product of the spin and U(1) component bundles, are either 12 = 1 or (−1)2 = 1 and so the spinC bundle satisfies the triple overlap condition and is therefore a legitimate bundle.
The details
The above intuitive geometric picture may be made concrete as follows. Consider the short exact sequence 0 → Z → Z → Z2 → 0, where the second arrow is multiplication by 2 and the third is reduction modulo 2. This induces a long exact sequence on cohomology, which contains
$\dots \longrightarrow {\textrm {H}}^{2}(M;\mathbf {Z} ){\stackrel {2}{\longrightarrow }}{\textrm {H}}^{2}(M;\mathbf {Z} )\longrightarrow {\textrm {H}}^{2}(M;\mathbf {Z} _{2}){\stackrel {\beta }{\longrightarrow }}{\textrm {H}}^{3}(M;\mathbf {Z} )\longrightarrow \dots ,$
where the second arrow is induced by multiplication by 2, the third is induced by restriction modulo 2 and the fourth is the associated Bockstein homomorphism β.
The obstruction to the existence of a spin bundle is an element w2 of H2(M,Z2). It reflects the fact that one may always locally lift an SO(n) bundle to a spin bundle, but one needs to choose a Z2 lift of each transition function, which is a choice of sign. The lift does not exist when the product of these three signs on a triple overlap is −1, which yields the Čech cohomology picture of w2.
To cancel this obstruction, one tensors this spin bundle with a U(1) bundle with the same obstruction w2. Notice that this is an abuse of the word bundle, as neither the spin bundle nor the U(1) bundle satisfies the triple overlap condition and so neither is actually a bundle.
A legitimate U(1) bundle is classified by its Chern class, which is an element of H2(M,Z). Identify this class with the first element in the above exact sequence. The next arrow doubles this Chern class, and so legitimate bundles will correspond to even elements in the second H2(M, Z), while odd elements will correspond to bundles that fail the triple overlap condition. The obstruction then is classified by the failure of an element in the second H2(M,Z) to be in the image of the arrow, which, by exactness, is classified by its image in H2(M,Z2) under the next arrow.
To cancel the corresponding obstruction in the spin bundle, this image needs to be w2. In particular, if w2 is not in the image of the arrow, then there does not exist any U(1) bundle with obstruction equal to w2 and so the obstruction cannot be cancelled. By exactness, w2 is in the image of the preceding arrow only if it is in the kernel of the next arrow, which we recall is the Bockstein homomorphism β. That is, the condition for the cancellation of the obstruction is
$W_{3}=\beta w_{2}=0$
where we have used the fact that the third integral Stiefel–Whitney class W3 is the Bockstein of the second Stiefel–Whitney class w2 (this can be taken as a definition of W3).
Integral lifts of Stiefel–Whitney classes
This argument also demonstrates that second Stiefel–Whitney class defines elements not only of Z2 cohomology but also of integral cohomology in one higher degree. In fact this is the case for all even Stiefel–Whitney classes. It is traditional to use an uppercase W for the resulting classes in odd degree, which are called the integral Stiefel–Whitney classes, and are labeled by their degree (which is always odd).
Examples
1. All oriented smooth manifolds of dimension 4 or less are spinC.[12]
2. All almost complex manifolds are spinC.
3. All spin manifolds are spinC.
Application to particle physics
In particle physics the spin–statistics theorem implies that the wavefunction of an uncharged fermion is a section of the associated vector bundle to the spin lift of an SO(N) bundle E. Therefore, the choice of spin structure is part of the data needed to define the wavefunction, and one often needs to sum over these choices in the partition function. In many physical theories E is the tangent bundle, but for the fermions on the worldvolumes of D-branes in string theory it is a normal bundle.
In quantum field theory charged spinors are sections of associated spinc bundles, and in particular no charged spinors can exist on a space that is not spinc. An exception arises in some supergravity theories where additional interactions imply that other fields may cancel the third Stiefel–Whitney class. The mathematical description of spinors in supergravity and string theory is a particularly subtle open problem, which was recently addressed in references.[13][14] It turns out that the standard notion of spin structure is too restrictive for applications to supergravity and string theory, and that the correct notion of spinorial structure for the mathematical formulation of these theories is a "Lipschitz structure".[13][15]
See also
• Metaplectic structure
• Orthonormal frame bundle
• Spinor
References
1. Haefliger, A. (1956). "Sur l'extension du groupe structural d'un espace fibré". C. R. Acad. Sci. Paris. 243: 558–560.
2. J. Milnor (1963). "Spin structures on manifolds". L'Enseignement Mathématique. 9: 198–203.
3. Lichnerowicz, A. (1964). "Champs spinoriels et propagateurs en rélativité générale". Bull. Soc. Math. Fr. 92: 11–100. doi:10.24033/bsmf.1604.
4. Karoubi, M. (1968). "Algèbres de Clifford et K-théorie". Ann. Sci. Éc. Norm. Supér. 1 (2): 161–270. doi:10.24033/asens.1163.
5. Alagia, H. R.; Sánchez, C. U. (1985), "Spin structures on pseudo-Riemannian manifolds" (PDF), Revista de la Unión Matemática Argentina, 32: 64–78
6. Borel, A.; Hirzebruch, F. (1958). "Characteristic classes and homogeneous spaces I". American Journal of Mathematics. 80 (2): 97–136. doi:10.2307/2372795. JSTOR 2372795.
7. Pati, Vishwambhar. "Elliptic complexes and index theory" (PDF). Archived (PDF) from the original on 20 Aug 2018.
8. "Spin manifold and the second Stiefel-Whitney class". Math.Stachexchange.
9. Lawson, H. Blaine; Michelsohn, Marie-Louise (1989). Spin Geometry. Princeton University Press. p. 391. ISBN 978-0-691-08542-5.
10. R. Gompf (1997). "Spinc–structures and homotopy equivalences". Geometry & Topology. 1: 41–50. arXiv:math/9705218. Bibcode:1997math......5218G. doi:10.2140/gt.1997.1.41. S2CID 6906852.
11. Friedrich, Thomas (2000). Dirac Operators in Riemannian Geometry. American Mathematical Society. p. 26. ISBN 978-0-8218-2055-1.
12. Gompf, Robert E.; Stipsicz, Andras I. (1999). 4-Manifolds and Kirby Calculus. American Mathematical Society. pp. 55–58, 186–187. ISBN 0-8218-0994-6.
13. Lazaroiu, C.; Shahbazi, C.S. (2019). "Real pinor bundles and real Lipschitz structures". Asian Journal of Mathematics. 23 (5): 749–836. arXiv:1606.07894. doi:10.4310/AJM.2019.v23.n5.a3. S2CID 119598006..
14. Lazaroiu, C.; Shahbazi, C.S. (2019). "On the spin geometry of supergravity and string theory". Geometric Methods in Physics XXXVI. Trends in Mathematics. pp. 229–235. arXiv:1607.02103. doi:10.1007/978-3-030-01156-7_25. ISBN 978-3-030-01155-0. S2CID 104292702.
15. Friedrich, Thomas; Trautman, Andrzej (2000). "Spin spaces, Lipschitz groups, and spinor bundles". Annals of Global Analysis and Geometry. 18 (3): 221–240. arXiv:math/9901137. doi:10.1023/A:1006713405277. S2CID 118698159.
Further reading
• Lawson, H. Blaine; Michelsohn, Marie-Louise (1989). Spin Geometry. Princeton University Press. ISBN 978-0-691-08542-5.
• Friedrich, Thomas (2000). Dirac Operators in Riemannian Geometry. American Mathematical Society. ISBN 978-0-8218-2055-1.
• Karoubi, Max (2008). K-Theory. Springer. pp. 212–214. ISBN 978-3-540-79889-7.
• Greub, Werner; Petry, Herbert-Rainer (2006) [1978]. "On the lifting of structure groups". Differential Geometrical Methods in Mathematical Physics II. Lecture Notes in Mathematics. Vol. 676. Springer-Verlag. pp. 217–246. doi:10.1007/BFb0063673. ISBN 9783540357216.
• Scorpan, Alexandru (2005). "4.5 Notes Spin structures, the structure group definition; Equivalence of the definitions of". The wild world of 4-manifolds. American Mathematical Society. pp. 174–189. ISBN 9780821837498.
External links
• Something on Spin Structures by Sven-S. Porst is a short introduction to orientation and spin structures for mathematics students.
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Wikipedia
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Spin-weighted spherical harmonics
In special functions, a topic in mathematics, spin-weighted spherical harmonics are generalizations of the standard spherical harmonics and—like the usual spherical harmonics—are functions on the sphere. Unlike ordinary spherical harmonics, the spin-weighted harmonics are U(1) gauge fields rather than scalar fields: mathematically, they take values in a complex line bundle. The spin-weighted harmonics are organized by degree l, just like ordinary spherical harmonics, but have an additional spin weight s that reflects the additional U(1) symmetry. A special basis of harmonics can be derived from the Laplace spherical harmonics Ylm, and are typically denoted by sYlm, where l and m are the usual parameters familiar from the standard Laplace spherical harmonics. In this special basis, the spin-weighted spherical harmonics appear as actual functions, because the choice of a polar axis fixes the U(1) gauge ambiguity. The spin-weighted spherical harmonics can be obtained from the standard spherical harmonics by application of spin raising and lowering operators. In particular, the spin-weighted spherical harmonics of spin weight s = 0 are simply the standard spherical harmonics:
${}_{0}Y_{lm}=Y_{lm}\ .$
Spaces of spin-weighted spherical harmonics were first identified in connection with the representation theory of the Lorentz group (Gelfand, Minlos & Shapiro 1958). They were subsequently and independently rediscovered by Newman & Penrose (1966) and applied to describe gravitational radiation, and again by Wu & Yang (1976) as so-called "monopole harmonics" in the study of Dirac monopoles.
Spin-weighted functions
Regard the sphere S2 as embedded into the three-dimensional Euclidean space R3. At a point x on the sphere, a positively oriented orthonormal basis of tangent vectors at x is a pair a, b of vectors such that
${\begin{aligned}\mathbf {x} \cdot \mathbf {a} =\mathbf {x} \cdot \mathbf {b} &=0\\\mathbf {a} \cdot \mathbf {a} =\mathbf {b} \cdot \mathbf {b} &=1\\\mathbf {a} \cdot \mathbf {b} &=0\\\mathbf {x} \cdot (\mathbf {a} \times \mathbf {b} )&>0,\end{aligned}}$
where the first pair of equations states that a and b are tangent at x, the second pair states that a and b are unit vectors, the penultimate equation that a and b are orthogonal, and the final equation that (x, a, b) is a right-handed basis of R3.
A spin-weight s function f is a function accepting as input a point x of S2 and a positively oriented orthonormal basis of tangent vectors at x, such that
$f{\bigl (}\mathbf {x} ,(\cos \theta )\mathbf {a} -(\sin \theta )\mathbf {b} ,(\sin \theta )\mathbf {a} +(\cos \theta )\mathbf {b} {\bigr )}=e^{is\theta }f(\mathbf {x} ,\mathbf {a} ,\mathbf {b} )$
for every rotation angle θ.
Following Eastwood & Tod (1982), denote the collection of all spin-weight s functions by B(s). Concretely, these are understood as functions f on C2\{0} satisfying the following homogeneity law under complex scaling
$f\left(\lambda z,{\overline {\lambda }}{\bar {z}}\right)=\left({\frac {\overline {\lambda }}{\lambda }}\right)^{s}f\left(z,{\bar {z}}\right).$
This makes sense provided s is a half-integer.
Abstractly, B(s) is isomorphic to the smooth vector bundle underlying the antiholomorphic vector bundle O(2s) of the Serre twist on the complex projective line CP1. A section of the latter bundle is a function g on C2\{0} satisfying
$g\left(\lambda z,{\overline {\lambda }}{\bar {z}}\right)={\overline {\lambda }}^{2s}g\left(z,{\bar {z}}\right).$
Given such a g, we may produce a spin-weight s function by multiplying by a suitable power of the hermitian form
$P\left(z,{\bar {z}}\right)=z\cdot {\bar {z}}.$
Specifically, f = P−sg is a spin-weight s function. The association of a spin-weighted function to an ordinary homogeneous function is an isomorphism.
The operator ð
The spin weight bundles B(s) are equipped with a differential operator ð (eth). This operator is essentially the Dolbeault operator, after suitable identifications have been made,
$\partial :{\overline {\mathbf {O} (2s)}}\to {\mathcal {E}}^{1,0}\otimes {\overline {\mathbf {O} (2s)}}\cong {\overline {\mathbf {O} (2s)}}\otimes \mathbf {O} (-2).$ :{\overline {\mathbf {O} (2s)}}\to {\mathcal {E}}^{1,0}\otimes {\overline {\mathbf {O} (2s)}}\cong {\overline {\mathbf {O} (2s)}}\otimes \mathbf {O} (-2).}
Thus for f ∈ B(s),
$\eth f\ {\stackrel {\text{def}}{=}}\ P^{-s+1}\partial \left(P^{s}f\right)$
defines a function of spin-weight s + 1.
Spin-weighted harmonics
Just as conventional spherical harmonics are the eigenfunctions of the Laplace-Beltrami operator on the sphere, the spin-weight s harmonics are the eigensections for the Laplace-Beltrami operator acting on the bundles E(s) of spin-weight s functions.
Representation as functions
The spin-weighted harmonics can be represented as functions on a sphere once a point on the sphere has been selected to serve as the North pole. By definition, a function η with spin weight s transforms under rotation about the pole via
$\eta \rightarrow e^{is\psi }\eta .$
Working in standard spherical coordinates, we can define a particular operator ð acting on a function η as:
$\eth \eta =-\left(\sin {\theta }\right)^{s}\left\{{\frac {\partial }{\partial \theta }}+{\frac {i}{\sin {\theta }}}{\frac {\partial }{\partial \phi }}\right\}\left[\left(\sin {\theta }\right)^{-s}\eta \right].$
This gives us another function of θ and φ. (The operator ð is effectively a covariant derivative operator in the sphere.)
An important property of the new function ðη is that if η had spin weight s, ðη has spin weight s + 1. Thus, the operator raises the spin weight of a function by 1. Similarly, we can define an operator ð which will lower the spin weight of a function by 1:
${\bar {\eth }}\eta =-\left(\sin {\theta }\right)^{-s}\left\{{\frac {\partial }{\partial \theta }}-{\frac {i}{\sin {\theta }}}{\frac {\partial }{\partial \phi }}\right\}\left[\left(\sin {\theta }\right)^{s}\eta \right].$
The spin-weighted spherical harmonics are then defined in terms of the usual spherical harmonics as:
${}_{s}Y_{lm}={\begin{cases}{\sqrt {\frac {(l-s)!}{(l+s)!}}}\ \eth ^{s}Y_{lm},&&0\leq s\leq l;\\{\sqrt {\frac {(l+s)!}{(l-s)!}}}\ \left(-1\right)^{s}{\bar {\eth }}^{-s}Y_{lm},&&-l\leq s\leq 0;\\0,&&l<|s|.\end{cases}}$
The functions sYlm then have the property of transforming with spin weight s.
Other important properties include the following:
${\begin{aligned}\eth \left({}_{s}Y_{lm}\right)&=+{\sqrt {(l-s)(l+s+1)}}\,{}_{s+1}Y_{lm};\\{\bar {\eth }}\left({}_{s}Y_{lm}\right)&=-{\sqrt {(l+s)(l-s+1)}}\,{}_{s-1}Y_{lm};\end{aligned}}$
Orthogonality and completeness
The harmonics are orthogonal over the entire sphere:
$\int _{S^{2}}{}_{s}Y_{lm}\,{}_{s}{\bar {Y}}_{l'm'}\,dS=\delta _{ll'}\delta _{mm'},$
and satisfy the completeness relation
$\sum _{lm}{}_{s}{\bar {Y}}_{lm}\left(\theta ',\phi '\right){}_{s}Y_{lm}(\theta ,\phi )=\delta \left(\phi '-\phi \right)\delta \left(\cos \theta '-\cos \theta \right)$
Calculating
These harmonics can be explicitly calculated by several methods. The obvious recursion relation results from repeatedly applying the raising or lowering operators. Formulae for direct calculation were derived by Goldberg et al. (1967) harvtxt error: no target: CITEREFGoldbergMacfarlaneNewmanRohlich1967 (help). Note that their formulae use an old choice for the Condon–Shortley phase. The convention chosen below is in agreement with Mathematica, for instance.
The more useful of the Goldberg, et al., formulae is the following:
${}_{s}Y_{lm}(\theta ,\phi )=\left(-1\right)^{l+m-s}{\sqrt {\frac {(l+m)!(l-m)!(2l+1)}{4\pi (l+s)!(l-s)!}}}\sin ^{2l}\left({\frac {\theta }{2}}\right)e^{im\phi }\times \sum _{r=0}^{l-s}\left(-1\right)^{r}{l-s \choose r}{l+s \choose r+s-m}\cot ^{2r+s-m}\left({\frac {\theta }{2}}\right)\,.$
A Mathematica notebook using this formula to calculate arbitrary spin-weighted spherical harmonics can be found here.
With the phase convention here:
${\begin{aligned}{}_{s}{\bar {Y}}_{lm}&=\left(-1\right)^{s+m}{}_{-s}Y_{l(-m)}\\{}_{s}Y_{lm}(\pi -\theta ,\phi +\pi )&=\left(-1\right)^{l}{}_{-s}Y_{lm}(\theta ,\phi ).\end{aligned}}$
First few spin-weighted spherical harmonics
Analytic expressions for the first few orthonormalized spin-weighted spherical harmonics:
Spin-weight s = 1, degree l = 1
${\begin{aligned}{}_{1}Y_{10}(\theta ,\phi )&={\sqrt {\frac {3}{8\pi }}}\,\sin \theta \\{}_{1}Y_{1\pm 1}(\theta ,\phi )&=-{\sqrt {\frac {3}{16\pi }}}(1\mp \cos \theta )\,e^{\pm i\phi }\end{aligned}}$
Relation to Wigner rotation matrices
$D_{-ms}^{l}(\phi ,\theta ,-\psi )=\left(-1\right)^{m}{\sqrt {\frac {4\pi }{2l+1}}}{}_{s}Y_{lm}(\theta ,\phi )e^{is\psi }$
This relation allows the spin harmonics to be calculated using recursion relations for the D-matrices.
Triple integral
The triple integral in the case that s1 + s2 + s3 = 0 is given in terms of the 3-j symbol:
$\int _{S^{2}}\,{}_{s_{1}}Y_{j_{1}m_{1}}\,{}_{s_{2}}Y_{j_{2}m_{2}}\,{}_{s_{3}}Y_{j_{3}m_{3}}={\sqrt {\frac {\left(2j_{1}+1\right)\left(2j_{2}+1\right)\left(2j_{3}+1\right)}{4\pi }}}{\begin{pmatrix}j_{1}&j_{2}&j_{3}\\m_{1}&m_{2}&m_{3}\end{pmatrix}}{\begin{pmatrix}j_{1}&j_{2}&j_{3}\\-s_{1}&-s_{2}&-s_{3}\end{pmatrix}}$
See also
• Spherical basis
References
• Dray, Tevian (May 1985), "The relationship between monopole harmonics and spin-weighted spherical harmonics", J. Math. Phys., American Institute of Physics, 26 (5): 1030–1033, Bibcode:1985JMP....26.1030D, doi:10.1063/1.526533.
• Eastwood, Michael; Tod, Paul (1982), "Edth-a differential operator on the sphere", Mathematical Proceedings of the Cambridge Philosophical Society, 92 (2): 317–330, Bibcode:1982MPCPS..92..317E, doi:10.1017/S0305004100059971, S2CID 121025245.
• Gelfand, I. M.; Minlos, Robert A.; Shapiro, Z. Ja. (1958), Predstavleniya gruppy vrashcheni i gruppy Lorentsa, ikh primeneniya, Gosudarstv. Izdat. Fiz.-Mat. Lit., Moscow, MR 0114876; (1963) Representations of the rotation and Lorentz groups and their applications (translation). Macmillan Publishers.
• Goldberg, J. N.; Macfarlane, A. J.; Newman, E. T.; Rohrlich, F.; Sudarshan, E. C. G. (November 1967), "Spin-s Spherical Harmonics and ð", J. Math. Phys., American Institute of Physics, 8 (11): 2155–2161, Bibcode:1967JMP.....8.2155G, doi:10.1063/1.1705135 (Note: As mentioned above, this paper uses a choice for the Condon-Shortley phase that is no longer standard.)
• Newman, E. T.; Penrose, R. (May 1966), "Note on the Bondi-Metzner-Sachs Group", J. Math. Phys., American Institute of Physics, 7 (5): 863–870, Bibcode:1966JMP.....7..863N, doi:10.1063/1.1931221.
• Wu, Tai Tsun; Yang, Chen Ning (1976), "Dirac monopole without strings: monopole harmonics", Nuclear Physics B, 107 (3): 365–380, Bibcode:1976NuPhB.107..365W, doi:10.1016/0550-3213(76)90143-7, MR 0471791.
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Wikipedia
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Spinor bundle
In differential geometry, given a spin structure on an $n$-dimensional orientable Riemannian manifold $(M,g),\,$ one defines the spinor bundle to be the complex vector bundle $\pi _{\mathbf {S} }\colon {\mathbf {S} }\to M\,$ associated to the corresponding principal bundle $\pi _{\mathbf {P} }\colon {\mathbf {P} }\to M\,$ of spin frames over $M$ and the spin representation of its structure group ${\mathrm {Spin} }(n)\,$ on the space of spinors $\Delta _{n}$.
A section of the spinor bundle ${\mathbf {S} }\,$ is called a spinor field.
Formal definition
Let $({\mathbf {P} },F_{\mathbf {P} })$ be a spin structure on a Riemannian manifold $(M,g),\,$that is, an equivariant lift of the oriented orthonormal frame bundle $\mathrm {F} _{SO}(M)\to M$ with respect to the double covering $\rho \colon {\mathrm {Spin} }(n)\to {\mathrm {SO} }(n)$ of the special orthogonal group by the spin group.
The spinor bundle ${\mathbf {S} }\,$ is defined [1] to be the complex vector bundle
${\mathbf {S} }={\mathbf {P} }\times _{\kappa }\Delta _{n}\,$
associated to the spin structure ${\mathbf {P} }$ via the spin representation $\kappa \colon {\mathrm {Spin} }(n)\to {\mathrm {U} }(\Delta _{n}),\,$ where ${\mathrm {U} }({\mathbf {W} })\,$ denotes the group of unitary operators acting on a Hilbert space ${\mathbf {W} }.\,$ It is worth noting that the spin representation $\kappa $ is a faithful and unitary representation of the group ${\mathrm {Spin} }(n).$[2]
See also
• Clifford bundle
• Clifford module bundle
• Orthonormal frame bundle
• Spin geometry
• Spinor
• Spinor representation
Notes
1. Friedrich, Thomas (2000), Dirac Operators in Riemannian Geometry, American Mathematical Society, ISBN 978-0-8218-2055-1 page 53
2. Friedrich, Thomas (2000), Dirac Operators in Riemannian Geometry, American Mathematical Society, ISBN 978-0-8218-2055-1 pages 20 and 24
Further reading
• Lawson, H. Blaine; Michelsohn, Marie-Louise (1989). Spin Geometry. Princeton University Press. ISBN 978-0-691-08542-5.
• Friedrich, Thomas (2000), Dirac Operators in Riemannian Geometry, American Mathematical Society, ISBN 978-0-8218-2055-1
Manifolds (Glossary)
Basic concepts
• Topological manifold
• Atlas
• Differentiable/Smooth manifold
• Differential structure
• Smooth atlas
• Submanifold
• Riemannian manifold
• Smooth map
• Submersion
• Pushforward
• Tangent space
• Differential form
• Vector field
Main results (list)
• Atiyah–Singer index
• Darboux's
• De Rham's
• Frobenius
• Generalized Stokes
• Hopf–Rinow
• Noether's
• Sard's
• Whitney embedding
Maps
• Curve
• Diffeomorphism
• Local
• Geodesic
• Exponential map
• in Lie theory
• Foliation
• Immersion
• Integral curve
• Lie derivative
• Section
• Submersion
Types of
manifolds
• Closed
• (Almost) Complex
• (Almost) Contact
• Fibered
• Finsler
• Flat
• G-structure
• Hadamard
• Hermitian
• Hyperbolic
• Kähler
• Kenmotsu
• Lie group
• Lie algebra
• Manifold with boundary
• Oriented
• Parallelizable
• Poisson
• Prime
• Quaternionic
• Hypercomplex
• (Pseudo−, Sub−) Riemannian
• Rizza
• (Almost) Symplectic
• Tame
Tensors
Vectors
• Distribution
• Lie bracket
• Pushforward
• Tangent space
• bundle
• Torsion
• Vector field
• Vector flow
Covectors
• Closed/Exact
• Covariant derivative
• Cotangent space
• bundle
• De Rham cohomology
• Differential form
• Vector-valued
• Exterior derivative
• Interior product
• Pullback
• Ricci curvature
• flow
• Riemann curvature tensor
• Tensor field
• density
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Connections
• Affine
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Related
• Classification of manifolds
• Gauge theory
• History
• Morse theory
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Generalizations
• Banach manifold
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• Fréchet manifold
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• over commutative algebras
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Tensors
Glossary of tensor theory
Scope
Mathematics
• Coordinate system
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Notation
• Abstract index notation
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Tensor
definitions
• Tensor (intrinsic definition)
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Operations
• Covariant derivative
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Related
abstractions
• Affine connection
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Notable tensors
Mathematics
• Kronecker delta
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Physics
• Moment of inertia
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• stress–energy tensor
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Mathematicians
• Élie Cartan
• Augustin-Louis Cauchy
• Elwin Bruno Christoffel
• Albert Einstein
• Leonhard Euler
• Carl Friedrich Gauss
• Hermann Grassmann
• Tullio Levi-Civita
• Gregorio Ricci-Curbastro
• Bernhard Riemann
• Jan Arnoldus Schouten
• Woldemar Voigt
• Hermann Weyl
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Wikipedia
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Spinor genus
In mathematics, the spinor genus is a classification of quadratic forms and lattices over the ring of integers, introduced by Martin Eichler. It refines the genus but may be coarser than proper equivalence.
Definitions
We define two Z-lattices L and M in a quadratic space V over Q to be spinor equivalent if there exists a transformation g in the proper orthogonal group O+(V) and for every prime p there exists a local transformation fp of Vp of spinor norm 1 such that M = g fpLp.
A spinor genus is an equivalence class for this equivalence relation. Properly equivalent lattices are in the same spinor genus, and lattices in the same spinor genus are in the same genus. The number of spinor genera in a genus is a power of two, and can be determined effectively.
Results
An important result is that for indefinite forms of dimension at least three, each spinor genus contains exactly one proper equivalence class.
See also
• Genus of a quadratic form
References
• Cassels, J. W. S. (1978). Rational Quadratic Forms. London Mathematical Society Monographs. Vol. 13. Academic Press. ISBN 0-12-163260-1. Zbl 0395.10029.
• Conway, J. H.; Sloane, N. J. A. Sphere packings, lattices and groups. Grundlehren der Mathematischen Wissenschaften. Vol. 290. With contributions by Bannai, E.; Borcherds, R. E.; Leech, J.; Norton, S. P.; Odlyzko, A. M.; Parker, R. A.; Queen, L.; Venkov, B. B. (3rd ed.). New York, NY: Springer-Verlag. ISBN 0-387-98585-9. Zbl 0915.52003.
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Wikipedia
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Spinors in three dimensions
In mathematics, the spinor concept as specialised to three dimensions can be treated by means of the traditional notions of dot product and cross product. This is part of the detailed algebraic discussion of the rotation group SO(3).
Formulation
The association of a spinor with a 2×2 complex Hermitian matrix was formulated by Élie Cartan.[1]
In detail, given a vector x = (x1, x2, x3) of real (or complex) numbers, one can associate the complex matrix
${\vec {x}}\rightarrow X\ =\left({\begin{matrix}x_{3}&x_{1}-ix_{2}\\x_{1}+ix_{2}&-x_{3}\end{matrix}}\right).$
In physics, this is often written as a dot product $X\equiv {\vec {\sigma }}\cdot {\vec {x}}$, where ${\vec {\sigma }}\equiv (\sigma _{1},\sigma _{2},\sigma _{3})$ is the vector form of Pauli matrices. Matrices of this form have the following properties, which relate them intrinsically to the geometry of 3-space:
• $\det X=-|{\vec {x}}|^{2}$, where $\det $ denotes the determinant.
• $X^{2}=|{\vec {x}}|^{2}I$, where I is the identity matrix.
• ${\frac {1}{2}}(XY+YX)=({\vec {x}}\cdot {\vec {y}})I$ [1]: 43
• ${\frac {1}{2}}(XY-YX)=iZ$ where Z is the matrix associated to the cross product ${\vec {z}}={\vec {x}}\times {\vec {y}}$.
• If ${\vec {u}}$ is a unit vector, then $-UXU$ is the matrix associated with the vector that results from reflecting ${\vec {x}}$ in the plane orthogonal to ${\vec {u}}$.
The last property can be used to simplify rotational operations. It is an elementary fact from linear algebra that any rotation in 3-space factors as a composition of two reflections. (More generally, any orientation-reversing orthogonal transformation is either a reflection or the product of three reflections.) Thus if R is a rotation which decomposes as the reflection in the plane perpendicular to a unit vector ${\vec {u}}_{1}$ followed by the reflection in the plane perpendicular to ${\vec {u}}_{2}$, then the matrix $U_{2}U_{1}XU_{1}U_{2}$ represents the rotation of the vector ${\vec {x}}$ through R.
Having effectively encoded all the rotational linear geometry of 3-space into a set of complex 2×2 matrices, it is natural to ask what role, if any, the 2×1 matrices (i.e., the column vectors) play. Provisionally, a spinor is a column vector
$\xi =\left[{\begin{matrix}\xi _{1}\\\xi _{2}\end{matrix}}\right],$ with complex entries ξ1 and ξ2.
The space of spinors is evidently acted upon by complex 2×2 matrices. As shown above, the product of two reflections in a pair of unit vectors defines a 2×2 matrix whose action on euclidean vectors is a rotation. So there is an action of rotations on spinors. However, there is one important caveat: the factorization of a rotation is not unique. Clearly, if $X\mapsto RXR^{-1}$ is a representation of a rotation, then replacing R by −R will yield the same rotation. In fact, one can easily show that this is the only ambiguity which arises. Thus the action of a rotation on a spinor is always double-valued.
History
There were some precursors to Cartan's work with 2×2 complex matrices: Wolfgang Pauli had used these matrices so intensively that elements of a certain basis of a four-dimensional subspace are called Pauli matrices σi, so that the Hermitian matrix is written as a Pauli vector ${\vec {x}}\cdot {\vec {\sigma }}.$[2] In the mid 19th century the algebraic operations of this algebra of four complex dimensions were studied as biquaternions.
Michael Stone and Paul Goldbar, in Mathematics for Physics, contest this, saying, "The spin representations were discovered by ´Elie Cartan in 1913, some years before they were needed in physics."
Formulation using isotropic vectors
Spinors can be constructed directly from isotropic vectors in 3-space without using the quaternionic construction. To motivate this introduction of spinors, suppose that X is a matrix representing a vector x in complex 3-space. Suppose further that x is isotropic: i.e.,
${\mathbf {x} }\cdot {\mathbf {x} }=x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=0.$
Then since the determinant of X is zero there is a proportionality between its rows or columns. Thus the matrix may be written as an outer product of two complex 2-vectors:
$X=2\left[{\begin{matrix}\xi _{1}\\\xi _{2}\end{matrix}}\right]\left[{\begin{matrix}-\xi _{2}&\xi _{1}\end{matrix}}\right].$
This factorization yields an overdetermined system of equations in the coordinates of the vector x:
$\left.{\begin{matrix}\xi _{1}^{2}-\xi _{2}^{2}&=x_{1}\\i(\xi _{1}^{2}+\xi _{2}^{2})&=x_{2}\\-2\xi _{1}\xi _{2}&=x_{3}\end{matrix}}\right\}$
(1)
subject to the constraint
$x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=0.$
(2)
This system admits the solutions
$\xi _{1}=\pm {\sqrt {\frac {x_{1}-ix_{2}}{2}}},\quad \xi _{2}=\pm {\sqrt {\frac {-x_{1}-ix_{2}}{2}}}.$
(3)
Either choice of sign solves the system (1). Thus a spinor may be viewed as an isotropic vector, along with a choice of sign. Note that because of the logarithmic branching, it is impossible to choose a sign consistently so that (3) varies continuously along a full rotation among the coordinates x. In spite of this ambiguity of the representation of a rotation on a spinor, the rotations do act unambiguously by a fractional linear transformation on the ratio ξ1:ξ2 since one choice of sign in the solution (3) forces the choice of the second sign. In particular, the space of spinors is a projective representation of the orthogonal group.
As a consequence of this point of view, spinors may be regarded as a kind of "square root" of isotropic vectors. Specifically, introducing the matrix
$C=\left({\begin{matrix}0&1\\-1&0\end{matrix}}\right),$
the system (1) is equivalent to solving X = 2 ξ tξ C for the undetermined spinor ξ.
A fortiori, if the roles of ξ and x are now reversed, the form Q(ξ) = x defines, for each spinor ξ, a vector x quadratically in the components of ξ. If this quadratic form is polarized, it determines a bilinear vector-valued form on spinors Q(μ, ξ). This bilinear form then transforms tensorially under a reflection or a rotation.
Reality
The above considerations apply equally well whether the original euclidean space under consideration is real or complex. When the space is real, however, spinors possess some additional structure which in turn facilitates a complete description of the representation of the rotation group. Suppose, for simplicity, that the inner product on 3-space has positive-definite signature:
$\left|x\right|^{2}=x_{1}^{2}+x_{2}^{2}+x_{3}^{2}$
(4)
With this convention, real vectors correspond to Hermitian matrices. Furthermore, real rotations preserving the form (4) correspond (in the double-valued sense) to unitary matrices of determinant one. In modern terms, this presents the special unitary group SU(2) as a double cover of SO(3). As a consequence, the spinor Hermitian product
$\langle \mu |\xi \rangle ={\bar {\mu }}_{1}\xi _{1}+{\bar {\mu }}_{2}\xi _{2}$
(5)
is preserved by all rotations, and therefore is canonical.
If, however, the signature of the inner product on 3-space is indefinite (i.e., non-degenerate, but also not positive definite), then the foregoing analysis must be adjusted to reflect this. Suppose then that the length form on 3-space is given by:
$\left|\mathbf {x} \right|^{2}=x_{1}^{2}-x_{2}^{2}+x_{3}^{2}$
(4′)
Then the construction of spinors of the preceding sections proceeds, but with $x_{2}$ replacing $i$ $x_{2}$ in all the formulas. With this new convention, the matrix associated to a real vector $(x_{1},x_{2},x_{3})$ is itself real:
$\left({\begin{matrix}x_{3}&x_{1}-x_{2}\\x_{1}+x_{2}&-x_{3}\end{matrix}}\right)$.
The form (5) is no longer invariant under a real rotation (or reversal), since the group stabilizing (4′) is now a Lorentz group O(2,1). Instead, the anti-Hermitian form
$\langle \mu |\xi \rangle ={\bar {\mu }}_{1}\xi _{2}-{\bar {\mu }}_{2}\xi _{1}$
defines the appropriate notion of inner product for spinors in this metric signature. This form is invariant under transformations in the connected component of the identity of O(2,1).
In either case, the quartic form
$\langle \mu |\xi \rangle ^{2}={\hbox{length}}\left(Q({\bar {\mu }},\xi )\right)^{2}$
is fully invariant under O(3) (or O(2,1), respectively), where Q is the vector-valued bilinear form described in the previous section. The fact that this is a quartic invariant, rather than quadratic, has an important consequence. If one confines attention to the group of special orthogonal transformations, then it is possible unambiguously to take the square root of this form and obtain an identification of spinors with their duals. In the language of representation theory, this implies that there is only one irreducible spin representation of SO(3) (or SO(2,1)) up to isomorphism. If, however, reversals (e.g., reflections in a plane) are also allowed, then it is no longer possible to identify spinors with their duals owing to a change of sign on the application of a reflection. Thus there are two irreducible spin representations of O(3) (or O(2,1)), sometimes called the pin representations.
Reality structures
The differences between these two signatures can be codified by the notion of a reality structure on the space of spinors. Informally, this is a prescription for taking a complex conjugate of a spinor, but in such a way that this may not correspond to the usual conjugate per the components of a spinor. Specifically, a reality structure is specified by a Hermitian 2 × 2 matrix K whose product with itself is the identity matrix: K2 = Id. The conjugate of a spinor with respect to a reality structure K is defined by
$\xi ^{*}=K{\bar {\xi }}.$
The particular form of the inner product on vectors (e.g., (4) or (4′)) determines a reality structure (up to a factor of -1) by requiring
${\bar {X}}=KXK\,$, whenever X is a matrix associated to a real vector.
Thus K = i C is the reality structure in Euclidean signature (4), and K = Id is that for signature (4′). With a reality structure in hand, one has the following results:
• X is the matrix associated to a real vector if, and only if,
${\bar {X}}=KXK\,.$
• If μ and ξ is a spinor, then the inner product
$\langle \mu |\xi \rangle =i\,^{t}\mu ^{*}C\xi $
determines a Hermitian form which is invariant under proper orthogonal transformations.
Examples in physics
Spinors of the Pauli spin matrices
See also: quaternions and spatial rotation
Often, the first example of spinors that a student of physics encounters are the 2×1 spinors used in Pauli's theory of electron spin. The Pauli matrices are a vector of three 2×2 matrices that are used as spin operators.
Given a unit vector in 3 dimensions, for example (a, b, c), one takes a dot product with the Pauli spin matrices to obtain a spin matrix for spin in the direction of the unit vector.
The eigenvectors of that spin matrix are the spinors for spin-1/2 oriented in the direction given by the vector.
Example: u = (0.8, -0.6, 0) is a unit vector. Dotting this with the Pauli spin matrices gives the matrix:
$S_{u}=(0.8,-0.6,0.0)\cdot {\vec {\sigma }}=0.8\sigma _{1}-0.6\sigma _{2}+0.0\sigma _{3}={\begin{bmatrix}0.0&0.8+0.6i\\0.8-0.6i&0.0\end{bmatrix}}$
The eigenvectors may be found by the usual methods of linear algebra, but a convenient trick is to note that a Pauli spin matrix is an involutory matrix, that is, the square of the above matrix is the identity matrix.
Thus a (matrix) solution to the eigenvector problem with eigenvalues of ±1 is simply 1 ± Su. That is,
$S_{u}(1\pm S_{u})=\pm 1(1\pm S_{u})$
One can then choose either of the columns of the eigenvector matrix as the vector solution, provided that the column chosen is not zero. Taking the first column of the above, eigenvector solutions for the two eigenvalues are:
${\begin{bmatrix}1.0+(0.0)\\0.0+(0.8-0.6i)\end{bmatrix}},{\begin{bmatrix}1.0-(0.0)\\0.0-(0.8-0.6i)\end{bmatrix}}$
The trick used to find the eigenvectors is related to the concept of ideals, that is, the matrix eigenvectors (1 ± Su)/2 are projection operators or idempotents and therefore each generates an ideal in the Pauli algebra. The same trick works in any Clifford algebra, in particular the Dirac algebra that is discussed below. These projection operators are also seen in density matrix theory where they are examples of pure density matrices.
More generally, the projection operator for spin in the (a, b, c) direction is given by
${\frac {1}{2}}{\begin{bmatrix}1+c&a-ib\\a+ib&1-c\end{bmatrix}}$
and any non zero column can be taken as the projection operator. While the two columns appear different, one can use a2 + b2 + c2 = 1 to show that they are multiples (possibly zero) of the same spinor.
General remarks
In atomic physics and quantum mechanics, the property of spin plays a major role. In addition to their other properties all particles possess a non-classical property, i.e., which has no correspondence at all in conventional physics, namely the spin, which is a kind of intrinsic angular momentum. In the position representation, instead of a wavefunction without spin, ψ = ψ(r), one has with spin: ψ = ψ(r, σ), where σ takes the following discrete set of values:
$\sigma =-S\cdot \hbar ,-(S-1)\cdot \hbar ,...,+(S-1)\cdot \hbar ,+S\cdot \hbar $.
The total angular momentum operator, ${\vec {\mathbb {J} }}$, of a particle corresponds to the sum of the orbital angular momentum (i.e., there only integers are allowed) and the intrinsic part, the spin. One distinguishes bosons (S = 0, ±1, ±2, ...) and fermions (S = ±1/2, ±3/2, ±5/2, ...).
See also
• Bloch sphere
• Pauli equation
References
1. Cartan, Élie (1981) [1938], The Theory of Spinors, New York: Dover Publications, ISBN 978-0-486-64070-9, MR 0631850
2. The Pauli vector is a formal device. It may be thought of as an element of M2(ℂ) ⊗ ℝ3, where the tensor product space is endowed with a mapping ⋅: ℝ3 × M2(ℂ) ⊗ ℝ3 → M2(ℂ).
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Wikipedia
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Spiral of Theodorus
In geometry, the spiral of Theodorus (also called square root spiral, Spiral of Einstein, Pythagorean spiral, or Pythagoras's snail)[1] is a spiral composed of right triangles, placed edge-to-edge. It was named after Theodorus of Cyrene.
Construction
The spiral is started with an isosceles right triangle, with each leg having unit length. Another right triangle is formed, an automedian right triangle with one leg being the hypotenuse of the prior triangle (with length the square root of 2) and the other leg having length of 1; the length of the hypotenuse of this second triangle is the square root of 3. The process then repeats; the $n$th triangle in the sequence is a right triangle with the side lengths ${\sqrt {n}}$ and 1, and with hypotenuse ${\sqrt {n+1}}$. For example, the 16th triangle has sides measuring $4={\sqrt {16}}$, 1 and hypotenuse of ${\sqrt {17}}$.
History and uses
Although all of Theodorus' work has been lost, Plato put Theodorus into his dialogue Theaetetus, which tells of his work. It is assumed that Theodorus had proved that all of the square roots of non-square integers from 3 to 17 are irrational by means of the Spiral of Theodorus.[2]
Plato does not attribute the irrationality of the square root of 2 to Theodorus, because it was well known before him. Theodorus and Theaetetus split the rational numbers and irrational numbers into different categories.[3]
Hypotenuse
Each of the triangles' hypotenuses $h_{n}$ gives the square root of the corresponding natural number, with $h_{1}={\sqrt {2}}$.
Plato, tutored by Theodorus, questioned why Theodorus stopped at ${\sqrt {17}}$. The reason is commonly believed to be that the ${\sqrt {17}}$ hypotenuse belongs to the last triangle that does not overlap the figure.[4]
Overlapping
In 1958, Kaleb Williams proved that no two hypotenuses will ever coincide, regardless of how far the spiral is continued. Also, if the sides of unit length are extended into a line, they will never pass through any of the other vertices of the total figure.[4][5]
Extension
Theodorus stopped his spiral at the triangle with a hypotenuse of ${\sqrt {17}}$. If the spiral is continued to infinitely many triangles, many more interesting characteristics are found.
Angle
If $\varphi _{n}$ is the angle of the $n$th triangle (or spiral segment), then:
$\tan \left(\varphi _{n}\right)={\frac {1}{\sqrt {n}}}.$
Therefore, the growth of the angle $\varphi _{n}$ of the next triangle $n$ is:[1]
$\varphi _{n}=\arctan \left({\frac {1}{\sqrt {n}}}\right).$
The sum of the angles of the first $k$ triangles is called the total angle $\varphi (k)$ for the $k$th triangle. It grows proportionally to the square root of $k$, with a bounded correction term $c_{2}$:[1]
$\varphi \left(k\right)=\sum _{n=1}^{k}\varphi _{n}=2{\sqrt {k}}+c_{2}(k)$
where
$\lim _{k\to \infty }c_{2}(k)=-2.157782996659\ldots $
(OEIS: A105459).
Radius
The growth of the radius of the spiral at a certain triangle $n$ is
$\Delta r={\sqrt {n+1}}-{\sqrt {n}}.$
Archimedean spiral
The Spiral of Theodorus approximates the Archimedean spiral.[1] Just as the distance between two windings of the Archimedean spiral equals mathematical constant $\pi $, as the number of spins of the spiral of Theodorus approaches infinity, the distance between two consecutive windings quickly approaches $\pi $.[6]
The following is a table showing of two windings of the spiral approaching pi:
Winding No.: Calculated average winding-distance Accuracy of average winding-distance in comparison to π
2 3.1592037 99.44255%
3 3.1443455 99.91245%
4 3.14428 99.91453%
5 3.142395 99.97447%
$\to \infty $ $\to \pi $ $\to 100\%$
As shown, after only the fifth winding, the distance is a 99.97% accurate approximation to $\pi $.[1]
Continuous curve
The question of how to interpolate the discrete points of the spiral of Theodorus by a smooth curve was proposed and answered in (Davis 2001, pp. 37–38) by analogy with Euler's formula for the gamma function as an interpolant for the factorial function. Davis found the function
$T(x)=\prod _{k=1}^{\infty }{\frac {1+i/{\sqrt {k}}}{1+i/{\sqrt {x+k}}}}\qquad (-1<x<\infty )$
which was further studied by his student Leader[7] and by Iserles (in an appendix to (Davis 2001) ). An axiomatic characterization of this function is given in (Gronau 2004) as the unique function that satisfies the functional equation
$f(x+1)=\left(1+{\frac {i}{\sqrt {x+1}}}\right)\cdot f(x),$
the initial condition $f(0)=1,$ and monotonicity in both argument and modulus; alternative conditions and weakenings are also studied therein. An alternative derivation is given in (Heuvers, Moak & Boursaw 2000).
An analytic continuation of Davis' continuous form of the Spiral of Theodorus which extends in the opposite direction from the origin is given in (Waldvogel 2009).
In the figure the nodes of the original (discrete) Theodorus spiral are shown as small green circles. The blue ones are those, added in the opposite direction of the spiral. Only nodes $n$ with the integer value of the polar radius $r_{n}=\pm {\sqrt {|n|}}$ are numbered in the figure. The dashed circle in the coordinate origin $O$ is the circle of curvature at $O$.
See also
• Fermat's spiral
• List of spirals
References
1. Hahn, Harry K. (2007), The ordered distribution of natural numbers on the square root spiral, arXiv:0712.2184
2. Nahin, Paul J. (1998), An Imaginary Tale: The Story of ${\sqrt {-1}}$, Princeton University Press, p. 33, ISBN 0-691-02795-1
3. Plato; Dyde, Samuel Walters (1899), The Theaetetus of Plato, J. Maclehose, pp. 86–87
4. Long, Kate, A Lesson on The Root Spiral, archived from the original on 11 April 2013, retrieved 30 April 2008
5. Teuffel, Erich (1958), "Eine Eigenschaft der Quadratwurzelschnecke", Mathematisch-Physikalische Semesterberichte zur Pflege des Zusammenhangs von Schule und Universität, 6: 148–152, MR 0096160
6. Hahn, Harry K. (2008), The distribution of natural numbers divisible by 2, 3, 5, 7, 11, 13, and 17 on the square root spiral, arXiv:0801.4422
7. Leader, Jeffery James (1990), The generalized Theodorus iteration (PhD thesis), Brown University, p. 173, MR 2685516, ProQuest 303808219
Further reading
• Davis, P. J. (2001), Spirals from Theodorus to Chaos, A K Peters/CRC Press
• Gronau, Detlef (March 2004), "The Spiral of Theodorus", The American Mathematical Monthly, Mathematical Association of America, 111 (3): 230–237, doi:10.2307/4145130, JSTOR 4145130
• Heuvers, J.; Moak, D. S.; Boursaw, B (2000), "The functional equation of the square root spiral", in T. M. Rassias (ed.), Functional Equations and Inequalities, pp. 111–117
• Waldvogel, Jörg (2009), Analytic Continuation of the Theodorus Spiral (PDF)
Ancient Greek mathematics
Mathematicians
(timeline)
• Anaxagoras
• Anthemius
• Archytas
• Aristaeus the Elder
• Aristarchus
• Aristotle
• Apollonius
• Archimedes
• Autolycus
• Bion
• Bryson
• Callippus
• Carpus
• Chrysippus
• Cleomedes
• Conon
• Ctesibius
• Democritus
• Dicaearchus
• Diocles
• Diophantus
• Dinostratus
• Dionysodorus
• Domninus
• Eratosthenes
• Eudemus
• Euclid
• Eudoxus
• Eutocius
• Geminus
• Heliodorus
• Heron
• Hipparchus
• Hippasus
• Hippias
• Hippocrates
• Hypatia
• Hypsicles
• Isidore of Miletus
• Leon
• Marinus
• Menaechmus
• Menelaus
• Metrodorus
• Nicomachus
• Nicomedes
• Nicoteles
• Oenopides
• Pappus
• Perseus
• Philolaus
• Philon
• Philonides
• Plato
• Porphyry
• Posidonius
• Proclus
• Ptolemy
• Pythagoras
• Serenus
• Simplicius
• Sosigenes
• Sporus
• Thales
• Theaetetus
• Theano
• Theodorus
• Theodosius
• Theon of Alexandria
• Theon of Smyrna
• Thymaridas
• Xenocrates
• Zeno of Elea
• Zeno of Sidon
• Zenodorus
Treatises
• Almagest
• Archimedes Palimpsest
• Arithmetica
• Conics (Apollonius)
• Catoptrics
• Data (Euclid)
• Elements (Euclid)
• Measurement of a Circle
• On Conoids and Spheroids
• On the Sizes and Distances (Aristarchus)
• On Sizes and Distances (Hipparchus)
• On the Moving Sphere (Autolycus)
• Optics (Euclid)
• On Spirals
• On the Sphere and Cylinder
• Ostomachion
• Planisphaerium
• Sphaerics
• The Quadrature of the Parabola
• The Sand Reckoner
Problems
• Constructible numbers
• Angle trisection
• Doubling the cube
• Squaring the circle
• Problem of Apollonius
Concepts
and definitions
• Angle
• Central
• Inscribed
• Axiomatic system
• Axiom
• Chord
• Circles of Apollonius
• Apollonian circles
• Apollonian gasket
• Circumscribed circle
• Commensurability
• Diophantine equation
• Doctrine of proportionality
• Euclidean geometry
• Golden ratio
• Greek numerals
• Incircle and excircles of a triangle
• Method of exhaustion
• Parallel postulate
• Platonic solid
• Lune of Hippocrates
• Quadratrix of Hippias
• Regular polygon
• Straightedge and compass construction
• Triangle center
Results
In Elements
• Angle bisector theorem
• Exterior angle theorem
• Euclidean algorithm
• Euclid's theorem
• Geometric mean theorem
• Greek geometric algebra
• Hinge theorem
• Inscribed angle theorem
• Intercept theorem
• Intersecting chords theorem
• Intersecting secants theorem
• Law of cosines
• Pons asinorum
• Pythagorean theorem
• Tangent-secant theorem
• Thales's theorem
• Theorem of the gnomon
Apollonius
• Apollonius's theorem
Other
• Aristarchus's inequality
• Crossbar theorem
• Heron's formula
• Irrational numbers
• Law of sines
• Menelaus's theorem
• Pappus's area theorem
• Problem II.8 of Arithmetica
• Ptolemy's inequality
• Ptolemy's table of chords
• Ptolemy's theorem
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Wikipedia
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Spiral optimization algorithm
In mathematics, the spiral optimization (SPO) algorithm is a metaheuristic inspired by spiral phenomena in nature.
The first SPO algorithm was proposed for two-dimensional unconstrained optimization[1] based on two-dimensional spiral models. This was extended to n-dimensional problems by generalizing the two-dimensional spiral model to an n-dimensional spiral model.[2] There are effective settings for the SPO algorithm: the periodic descent direction setting[3] and the convergence setting.[4]
Metaphor
The motivation for focusing on spiral phenomena was due to the insight that the dynamics that generate logarithmic spirals share the diversification and intensification behavior. The diversification behavior can work for a global search (exploration) and the intensification behavior enables an intensive search around a current found good solution (exploitation).
Algorithm
The SPO algorithm is a multipoint search algorithm that has no objective function gradient, which uses multiple spiral models that can be described as deterministic dynamical systems. As search points follow logarithmic spiral trajectories towards the common center, defined as the current best point, better solutions can be found and the common center can be updated.
The general SPO algorithm for a minimization problem under the maximum iteration $k_{\max }$ (termination criterion) is as follows:
0) Set the number of search points $m\geq 2$ and the maximum iteration number $k_{\max }$.
1) Place the initial search points $x_{i}(0)\in \mathbb {R} ^{n}~(i=1,\ldots ,m)$ and determine the center $x^{\star }(0)=x_{i_{\text{b}}}(0)$, $\displaystyle i_{\text{b}}=\mathop {\text{argmin}} _{i=1,\ldots ,m}\{f(x_{i}(0))\}$,and then set $k=0$.
2) Decide the step rate $r(k)$ by a rule.
3) Update the search points: $x_{i}(k+1)=x^{\star }(k)+r(k)R(\theta )(x_{i}(k)-x^{\star }(k))\quad (i=1,\ldots ,m).$
4) Update the center: $x^{\star }(k+1)={\begin{cases}x_{i_{\text{b}}}(k+1)&{\big (}{\text{if }}f(x_{i_{\text{b}}}(k+1))<f(x^{\star }(k)){\big )},\\x^{\star }(k)&{\big (}{\text{otherwise}}{\big )},\end{cases}}$ where $\displaystyle i_{\text{b}}=\mathop {\text{argmin}} _{i=1,\ldots ,m}\{f(x_{i}(k+1))\}$.
5) Set $k:=k+1$. If $k=k_{\max }$ is satisfied then terminate and output $x^{\star }(k)$. Otherwise, return to Step 2).
Setting
The search performance depends on setting the composite rotation matrix $R(\theta )$, the step rate $r(k)$, and the initial points $x_{i}(0)~(i=1,\ldots ,m)$. The following settings are new and effective.
Setting 1 (Periodic Descent Direction Setting)
This setting is an effective setting for high dimensional problems under the maximum iteration $k_{\max }$. The conditions on $R(\theta )$ and $x_{i}(0)~(i=1,\ldots ,m)$ together ensure that the spiral models generate descent directions periodically. The condition of $r(k)$ works to utilize the periodic descent directions under the search termination $k_{\max }$.
• Set $R(\theta )$ as follows:$R(\theta )={\begin{bmatrix}0_{n-1}^{\top }&-1\\I_{n-1}&0_{n-1}\\\end{bmatrix}}$ where $I_{n-1}$ is the $(n-1)\times (n-1)$ identity matrix and $0_{n-1}$ is the $(n-1)\times 1$ zero vector.
• Place the initial points $x_{i}(0)\in \mathbb {R} ^{n}$ $(i=1,\ldots ,m)$ at random to satisfy the following condition:
$\min _{i=1,\ldots ,m}\{\max _{j=1,\ldots ,m}{\bigl \{}{\text{rank}}{\bigl [}d_{j,i}(0)~R(\theta )d_{j,i}(0)~~\cdots ~~R(\theta )^{2n-1}d_{j,i}(0){\bigr ]}{\bigr \}}{\bigr \}}=n$ where $d_{j,i}(0)=x_{j}(0)-x_{i}(0)$. Note that this condition is almost all satisfied by a random placing and thus no check is actually fine.
• Set $r(k)$ at Step 2) as follows:$r(k)=r={\sqrt[{k_{\max }}]{\delta }}~~~~{\text{(constant value)}}$ where a sufficiently small $\delta >0$ such as $\delta =1/k_{\max }$ or $\delta =10^{-3}$.[3]
Setting 2 (Convergence Setting)
This setting ensures that the SPO algorithm converges to a stationary point under the maximum iteration $k_{\max }=\infty $. The settings of $R(\theta )$ and the initial points $x_{i}(0)~(i=1,\ldots ,m)$ are the same with the above Setting 1. The setting of $r(k)$ is as follows.
• Set $r(k)$ at Step 2) as follows:$r(k)={\begin{cases}1&(k^{\star }\leqq k\leqq k^{\star }+2n-1),\\h&(k\geqq k^{\star }+2n),\end{cases}}$ where $k^{\star }$ is an iteration when the center is newly updated at Step 4) and $h={\sqrt[{2n}]{\delta }},\delta \in (0,1)$ such as $\delta =0.5$. Thus we have to add the following rules about $k^{\star }$ to the Algorithm:
•(Step 1) $k^{\star }=0$.
•(Step 4) If $x^{\star }(k+1)\neq x^{\star }(k)$ then $k^{\star }=k+1$.[4]
Future works
• The algorithms with the above settings are deterministic. Thus, incorporating some random operations make this algorithm powerful for global optimization. Cruz-Duarte et al.[5] demonstrated it by including stochastic disturbances in spiral searching trajectories. However, this door remains open to further studies.
• To find an appropriate balance between diversification and intensification spirals depending on the target problem class (including $k_{\max }$) is important to enhance the performance.
Extended works
Many extended studies have been conducted on the SPO due to its simple structure and concept; these studies have helped improve its global search performance and proposed novel applications.[6][7][8][9][10][11]
References
1. Tamura, K.; Yasuda, K. (2011). "Primary Study of Spiral Dynamics Inspired Optimization". IEEJ Transactions on Electrical and Electronic Engineering. 6 (S1): 98–100. doi:10.1002/tee.20628. S2CID 109093423.
2. Tamura, K.; Yasuda, K. (2011). "Spiral Dynamics Inspired Optimization". Journal of Advanced Computational Intelligence and Intelligent Informatics. 132 (5): 1116–1121. doi:10.20965/jaciii.2011.p1116.
3. Tamura, K.; Yasuda, K. (2016). "Spiral Optimization Algorithm Using Periodic Descent Directions". SICE Journal of Control, Measurement, and System Integration. 6 (3): 133–143. Bibcode:2016JCMSI...9..134T. doi:10.9746/jcmsi.9.134.
4. Tamura, K.; Yasuda, K. (2020). "The Spiral Optimization Algorithm: Convergence Conditions and Settings". IEEE Transactions on Systems, Man, and Cybernetics: Systems. 50 (1): 360–375. doi:10.1109/TSMC.2017.2695577. S2CID 126109444.
5. Cruz-Duarte, Jorge M.; Martin-Diaz, Ignacio; Munoz-Minjares, J. U.; Sanchez-Galindo, Luis A.; Avina-Cervantes, Juan G.; Garcia-Perez, Arturo; Correa-Cely, C. Rodrigo (2017). "Primary study on the stochastic spiral optimization algorithm". 2017 IEEE International Autumn Meeting on Power, Electronics and Computing (ROPEC). pp. 1–6. doi:10.1109/ROPEC.2017.8261609. ISBN 978-1-5386-0819-7. S2CID 37580653.
6. Nasir, A. N. K.; Tokhi, M. O. (2015). "An improved spiral dynamic optimization algorithm with engineering application". IEEE Transactions on Systems, Man, and Cybernetics: Systems. 45 (6): 943–954. doi:10.1109/tsmc.2014.2383995. S2CID 24253496.
7. Nasir, A. N. K.; Ismail, R.M.T.R.; Tokhi, M. O. (2016). "Adaptive spiral dynamics metaheuristic algorithm for global optimisation with application to modelling of a flexible system" (PDF). Applied Mathematical Modelling. 40 (9–10): 5442–5461. doi:10.1016/j.apm.2016.01.002.
8. Ouadi, A.; Bentarzi, H.; Recioui, A. (2013). "multiobjective design of digital filters using spiral optimization technique". SpringerPlus. 2 (461): 697–707. doi:10.1186/2193-1801-2-461. PMC 3786071. PMID 24083108.
9. Benasla, L.; Belmadani, A.; Rahli, M. (2014). "Spiral optimization algorithm for solving combined economic and Emission Dispatch". International Journal of Electrical Power & Energy Systems. 62: 163–174. doi:10.1016/j.ijepes.2014.04.037.
10. Sidarto, K. A.; Kania, A. (2015). "Finding all solutions of systems of nonlinear equations using spiral dynamics inspired optimization with clustering". Journal of Advanced Computational Intelligence and Intelligent Informatics. 19 (5): 697–707. doi:10.20965/jaciii.2015.p0697.
11. Kaveh, A.; Mahjoubi, S. (October 2019). "Hypotrochoid spiral optimization approach for sizing and layout optimization of truss structures with multiple frequency constraints". Engineering with Computers. 35 (4): 1443–1462. doi:10.1007/s00366-018-0675-6. S2CID 54457145.
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Wikipedia
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Spiral similarity
Spiral similarity is a plane transformation in mathematics composed of a rotation and a dilation.[1] It is used widely in Euclidean geometry to facilitate the proofs of many theorems and other results in geometry, especially in mathematical competitions and Olympiads. Though the origin of this idea is not known, it was documented in 1967 by Coxeter in his book Geometry Revisited.[2]
The following theorem is important for the Euclidean plane:
Any two directly similar figures are related either by a translation or by a spiral similarity. [3]
(Hint: Directly similar figures are similar and have the same orientation)
Definition
A spiral similarity $S$ is composed of a rotation of the plane followed a dilation about a center $O$ with coordinates $c$ in the plane.[4] Expressing the rotation by a linear transformation $T(x)$ and the dilation as multiplying by a scale factor $d$, a point $p$ gets mapped to
$S(p)=d(T(p-c))+c.$
On the complex plane, any spiral similarity can be expressed in the form $T(x)=x_{0}+\alpha (x-x_{0})$, where $\alpha $ is a complex number. The magnitude $|\alpha |$ is the dilation factor of the spiral similarity, and the argument ${\text{arg}}(\alpha )$ is the angle of rotation.[5]
Properties
Center of a spiral similarity for two line segments
Through a dilation of a line, rotation, and translation, any line segment can be mapped into any other through the series of plane transformations. We can find the center of the spiral similarity through the following construction: [1]
• Draw lines ${\overline {AC}}$ and ${\overline {BD}}$, and let $P$ be the intersection of the two lines.
• Draw the circumcircles of triangles $\triangle PAB$ and $\triangle PCD$.
• The circumcircles intersect at a second point $X\neq P$. Then $X$ is the spiral center mapping ${\overline {AB}}$ to ${\overline {CD}}.$
Proof: Note that $ABPX$ and $XPCD$ are cyclic quadrilaterals. Thus, $\angle XAB=180^{\circ }-\angle BPX=\angle XPD=\angle XCD$. Similarly, $\angle ABX=\angle APX=180^{\circ }-\angle XPC=\angle XDC$. Therefore, by AA similarity, triangles $XAB$ and $XCD$ are similar. Thus, $\angle AXB=\angle CXD,$ so a rotation angle mapping $A$ to $B$ also maps $C$ to $D$. The dilation factor is then just the ratio of side lengths ${\overline {CD}}$ to ${\overline {AB}}$.[4]
Solution with complex numbers
If we express $A,B,C,$ and $D$ as points on the complex plane with corresponding complex numbers $a,b,c,$ and $d$, we can solve for the expression of the spiral similarity which takes $A$ to $C$ and $B$ to $D$. Note that $T(a)=x_{0}+\alpha (a-x_{0})$ and $T(b)=x_{0}+\alpha (b-x_{0})$, so ${\frac {T(b)-T(a)}{b-a}}=\alpha $. Since $T(a)=c$ and $T(b)=d$, we plug in to obtain $\alpha ={\frac {d-c}{b-a}}$, from which we obtain $x_{0}={\frac {ad-bc}{a+d-b-c}}$.[4]
Pairs of spiral similarities
For any points $A,B,C,$ and $D$, the center of the spiral similarity taking ${\overline {AB}}$ to ${\overline {CD}}$ is also the center of a spiral similarity taking ${\overline {AC}}$ to ${\overline {BD}}$.
This can be seen through the above construction. If we let $X$ be the center of spiral similarity taking ${\overline {AB}}$ to ${\overline {CD}}$, then $\triangle XAB\sim \triangle XCD$. Therefore, $\angle AXC=\angle AXB+\angle BXC=\angle CXD+\angle BXC=\angle BXD$. Also, ${\frac {AX}{BX}}={\frac {CX}{DX}}$ implies that ${\frac {AX}{CX}}={\frac {BX}{DX}}$. So, by SAS similarity, we see that $\triangle AXC\sim \triangle BXD$. Thus $X$ is also the center of the spiral similarity which takes ${\overline {AC}}$ to ${\overline {BD}}$.[4][5]
Corollaries
Proof of Miquel's Quadrilateral Theorem
Spiral similarity can be used to prove Miquel's Quadrilateral Theorem: given four noncollinear points $A,B,C,$ and $D$, the circumcircles of the four triangles $\triangle PAB,\triangle PDC,\triangle QAD,$ and $\triangle QBC$ intersect at one point, where $P$ is the intersection of $AD$ and $BC$ and $Q$ is the intersection of $AB$ and $CD$ (see diagram). [1]
Let $M$ be the center of the spiral similarity which takes $AB$ to $DC$. By the above construction, the circumcircles of $\triangle PAB$ and $\triangle PDC$ intersect at $M$ and $P$. Since $M$ is also the center of the spiral similarity taking $DA$ to $BC$, by similar reasoning the circumcircles of $\triangle QAD$ and $\triangle QBC$ meet at $Q$ and $M$. Thus, all four circles intersect at $M$.[1]
Example problem
Here is an example problem on the 2018 Japan MO Finals which can be solved using spiral similarity:
Given a scalene triangle $ABC$, let $D$ and $E$ be points on segments $AB$ and $AC$, respectively, so that $CA=CD,BA=BE$. Let $\omega $ be the circumcircle of triangle $ADE$ and $P$ the reflection of $A$ across $BC$. Lines $PD$ and $PE$ meet $\omega $ again at $X$ and $Y$, respectively. Prove that $BX$ and $CY$ intersect on $\omega $.[4]
Proof: We first prove the following claims:
Claim 1: Quadrilateral $PBEC$ is cyclic.
Proof: Since $\triangle BAE$ is isosceles, we note that $\angle BPC=\angle BAC=180^{\circ }-\angle BEC,$ thus proving that quadrilateral $PBEC$ is cyclic, as desired. By symmetry, we can prove that quadrilateral $PBDC$ is cyclic.
Claim 2: $\triangle AXY\sim \triangle ABC.$
Proof: We have that $\angle AXY=180^{\circ }-\angle AEY=\angle YEC=\angle PEC=\angle PBC=\angle ABC.$ By similar reasoning, $\angle AYX=\angle ACB,$ so by AA similarity, $\triangle AXY\sim \triangle ABC,$ as desired.
We now note that $A$ is the spiral center that maps $XY$ to $BC$. Let $F$ be the intersection of $BX$ and $CY$. By the spiral similarity construction above, the spiral center must be the intersection of the circumcircles of $\triangle FXY$ and $\triangle FBC$. However, this point is $A$, so thus points $A,F,X,Y$ must be concyclic. Hence, $F$ must lie on $\omega $, as desired.
References
1. Chen, Evan (2016). Euclidean Geometry in Mathematical Olympiads. United States: MAA Press. pp. 196–200. ISBN 978-0-88385-839-4.
2. Coxeter, H.S.M. (1967). Geometry Revisited. Toronto and New York: Mathematical Association of America. pp. 95–100. ISBN 978-0-88385-619-2.
3. Coxeter, H.S.M. (1967). Geometry Revisited. Mathematical Association of America. p. 97]. ISBN 978-0-88385-619-2.
4. Baca, Jafet (2019). "On a special center of spiral similarity". Mathematical Reflections. 1: 1–9.
5. Zhao, Y. (2010). Three Lemmas in Geometry.
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Wikipedia
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Spiral
In mathematics, a spiral is a curve which emanates from a point, moving farther away as it revolves around the point.[1][2][3][4] It is a subtype of whorled patterns, a broad group that also includes concentric objects.
Helices
Two major definitions of "spiral" in the American Heritage Dictionary are:[5]
1. a curve on a plane that winds around a fixed center point at a continuously increasing or decreasing distance from the point.
2. a three-dimensional curve that turns around an axis at a constant or continuously varying distance while moving parallel to the axis; a helix.
The first definition describes a planar curve, that extends in both of the perpendicular directions within its plane; the groove on one side of a record closely approximates a plane spiral (and it is by the finite width and depth of the groove, but not by the wider spacing between than within tracks, that it falls short of being a perfect example); note that successive loops differ in diameter. In another example, the "center lines" of the arms of a spiral galaxy trace logarithmic spirals.
The second definition includes two kinds of 3-dimensional relatives of spirals:
1. a conical or volute spring (including the spring used to hold and make contact with the negative terminals of AA or AAA batteries in a battery box), and the vortex that is created when water is draining in a sink is often described as a spiral, or as a conical helix.
2. quite explicitly, definition 2 also includes a cylindrical coil spring and a strand of DNA, both of which are quite helical, so that "helix" is a more useful description than "spiral" for each of them; in general, "spiral" is seldom applied if successive "loops" of a curve have the same diameter.[5]
In the side picture, the black curve at the bottom is an Archimedean spiral, while the green curve is a helix. The curve shown in red is a conic helix.
Two-dimensional
Main article: List of spirals
A two-dimensional, or plane, spiral may be described most easily using polar coordinates, where the radius $r$ is a monotonic continuous function of angle $\varphi $:
• $r=r(\varphi )\;.$
The circle would be regarded as a degenerate case (the function not being strictly monotonic, but rather constant).
In $x$-$y$-coordinates the curve has the parametric representation:
• $x=r(\varphi )\cos \varphi \ ,\qquad y=r(\varphi )\sin \varphi \;.$
Examples
Some of the most important sorts of two-dimensional spirals include:
• The Archimedean spiral: $r=a\varphi $
• The hyperbolic spiral: $r=a/\varphi $
• Fermat's spiral: $r=a\varphi ^{1/2}$
• The lituus: $r=a\varphi ^{-1/2}$
• The logarithmic spiral: $r=ae^{k\varphi }$
• The Cornu spiral or clothoid
• The Fibonacci spiral and golden spiral
• The Spiral of Theodorus: an approximation of the Archimedean spiral composed of contiguous right triangles
• The involute of a circle, used twice on each tooth of almost every modern gear
• Archimedean spiral
• hyperbolic spiral
• Fermat's spiral
• lituus
• logarithmic spiral
• Cornu spiral
• spiral of Theodorus
• Fibonacci Spiral (golden spiral)
• The involute of a circle (black) is not identical to the Archimedean spiral (red).
An Archimedean spiral is, for example, generated while coiling a carpet.[6]
A hyperbolic spiral appears as image of a helix with a special central projection (see diagram). A hyperbolic spiral is some times called reciproke spiral, because it is the image of an Archimedean spiral with a circle-inversion (see below).[7]
The name logarithmic spiral is due to the equation $\varphi ={\tfrac {1}{k}}\cdot \ln {\tfrac {r}{a}}$. Approximations of this are found in nature.
Spirals which do not fit into this scheme of the first 5 examples:
A Cornu spiral has two asymptotic points.
The spiral of Theodorus is a polygon.
The Fibonacci Spiral consists of a sequence of circle arcs.
The involute of a circle looks like an Archimedean, but is not: see Involute#Examples.
Geometric properties
The following considerations are dealing with spirals, which can be described by a polar equation $r=r(\varphi )$, especially for the cases $r(\varphi )=a\varphi ^{n}$ (Archimedean, hyperbolic, Fermat's, lituus spirals) and the logarithmic spiral $r=ae^{k\varphi }$.
Polar slope angle
The angle $\alpha $ between the spiral tangent and the corresponding polar circle (see diagram) is called angle of the polar slope and $\tan \alpha $ the polar slope.
From vector calculus in polar coordinates one gets the formula
$\tan \alpha ={\frac {r'}{r}}\ .$
Hence the slope of the spiral $\;r=a\varphi ^{n}\;$ is
• $\tan \alpha ={\frac {n}{\varphi }}\ .$
In case of an Archimedean spiral ($n=1$) the polar slope is $\;\tan \alpha ={\tfrac {1}{\varphi }}\ .$
The logarithmic spiral is a special case, because of $\ \tan \alpha =k\ $ constant !
curvature
The curvature $\kappa $ of a curve with polar equation $r=r(\varphi )$ is
$\kappa ={\frac {r^{2}+2(r')^{2}-r\;r''}{(r^{2}+(r')^{2})^{3/2}}}\ .$
For a spiral with $r=a\varphi ^{n}$ one gets
• $\kappa =\dotsb ={\frac {1}{a\varphi ^{n-1}}}{\frac {\varphi ^{2}+n^{2}+n}{(\varphi ^{2}+n^{2})^{3/2}}}\ .$
In case of $n=1$ (Archimedean spiral) $\kappa ={\tfrac {\varphi ^{2}+2}{a(\varphi ^{2}+1)^{3/2}}}$.
Only for $-1<n<0$ the spiral has an inflection point.
The curvature of a logarithmic spiral $\;r=ae^{k\varphi }\;$ is $\;\kappa ={\tfrac {1}{r{\sqrt {1+k^{2}}}}}\;.$
Sector area
The area of a sector of a curve (see diagram) with polar equation $r=r(\varphi )$ is
$A={\frac {1}{2}}\int _{\varphi _{1}}^{\varphi _{2}}r(\varphi )^{2}\;d\varphi \ .$
For a spiral with equation $r=a\varphi ^{n}\;$ one gets
• $A={\frac {1}{2}}\int _{\varphi _{1}}^{\varphi _{2}}a^{2}\varphi ^{2n}\;d\varphi ={\frac {a^{2}}{2(2n+1)}}{\big (}\varphi _{2}^{2n+1}-\varphi _{1}^{2n+1}{\big )}\ ,\quad {\text{if}}\quad n\neq -{\frac {1}{2}},$
$A={\frac {1}{2}}\int _{\varphi _{1}}^{\varphi _{2}}{\frac {a^{2}}{\varphi }}\;d\varphi ={\frac {a^{2}}{2}}(\ln \varphi _{2}-\ln \varphi _{1})\ ,\quad {\text{if}}\quad n=-{\frac {1}{2}}\ .$
The formula for a logarithmic spiral $\;r=ae^{k\varphi }\;$ is $\ A={\tfrac {r(\varphi _{2})^{2}-r(\varphi _{1})^{2})}{4k}}\ .$
Arc length
The length of an arc of a curve with polar equation $r=r(\varphi )$ is
$L=\int \limits _{\varphi _{1}}^{\varphi _{2}}{\sqrt {\left(r^{\prime }(\varphi )\right)^{2}+r^{2}(\varphi )}}\,\mathrm {d} \varphi \ .$
For the spiral $r=a\varphi ^{n}\;$ the length is
• $L=\int _{\varphi _{1}}^{\varphi _{2}}{\sqrt {{\frac {n^{2}r^{2}}{\varphi ^{2}}}+r^{2}}}\;d\varphi =a\int \limits _{\varphi _{1}}^{\varphi _{2}}\varphi ^{n-1}{\sqrt {n^{2}+\varphi ^{2}}}d\varphi \ .$
Not all these integrals can be solved by a suitable table. In case of a Fermat's spiral the integral can be expressed by elliptic integrals only.
The arc length of a logarithmic spiral $\;r=ae^{k\varphi }\;$ is $\ L={\tfrac {\sqrt {k^{2}+1}}{k}}{\big (}r(\varphi _{2})-r(\varphi _{1}){\big )}\ .$
Circle inversion
The inversion at the unit circle has in polar coordinates the simple description: $\ (r,\varphi )\mapsto ({\tfrac {1}{r}},\varphi )\ $.
• The image of a spiral $\ r=a\varphi ^{n}\ $ under the inversion at the unit circle is the spiral with polar equation $\ r={\tfrac {1}{a}}\varphi ^{-n}\ $. For example: The inverse of an Archimedean spiral is a hyperbolic spiral.
A logarithmic spiral $\;r=ae^{k\varphi }\;$ is mapped onto the logarithmic spiral $\;r={\tfrac {1}{a}}e^{-k\varphi }\;.$
Bounded spirals
The function $r(\varphi )$ of a spiral is usually strictly monotonic, continuous and unbounded. For the standard spirals $r(\varphi )$ is either a power function or an exponential function. If one chooses for $r(\varphi )$ a bounded function the spiral is bounded, too. A suitable bounded function is the arctan function:
Example 1
Setting $\;r=a\arctan(k\varphi )\;$ and the choice $\;k=0.1,a=4,\;\varphi \geq 0\;$ gives a spiral, that starts at the origin (like an Archimedean spiral) and approaches the circle with radius $\;r=a\pi /2\;$ (diagram, left).
Example 2
For $\;r=a(\arctan(k\varphi )+\pi /2)\;$ and $\;k=0.2,a=2,\;-\infty <\varphi <\infty \;$ one gets a spiral, that approaches the origin (like a hyperbolic spiral) and approaches the circle with radius $\;r=a\pi \;$ (diagram, right).
Three-dimensional
Two well-known spiral space curves are conic spirals and spherical spirals, defined below. Another instance of space spirals is the toroidal spiral.[8] A spiral wound around a helix,[9] also known as double-twisted helix,[10] represents objects such as coiled coil filaments.
Conical spirals
Main article: conical spiral
If in the $x$-$y$-plane a spiral with parametric representation
$x=r(\varphi )\cos \varphi \ ,\qquad y=r(\varphi )\sin \varphi $
is given, then there can be added a third coordinate $z(\varphi )$, such that the now space curve lies on the cone with equation $\;m(x^{2}+y^{2})=(z-z_{0})^{2}\ ,\ m>0\;$:
• $x=r(\varphi )\cos \varphi \ ,\qquad y=r(\varphi )\sin \varphi \ ,\qquad \color {red}{z=z_{0}+mr(\varphi )}\ .$
Spirals based on this procedure are called conical spirals.
Example
Starting with an archimedean spiral $\;r(\varphi )=a\varphi \;$ one gets the conical spiral (see diagram)
$x=a\varphi \cos \varphi \ ,\qquad y=a\varphi \sin \varphi \ ,\qquad z=z_{0}+ma\varphi \ ,\quad \varphi \geq 0\ .$
Spherical spirals
If one represents a sphere of radius $r$ by:
${\begin{array}{cll}x&=&r\cdot \sin \theta \cdot \cos \varphi \\y&=&r\cdot \sin \theta \cdot \sin \varphi \\z&=&r\cdot \cos \theta \end{array}}$
and sets the linear dependency $\;\varphi =c\theta ,\;c>2\;,$ for the angle coordinates, one gets a spherical curve called spherical spiral[11] with the parametric representation (with $c$ equal to twice the number of turns)
• ${\begin{array}{cll}x&=&r\cdot \sin \theta \cdot \cos {\color {red}c\theta }\\y&=&r\cdot \sin \theta \cdot \sin {\color {red}c\theta }\\z&=&r\cdot \cos \theta \qquad \qquad 0\leq \theta \leq \pi \ .\end{array}}$
Spherical spirals were known to Pappus, too.
Remark: a rhumb line is not a spherical spiral in this sense.
• Spherical spiral
• Loxodrome
A rhumb line (also known as a loxodrome or "spherical spiral") is the curve on a sphere traced by a ship with constant bearing (e.g., travelling from one pole to the other while keeping a fixed angle with respect to the meridians). The loxodrome has an infinite number of revolutions, with the separation between them decreasing as the curve approaches either of the poles, unlike an Archimedean spiral which maintains uniform line-spacing regardless of radius.
In nature
The study of spirals in nature has a long history. Christopher Wren observed that many shells form a logarithmic spiral; Jan Swammerdam observed the common mathematical characteristics of a wide range of shells from Helix to Spirula; and Henry Nottidge Moseley described the mathematics of univalve shells. D’Arcy Wentworth Thompson's On Growth and Form gives extensive treatment to these spirals. He describes how shells are formed by rotating a closed curve around a fixed axis: the shape of the curve remains fixed but its size grows in a geometric progression. In some shells, such as Nautilus and ammonites, the generating curve revolves in a plane perpendicular to the axis and the shell will form a planar discoid shape. In others it follows a skew path forming a helico-spiral pattern. Thompson also studied spirals occurring in horns, teeth, claws and plants.[12]
A model for the pattern of florets in the head of a sunflower[13] was proposed by H. Vogel. This has the form
$\theta =n\times 137.5^{\circ },\ r=c{\sqrt {n}}$
where n is the index number of the floret and c is a constant scaling factor, and is a form of Fermat's spiral. The angle 137.5° is the golden angle which is related to the golden ratio and gives a close packing of florets.[14]
Spirals in plants and animals are frequently described as whorls. This is also the name given to spiral shaped fingerprints.
• An artist's rendering of a spiral galaxy.
• Sunflower head displaying florets in spirals of 34 and 55 around the outside.
As a symbol
A spiral like form has been found in Mezine, Ukraine, as part of a decorative object dated to 10,000 BCE.
The spiral and triple spiral motif is a Neolithic symbol in Europe (Megalithic Temples of Malta). The Celtic symbol the triple spiral is in fact a pre-Celtic symbol.[15] It is carved into the rock of a stone lozenge near the main entrance of the prehistoric Newgrange monument in County Meath, Ireland. Newgrange was built around 3200 BCE predating the Celts and the triple spirals were carved at least 2,500 years before the Celts reached Ireland but has long since been incorporated into Celtic culture.[16] The triskelion symbol, consisting of three interlocked spirals or three bent human legs, appears in many early cultures, including Mycenaean vessels, on coinage in Lycia, on staters of Pamphylia (at Aspendos, 370–333 BC) and Pisidia, as well as on the heraldic emblem on warriors' shields depicted on Greek pottery.[17]
Spirals can be found throughout pre-Columbian art in Latin and Central America. The more than 1,400 petroglyphs (rock engravings) in Las Plazuelas, Guanajuato Mexico, dating 750-1200 AD, predominantly depict spirals, dot figures and scale models.[18] In Colombia monkeys, frog and lizard like figures depicted in petroglyphs or as gold offering figures frequently includes spirals, for example on the palms of hands.[19] In Lower Central America spirals along with circles, wavy lines, crosses and points are universal petroglyphs characters.[20] Spirals can also be found among the Nazca Lines in the coastal desert of Peru, dating from 200 BC to 500 AD. The geoglyphs number in the thousands and depict animals, plants and geometric motifs, including spirals.[21]
Spiral shapes, including the swastika, triskele, etc., have often been interpreted as solar symbols. Roof tiles dating back to the Tang Dynasty with this symbol have been found west of the ancient city of Chang'an (modern-day Xi'an).
Spirals are also a symbol of hypnosis, stemming from the cliché of people and cartoon characters being hypnotized by staring into a spinning spiral (one example being Kaa in Disney's The Jungle Book). They are also used as a symbol of dizziness, where the eyes of a cartoon character, especially in anime and manga, will turn into spirals to show they are dizzy or dazed. The spiral is also found in structures as small as the double helix of DNA and as large as a galaxy. Because of this frequent natural occurrence, the spiral is the official symbol of the World Pantheist Movement.[22] The spiral is also a symbol of the dialectic process and Dialectical monism.
In art
The spiral has inspired artists throughout the ages. Among the most famous of spiral-inspired art is Robert Smithson's earthwork, "Spiral Jetty", at the Great Salt Lake in Utah.[23] The spiral theme is also present in David Wood's Spiral Resonance Field at the Balloon Museum in Albuquerque, as well as in the critically acclaimed Nine Inch Nails 1994 concept album The Downward Spiral. The Spiral is also a prominent theme in the anime Gurren Lagann, where it represents a philosophy and way of life. It also central in Mario Merz and Andy Goldsworthy's work. The spiral is the central theme of the horror manga Uzumaki by Junji Ito, where a small coastal town is afflicted by a curse involving spirals. 2012 A Piece of Mind By Wayne A Beale also depicts a large spiral in this book of dreams and images.[24][25] The coiled spiral is a central image in Australian artist Tanja Stark's Suburban Gothic iconography, that incorporates spiral electric stove top elements as symbols of domestic alchemy and spirituality.[26][27]
See also
• Celtic maze (straight-line spiral)
• Concentric circles
• DNA
• Fibonacci number
• Hypogeum of Ħal-Saflieni
• Megalithic Temples of Malta
• Patterns in nature
• Seashell surface
• Spirangle
• Spiral vegetable slicer
• Spiral stairs
• Triskelion
References
1. "Spiral | mathematics". Encyclopedia Britannica. Retrieved 2020-10-08.
2. "Spiral Definition (Illustrated Mathematics Dictionary)". www.mathsisfun.com. Retrieved 2020-10-08.
3. "spiral.htm". www.math.tamu.edu. Retrieved 2020-10-08.
4. "Math Patterns in Nature". The Franklin Institute. 2017-06-01. Retrieved 2020-10-08.
5. "Spiral, American Heritage Dictionary of the English Language, Houghton Mifflin Company, Fourth Edition, 2009.
6. Weisstein, Eric W. "Archimedean Spiral". mathworld.wolfram.com. Retrieved 2020-10-08.
7. Weisstein, Eric W. "Hyperbolic Spiral". mathworld.wolfram.com. Retrieved 2020-10-08.
8. von Seggern, D.H. (1994). Practical Handbook of Curve Design and Generation. Taylor & Francis. p. 241. ISBN 978-0-8493-8916-0. Retrieved 2022-03-03.
9. "Slinky -- from Wolfram MathWorld". Wolfram MathWorld. 2002-09-13. Retrieved 2022-03-03.
10. Ugajin, R.; Ishimoto, C.; Kuroki, Y.; Hirata, S.; Watanabe, S. (2001). "Statistical analysis of a multiply-twisted helix". Physica A: Statistical Mechanics and Its Applications. Elsevier BV. 292 (1–4): 437–451. Bibcode:2001PhyA..292..437U. doi:10.1016/s0378-4371(00)00572-0. ISSN 0378-4371.
11. Kuno Fladt: Analytische Geometrie spezieller Flächen und Raumkurven, Springer-Verlag, 2013, ISBN 3322853659, 9783322853653, S. 132
12. Thompson, D'Arcy (1942) [1917]. On Growth and Form. Cambridge : University Press ; New York : Macmillan. pp. 748–933.
13. Ben Sparks. "Geogebra: Sunflowers are Irrationally Pretty".
14. Prusinkiewicz, Przemyslaw; Lindenmayer, Aristid (1990). The Algorithmic Beauty of Plants. Springer-Verlag. pp. 101–107. ISBN 978-0-387-97297-8.
15. Anthony Murphy and Richard Moore, Island of the Setting Sun: In Search of Ireland's Ancient Astronomers, 2nd ed., Dublin: The Liffey Press, 2008, pp. 168-169
16. "Newgrange Ireland - Megalithic Passage Tomb - World Heritage Site". Knowth.com. 2007-12-21. Archived from the original on 2013-07-26. Retrieved 2013-08-16.
17. For example, the trislele on Achilles' round shield on an Attic late sixth-century hydria at the Boston Museum of Fine Arts, illustrated in John Boardman, Jasper Griffin and Oswyn Murray, Greece and the Hellenistic World (Oxford History of the Classical World) vol. I (1988), p. 50.
18. "Rock Art Of Latin America & The Caribbean" (PDF). International Council on Monuments & Sites. June 2006. p. 5. Archived (PDF) from the original on 5 January 2014. Retrieved 4 January 2014.
19. "Rock Art Of Latin America & The Caribbean" (PDF). International Council on Monuments & Sites. June 2006. p. 99. Archived (PDF) from the original on 5 January 2014. Retrieved 4 January 2014.
20. "Rock Art Of Latin America & The Caribbean" (PDF). International Council on Monuments & Sites. June 2006. p. 17. Archived (PDF) from the original on 5 January 2014. Retrieved 4 January 2014.
21. Jarus, Owen (14 August 2012). "Nazca Lines: Mysterious Geoglyphs in Peru". LiveScience. Archived from the original on 4 January 2014. Retrieved 4 January 2014.
22. Harrison, Paul. "Pantheist Art" (PDF). World Pantheist Movement. Retrieved 7 June 2012.
23. Israel, Nico (2015). Spirals : the whirled image in twentieth-century literature and art. New York Columbia University Press. pp. 161–186. ISBN 978-0-231-15302-7.
24. 2012 A Piece of Mind By Wayne A Beale
25. http://www.blurb.com/distribution?id=573100/#/project/573100/project-details/edit (subscription required)
26. Stark, Tanja (4 July 2012). "Spiral Journeys : Turning and Returning". tanjastark.com.
27. Stark, Tanja. "Lecture : Spiralling Undercurrents: Archetypal Symbols of Hurt, Hope and Healing". Jung Society Melbourne.
Related publications
• Cook, T., 1903. Spirals in nature and art. Nature 68 (1761), 296.
• Cook, T., 1979. The curves of life. Dover, New York.
• Habib, Z., Sakai, M., 2005. Spiral transition curves and their applications. Scientiae Mathematicae Japonicae 61 (2), 195 – 206.
• Dimulyo, Sarpono; Habib, Zulfiqar; Sakai, Manabu (2009). "Fair cubic transition between two circles with one circle inside or tangent to the other". Numerical Algorithms. 51 (4): 461–476. Bibcode:2009NuAlg..51..461D. doi:10.1007/s11075-008-9252-1. S2CID 22532724.
• Harary, G., Tal, A., 2011. The natural 3D spiral. Computer Graphics Forum 30 (2), 237 – 246 .
• Xu, L., Mould, D., 2009. Magnetic curves: curvature-controlled aesthetic curves using magnetic fields. In: Deussen, O., Hall, P. (Eds.), Computational Aesthetics in Graphics, Visualization, and Imaging. The Eurographics Association .
• Wang, Yulin; Zhao, Bingyan; Zhang, Luzou; Xu, Jiachuan; Wang, Kanchang; Wang, Shuchun (2004). "Designing fair curves using monotone curvature pieces". Computer Aided Geometric Design. 21 (5): 515–527. doi:10.1016/j.cagd.2004.04.001.
• Kurnosenko, A. (2010). "Applying inversion to construct planar, rational spirals that satisfy two-point G2 Hermite data". Computer Aided Geometric Design. 27 (3): 262–280. arXiv:0902.4834. doi:10.1016/j.cagd.2009.12.004. S2CID 14476206.
• A. Kurnosenko. Two-point G2 Hermite interpolation with spirals by inversion of hyperbola. Computer Aided Geometric Design, 27(6), 474–481, 2010.
• Miura, K.T., 2006. A general equation of aesthetic curves and its self-affinity. Computer-Aided Design and Applications 3 (1–4), 457–464 .
• Miura, K., Sone, J., Yamashita, A., Kaneko, T., 2005. Derivation of a general formula of aesthetic curves. In: 8th International Conference on Humans and Computers (HC2005). Aizu-Wakamutsu, Japan, pp. 166 – 171 .
• Meek, D.S.; Walton, D.J. (1989). "The use of Cornu spirals in drawing planar curves of controlled curvature". Journal of Computational and Applied Mathematics. 25: 69–78. doi:10.1016/0377-0427(89)90076-9.
• Thomas, Sunil (2017). "Potassium sulfate forms a spiral structure when dissolved in solution". Russian Journal of Physical Chemistry B. 11 (1): 195–198. Bibcode:2017RJPCB..11..195T. doi:10.1134/S1990793117010328. S2CID 99162341.
• Farin, Gerald (2006). "Class a Bézier curves". Computer Aided Geometric Design. 23 (7): 573–581. doi:10.1016/j.cagd.2006.03.004.
• Farouki, R.T., 1997. Pythagorean-hodograph quintic transition curves of monotone curvature. Computer-Aided Design 29 (9), 601–606.
• Yoshida, N., Saito, T., 2006. Interactive aesthetic curve segments. The Visual Computer 22 (9), 896–905 .
• Yoshida, N., Saito, T., 2007. Quasi-aesthetic curves in rational cubic Bézier forms. Computer-Aided Design and Applications 4 (9–10), 477–486 .
• Ziatdinov, R., Yoshida, N., Kim, T., 2012. Analytic parametric equations of log-aesthetic curves in terms of incomplete gamma functions. Computer Aided Geometric Design 29 (2), 129—140 .
• Ziatdinov, R., Yoshida, N., Kim, T., 2012. Fitting G2 multispiral transition curve joining two straight lines, Computer-Aided Design 44(6), 591—596 .
• Ziatdinov, R., 2012. Family of superspirals with completely monotonic curvature given in terms of Gauss hypergeometric function. Computer Aided Geometric Design 29(7): 510–518, 2012 .
• Ziatdinov, R., Miura K.T., 2012. On the Variety of Planar Spirals and Their Applications in Computer Aided Design. European Researcher 27(8-2), 1227—1232 .
External links
Wikimedia Commons has media related to Spiral.
• Archimedes' spiral transforms into Galileo's spiral. Mikhail Gaichenkov, OEIS
Spirals, curves and helices
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Wikipedia
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Spiric section
In geometry, a spiric section, sometimes called a spiric of Perseus, is a quartic plane curve defined by equations of the form
$(x^{2}+y^{2})^{2}=dx^{2}+ey^{2}+f.\,$
Equivalently, spiric sections can be defined as bicircular quartic curves that are symmetric with respect to the x and y-axes. Spiric sections are included in the family of toric sections and include the family of hippopedes and the family of Cassini ovals. The name is from σπειρα meaning torus in ancient Greek.
A spiric section is sometimes defined as the curve of intersection of a torus and a plane parallel to its rotational symmetry axis. However, this definition does not include all of the curves given by the previous definition unless imaginary planes are allowed.
Spiric sections were first described by the ancient Greek geometer Perseus in roughly 150 BC, and are assumed to be the first toric sections to be described. The name spiric is due to the ancient notation spira of a torus.,[1][2]
Equations
Start with the usual equation for the torus:
$(x^{2}+y^{2}+z^{2}+b^{2}-a^{2})^{2}=4b^{2}(x^{2}+y^{2}).\,$
Interchanging y and z so that the axis of revolution is now on the xy-plane, and setting z=c to find the curve of intersection gives
$(x^{2}+y^{2}-a^{2}+b^{2}+c^{2})^{2}=4b^{2}(x^{2}+c^{2}).\,$
In this formula, the torus is formed by rotating a circle of radius a with its center following another circle of radius b (not necessarily larger than a, self-intersection is permitted). The parameter c is the distance from the intersecting plane to the axis of revolution. There are no spiric sections with c > b + a, since there is no intersection; the plane is too far away from the torus to intersect it.
Expanding the equation gives the form seen in the definition
$(x^{2}+y^{2})^{2}=dx^{2}+ey^{2}+f\,$
where
$d=2(a^{2}+b^{2}-c^{2}),\ e=2(a^{2}-b^{2}-c^{2}),\ f=-(a+b+c)(a+b-c)(a-b+c)(a-b-c).\,$
In polar coordinates this becomes
$(r^{2}-a^{2}+b^{2}+c^{2})^{2}=4b^{2}(r^{2}\cos ^{2}\theta +c^{2})\,$
or
$r^{4}=r^{2}(d\cos ^{2}\theta +e\sin ^{2}\theta )+f.$
Spiric sections on a spindle torus
Spiric sections on a spindle torus, whose planes intersect the spindle (inner part), consist of an outer and an inner curve (s. picture).
Spriric sections as isoptics
Isoptics of ellipses and hyperbolas are spiric sections. (S. also weblink The Mathematics Enthusiast.)
Examples of spiric sections
Examples include the hippopede and the Cassini oval and their relatives, such as the lemniscate of Bernoulli. The Cassini oval has the remarkable property that the product of distances to two foci are constant. For comparison, the sum is constant in ellipses, the difference is constant in hyperbolae and the ratio is constant in circles.
References
• Weisstein, Eric W. "Spiric Section". MathWorld.
• MacTutor history
• 2Dcurves.com description
• MacTutor biography of Perseus
• The Mathematics Enthusiast Number 9, article 4
Specific
1. John Stillwell: Mathematics and Its History, Springer-Verlag, 2010, ISBN 978-1-4419-6053-5, p. 33.
2. Wilbur R. Knorr: The Ancient Tradition of Geometric Problems, Dover-Publ., New York, 1993, ISBN 0-486-67532-7, p. 268 .
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Wikipedia
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Spirolateral
In Euclidean geometry, a spirolateral is a polygon created by a sequence of fixed vertex internal angles and sequential edge lengths 1,2,3,...,n which repeat until the figure closes. The number of repeats needed is called its cycles.[1] A simple spirolateral has only positive angles. A simple spiral approximates of a portion of an archimedean spiral. A general spirolateral allows positive and negative angles.
Simple spirolaterals
390° (4 cycles)
3108° (5 cycles)
990° ccw spiral
990° (4 cycles)
100120° spiral
100120° (4 cycles)
A spirolateral which completes in one turn is a simple polygon, while requiring more than 1 turn is a star polygon and must be self-crossing.[2] A simple spirolateral can be an equangular simple polygon <p> with p vertices, or an equiangular star polygon <p/q> with p vertices and q turns.
Spirolaterals were invented and named by Frank C. Odds as a teenager in 1962, as square spirolaterals with 90° angles, drawn on graph paper. In 1970, Odds discovered triangular and hexagonal spirolateral, with 60° and 120° angles, can be drawn on isometric[3] (triangular) graph paper.[4] Odds wrote to Martin Gardner who encouraged him to publish the results in Mathematics Teacher[5] in 1973.[3]
The process can be represented in turtle graphics, alternating turn angle and move forward instructions, but limiting the turn to a fixed rational angle.[2]
The smallest golygon is a spirolateral, 790°4, made with 7 right angles, and length 4 follow concave turns. Golygons are different in that they must close with a single sequence 1,2,3,..n, while a spirolateral will repeat that sequence until it closes.
Classifications
Varied cases
Simple 690°, 2 cycle, 3 turn
Regular unexpected closed spirolateral, 890°1,5
Unexpectedly closed spirolateral 790°4
Crossed rectangle
(1,2,-1,-2)60°
Crossed hexagon
(1,1,2,-1,-1,-2)90°
(-1.2.4.3.2)60°
(2...4)90°
(2,1,-2,3,-4,3)120°
A simple spirolateral has turns all the same direction.[2] It is denoted by nθ, where n is the number of sequential integer edge lengths and θ is the internal angle, as any rational divisor of 360°. Sequential edge lengths can be expressed explicitly as (1,2,...,n)θ.
Note: The angle θ can be confusing because it represents the internal angle, while the supplementary turn angle can make more sense. These two angles are the same for 90°.
This defines an equiangular polygon of the form <kp/kq>, where angle θ = 180(1−2q/p), with k = n/d, and d = gcd(n,p). If d = n, the pattern never closes. Otherwise it has kp vertices and kq density. The cyclic symmetry of a simple spirolateral is p/d-fold.
A regular polygon, {p} is a special case of a spirolateral, 1180(1−2/p)°. A regular star polygon, {p/q}, is a special case of a spirolateral, 1180(1−2q/p)°. An isogonal polygon, is a special case spirolateral, 2180(1−2/p)° or 2180(1−2q/p)°.
A general spirolateral can turn left or right.[2] It is denoted by nθa1,...,ak, where ai are indices with negative or concave angles.[6] For example, 260°2 is a crossed rectangle with ±60° internal angles, bending left or right.
An unexpected closed spiralateral returns to the first vertex on a single cycle. Only general spirolaterals may not close. A golygon is a regular unexpected closed spiralateral that closes from the expected direction. An irregular unexpected closed spiralateral is one that returns to the first point but from the wrong direction. For example 790°4. It takes 4 cycles to return to the start in the correct direction.[2]
A modern spirolateral, also called a loop-de-loops[7] by Educator Anna Weltman, is denoted by (i1,...,in)θ, allowing any sequence of integers as the edge lengths, i1 to in.[8] For example, (2,3,4)90° has edge lengths 2,3,4 repeating. Opposite direction turns can be given a negative integer edge length. For example, a crossed rectangle can be given as (1,2,−1,−2)θ.
An open spirolateral never closes. A simple spirolateral, nθ, never closes if nθ is a multiple of 360°, gcd(p,n) = p. A general spirolateral can also be open if half of the angles are positive, half negative.
Closure
The number of cycles it takes to close a spirolateral, nθ, with k opposite turns, p/q=360/(180-θ) can be computed. Reduce fraction (p-2q)(n-2k)/2p = a/b. The figure repeats after b cycles, and complete a total turns. If b=1, the figure never closes.[1]
Explicitly, the number of cycles is 2p/d, where d=gcd((p-2q)(n-2k),2p). If d=2p, it closes on 1 cycle or never.
The number of cycles can be seen as the rotational symmetry order of the spirolateral.
n90°
• 190°, 4 cycle, 1 turn
• 290°, 2 cycle, 1 turn
• 390°, 4 cycle, 3 turn
• 490°, never closes
• 590°, 4 cycle, 5 turn
• 690°, 2 cycle, 3 turn
• 790°, 4 cycle, 6 turns
• 890°, never closes
• 990°, 4 cycle, 9 turn
• 1090°, 2 cycle, 5 turn
n60°
• 160°, 3 cycle, 1 turn
• 260°, 3 cycle, 2 turn
• 360°, never closes
• 460°, 3 cycle, 4 turn
• 560°, 3 cycle, 5 turn
• 660°, never closes
• 760°, 3 cycle, 7 turn
• 860°, 3 cycle, 8 turn
• 960°, never closes
• 1060°, 3 cycle, 10 turn
Small simple spirolaterals
Spirolaterals can be constructed from any rational divisor of 360°. The first table's columns sample angles from small regular polygons and second table from star polygons, with examples up to n = 6.
An equiangular polygon <p/q> has p vertices and q density. <np/nq> can be reduced by d = gcd(n,p).
Small whole divisor angles
Simple spirolaterals (whole divisors p) nθ or (1,2,...,n)θ
θ60°90°108°120°128 4/7°135°140°144°147 3/11°150°
180-θ
Turn angle
120°90°72°60°51 3/7°45°40°36°32 8/11°30°
nθ \ p3456789101112
1θ
Regular
{p}
160°
{3}
190°
{4}
1108°
{5}
1120°
{6}
1128.57°
{7}
1135°
{8}
1140°
{9}
1144°
{10}
1147.27°
{11}
1150°
{12}
2θ
Isogonal
<2p/2>
260°
<6/2>
290°
<8/2> → <4>
2108°
<10/2>
2120°
<12/2> → <6>
2128.57°
<14/2>
2135°
<16/2> → <8>
2140°
<18/2>
2144°
<20/2> → <10>
2147°
<22/2>
2150°
<24/2> → <12>
3θ
2-isogonal
<3p/3>
360°
open
390°
<12/3>
3108°
<15/3>
3120°
<18/3> → <6>
3128.57°
<21/3>
3135°
<24/3>
3140°
<27/3> → <9>
3144°
<30/3>
3147°
<33/3>
3150°
<36/3> → <12>
4θ
3-isogonal
<4p/4>
460°
<12/4>
490°
open
4108°
<20/4>
4120°
<24/4> → <12/2>
4128.57°
<28/4>
4135°
<32/4> → <8>
4140°
<36/4>
4144°
<40/4> → <20/2>
4147°
<44/4>
4150°
<48/4> → <12>
5θ
4-isogonal
<5p/5>
560°
<15/5>
590°
<20/5>
5108°
open
5120°
<30/5>
5128.57°
<35/5>
5135°
<40/5>
5140°
<45/5>
5144°
<50/5> → <10>
5147°
<55/5>
5150°
<60/5>
6θ
5-isogonal
<6p/6>
660°
Open
690°
<24/6> → <12/3>
6108°
<30/6>
6120°
Open
6128.57°
<42/6>
6135°
<48/6> → <24/3>
6140°
<54/6> → <18/2>
6144°
<60/6> → <30/3>
6147°
<66/6>
6150°
<72/6> → <12>
Small rational divisor angles
Simple spirolaterals (rational divisors p/q) nθ or (1,2,...,n)θ
θ15°16 4/11°20°25 5/7°30°36°45°49 1/11°72°77 1/7°81 9/11°100°114 6/11°
180-θ
Turn angle
165°163 7/11°160°154 2/7°150°144°135°130 10/11°108°102 6/7°98 2/11°80°65 5/11°
nθ \ p/q24/1111/59/47/312/55/28/311/410/37/211/39/211/2
1θ
Regular
{p/q}
115°
{24/11}
116.36°
{11/5}
120°
{9/4}
125.71°
{7/3}
130°
{12/5}
136°
{5/2}
145°
{8/3}
149.10°
{11/4}
172°
{10/3}
177.14°
{7/2}
181.82°
{11/3}
1100°
{9/2}
1114.55°
{11/2}
2θ
Isogonal
<2p/2q>
215°
<48/22> → <24/11>
216.36°
<22/10>
220°
<18/8>
225.71°
<14/6>
230°
<24/10> → <12/5>
236°
<10/4>
245°
<16/6> → <8/3>
249.10°
<22/8>
272°
<20/6> → <10/3>
277.14°
<14/4>
281.82°
<22/6>
2100°
<18/4>
2114.55°
<22/4>
3θ
2-isogonal
<3p/3q>
315°
<72/33> → <24/11>
316.36°
<33/15>
320°
<27/12> → <9/4>
325.71°
<21/9>
330°
<36/15> → <12/5>
336°
<15/6>
345°
<24/9>
349.10°
<33/12>
372°
<30/9>
377.14°
<21/6>
381.82°
<33/9>
3100°
<27/6> → <9/2>
3114.55°
<33/6>
4θ
3-isogonal
<4p/4q>
415°
<96/44> → <24/11>
416.36°
<44/20>
420°
<36/12>
425.71°
<28/4>
430°
<48/40> → <12/5>
436°
<20/8>
445°
<32/12> → <8/3>
449.10°
<44/16>
472°
<40/12> → <20/6>
477.14°
<28/8>
481.82°
<44/12>
4100°
<36/8>
4114.55°
<44/8>
5θ
4-isogonal
<5p/5q>
515°
<120/55>
516.36°
<55/25>
520°
<45/20>
525.71°
<35/15>
530°
<60/25>
536°
open
545°
<40/15>
549.10°
<55/20>
572°
<50/15> → <10/3>
577.14°
<35/10>
581.82°
<55/15>
5100°
<45/10>
5114.55°
<55/10>
6θ
5-isogonal
<6p/6q>
615°
<144/66> → <24/11>
616.36°
<66/30>
620°
<54/24> → <18/8>
625.71°
<42/18>
630°
<72/30> → <12/5>
636°
<30/12>
645°
<48/18> → <24/9>
649.10°
<66/24>
672°
<60/18> → <30/9>
677.14°
<42/12>
681.82°
<66/18>
6100°
<54/12> → <18/4>
6114.55°
<66/12>
See also
Wikimedia Commons has media related to Spirolaterals.
• Turtle graphics represent a computer language that defines an open or close path as move lengths and turn angles.
References
1. Gardner, M. Worm Paths Ch. 17 Knotted Doughnuts and Other Mathematical Entertainments New York: W. H. Freeman, pp. 205-221, 1986.
2. Abelson, Harold, diSessa, Andera, 1980, Turtle Geometry, MIT Press, pp.37-39, 120-122
3. Focus on...Spirolaterals Secondary Magazine Issue 78
4. Frank Odds, British biochemist, 8/29/1945-7/7/2020
5. Odds, Frank C. Spirolaterals, Mathematics Teacher, Feb 1973, Volume 66: Issue 2, pp. 121–124 DOI
6. Weisstein, Eric W. "Spirolateral". MathWorld.
7. Anna Weltman, This is Not a Math Book A Graphic Activity Book, Kane Miller; Act Csm edition, 2017
8. "Practice Multiplication with Simple Spirolateral Math Art". 23 July 2015.
• Alice Kaseberg Schwandt Spirolaterals: An advanced Investignation from an Elementary Standpoint, Mathematical Teacher, Vol 72, 1979, 166-169
• Margaret Kenney and Stanley Bezuszka, Square Spirolaterals Mathematics Teaching, Vol 95, 1981, pp. 22–27
• Gascoigne, Serafim Turtle Fun LOGO for the Spectrum 48K pp 42-46 | Spirolaterals 1985
• Wells, D. The Penguin Dictionary of Curious and Interesting Geometry London: Penguin, pp. 239–241, 1991.
• Krawczyk, Robert, "Hilbert's Building Blocks", Mathematics & Design, The University of the Basque Country, pp. 281–288, 1998.
• Krawczyk, Robert, Spirolaterals, Complexity from Simplicity, International Society of Arts, Mathematics and Architecture 99,The University of the Basque Country, pp. 293–299, 1999.
• Krawczyk, Robert J. The Art of Spirolateral reversals
External links
• Spirolaterals Javascript App
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Wikipedia
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Spitalfields Mathematical Society
The Spitalfields Mathematical Society was founded in 1717 by Joseph Middleton.[1] The society had 64 members when it was established, and at first meetings were held in the Monmouth's Head, a public house in the Spitalfields district of London.[2][3] Fellows of the society were drawn from artisans and craftsmen such as weavers, apothecaries, brewers, ironmongers, stockbrokers, and makers of optical and mathematical instruments. Well-known members included John Canton, John Dollond, Thomas Simpson, John Crosley, John Tatum, Francis Baily, and Benjamin Gompertz.[4]
It merged with the Royal Astronomical Society in 1846.[1]
The name lives on in the "Spitalfields Days" organised by, among others, the Isaac Newton Institute, Cambridge, Mathematics Research Centre, Warwick, and International Centre for Mathematical Sciences, Edinburgh.[5]
References
1. Sampson, R. A. (1923). "The Decade 1840–1850". In Dreyer, J. L. E.; Turner, H. H. (eds.). History of the Royal Astronomical Society, 1820–1920. London, United Kingdom: Royal Astronomical Society. pp. 99–104. ISBN 0-632-02173-X.
2. Stewart, Larry; Weindling, Paul (1995). "Philosophical threads: natural philosophy and public experiment among the weavers of Spitalfields". British Journal for the History of Science. 28: 40.
3. O'Connor, John J.; Robertson, Edmund F. "The Spitalfields Mathematical Society". Retrieved 24 February 2015.
4. Stewart, Larry; Weindling, Paul (1995). "Philosophical threads: natural philosophy and public experiment among the weavers of Spitalfields". British Journal for the History of Science. 28: 41–2.
5. "Spitalfields Days". London Mathematical Society. Retrieved 24 February 2015.
• http://www.mernick.org.uk/thhol/mathematical.html
• http://technicaleducationmatters.org/2009/05/14/the-spitalfields-mathematical-society-1717-to-1846/
• Cawthorne. H.H. ‘The Spitalfields Mathematical Society’. (1717 – 1845). Journal of Adult Education. Vol. 111. No. 2. (April 1929). Cassels.
• J.W.S. ‘The Spitalfields Mathematical Society’ Bulletin of LMS. 11 p. 241 – 258. 1979.
Authority control
International
• VIAF
National
• United States
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Wikipedia
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Splicing rule
In mathematics and computer science, a splicing rule is a transformation on formal languages which formalises the action of gene splicing in molecular biology. A splicing language is a language generated by iterated application of a splicing rule: the splicing languages form a proper subset of the regular languages.
Definition
Let A be an alphabet and L a language, that is, a subset of the free monoid A∗. A splicing rule is a quadruple r = (a,b,c,d) of elements of A∗, and the action of the rule r on L is to produce the language
$r(L)=\{xady:xabq,pcdy\in L\}\ .$
If R is a set of rules then R(L) is the union of the languages produced by the rules of R. We say that R respects L if R(L) is a subset of L. The R-closure of L is the union of L and all iterates of R on L: clearly it is respected by R. A splicing language is the R-closure of a finite language.[1]
A rule set R is reflexive if (a,b,c,d) in R implies that (a,b,a,b) and (c,d,c,d) are in R. A splicing language is reflexive if it is defined by a reflexive rule set.[2]
Examples
• Let A = {a,b,c}. The rule (caba,a,cab,a) applied to the finite set {cabb,cabab,cabaab} generates the regular language caba∗b.[3]
Properties
• All splicing languages are regular.[4]
• Not all regular languages are splicing.[5] An example is (aa)∗ over {a,b}.[4]
• If L is a regular language on the alphabet A, and z is a letter not in A, then the language { zw : w in L } is a splicing language.[3]
• There is an algorithm to determine whether a given regular language is a reflexive splicing language.[2]
• The set of splicing rules that respect a regular language can be determined from the syntactic monoid of the language.[6]
References
1. Anderson (2006) p. 236
2. Anderson (2006) p. 242
3. Anderson (2006) p. 238
4. Anderson (2006) p. 239
5. Anderson (2006) p. 240
6. Anderson (2006) p. 241
• Anderson, James A. (2006). Automata theory with modern applications. With contributions by Tom Head. Cambridge: Cambridge University Press. ISBN 0-521-61324-8. Zbl 1127.68049.
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Wikipedia
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Spline interpolation
In the mathematical field of numerical analysis, spline interpolation is a form of interpolation where the interpolant is a special type of piecewise polynomial called a spline. That is, instead of fitting a single, high-degree polynomial to all of the values at once, spline interpolation fits low-degree polynomials to small subsets of the values, for example, fitting nine cubic polynomials between each of the pairs of ten points, instead of fitting a single degree-ten polynomial to all of them. Spline interpolation is often preferred over polynomial interpolation because the interpolation error can be made small even when using low-degree polynomials for the spline.[1] Spline interpolation also avoids the problem of Runge's phenomenon, in which oscillation can occur between points when interpolating using high-degree polynomials.
For broader coverage of this topic, see Spline (mathematics).
Introduction
Originally, spline was a term for elastic rulers that were bent to pass through a number of predefined points, or knots. These were used to make technical drawings for shipbuilding and construction by hand, as illustrated in the figure.
We wish to model similar kinds of curves using a set of mathematical equations. Assume we have a sequence of $n+1$ knots, $(x_{0},y_{0})$ through $(x_{n},y_{n})$. There will be a cubic polynomial $q_{i}(x)=y$ between each successive pair of knots $(x_{i-1},y_{i-1})$ and $(x_{i},y_{i})$ connecting to both of them, where $i=1,2,\dots ,n$. So there will be $n$ polynomials, with the first polynomial starting at $(x_{0},y_{0})$, and the last polynomial ending at $(x_{n},y_{n})$.
The curvature of any curve $y=y(x)$ is defined as
$\kappa ={\frac {y''}{(1+y'^{2})^{3/2}}},$
where $y'$ and $y''$ are the first and second derivatives of $y(x)$ with respect to $x$. To make the spline take a shape that minimizes the bending (under the constraint of passing through all knots), we will define both $y'$ and $y''$ to be continuous everywhere, including at the knots. Each successive polynomial must have equal values (which are equal to the y-value of the corresponding datapoint), derivatives, and second derivatives at their joining knots, which is to say that
${\begin{cases}q_{i}(x_{i})=q_{i+1}(x_{i})=y_{i}\\q'_{i}(x_{i})=q'_{i+1}(x_{i})\\q''_{i}(x_{i})=q''_{i+1}(x_{i})\end{cases}}\qquad 1\leq i\leq n-1.$
This can only be achieved if polynomials of degree 3 (cubic polynomials) or higher are used. The classical approach is to use polynomials of exactly degree 3 — cubic splines.
In addition to the three conditions above, a 'natural cubic spline' has the condition that $q''_{1}(x_{0})=q''_{n}(x_{n})=0$.
In addition to the three main conditions above, a 'clamped cubic spline' has the conditions that $q'_{1}(x_{0})=f'(x_{0})$ and $q'_{n}(x_{n})=f'(x_{n})$ where $f'(x)$ is the derivative of the interpolated function.
In addition to the three main conditions above, a 'not-a-knot spline' has the conditions that $q'''_{1}(x_{1})=q'''_{2}(x_{1})$ and $q'''_{n-1}(x_{n-1})=q'''_{n}(x_{n-1})$.[2]
Algorithm to find the interpolating cubic spline
We wish to find each polynomial $q_{i}(x)$ given the points $(x_{0},y_{0})$ through $(x_{n},y_{n})$. To do this, we will consider just a single piece of the curve, $q(x)$, which will interpolate from $(x_{1},y_{1})$ to $(x_{2},y_{2})$. This piece will have slopes $k_{1}$ and $k_{2}$ at its endpoints. Or, more precisely,
$q(x_{1})=y_{1},$
$q(x_{2})=y_{2},$
$q'(x_{1})=k_{1},$
$q'(x_{2})=k_{2}.$
The full equation $q(x)$ can be written in the symmetrical form
$q(x)={\big (}1-t(x){\big )}\,y_{1}+t(x)\,y_{2}+t(x){\big (}1-t(x){\big )}{\Big (}{\big (}1-t(x){\big )}\,a+t(x)\,b{\Big )},$
(1)
where
$t(x)={\frac {x-x_{1}}{x_{2}-x_{1}}},$
(2)
$a=k_{1}(x_{2}-x_{1})-(y_{2}-y_{1}),$
(3)
$b=-k_{2}(x_{2}-x_{1})+(y_{2}-y_{1}).$
(4)
But what are $k_{1}$ and $k_{2}$? To derive these critical values, we must consider that
$q'={\frac {dq}{dx}}={\frac {dq}{dt}}{\frac {dt}{dx}}={\frac {dq}{dt}}{\frac {1}{x_{2}-x_{1}}}.$
It then follows that
$q'={\frac {y_{2}-y_{1}}{x_{2}-x_{1}}}+(1-2t){\frac {a(1-t)+bt}{x_{2}-x_{1}}}+t(1-t){\frac {b-a}{x_{2}-x_{1}}},$
(5)
$q''=2{\frac {b-2a+(a-b)3t}{{(x_{2}-x_{1})}^{2}}}.$
(6)
Setting t = 0 and t = 1 respectively in equations (5) and (6), one gets from (2) that indeed first derivatives q′(x1) = k1 and q′(x2) = k2, and also second derivatives
$q''(x_{1})=2{\frac {b-2a}{{(x_{2}-x_{1})}^{2}}},$
(7)
$q''(x_{2})=2{\frac {a-2b}{{(x_{2}-x_{1})}^{2}}}.$
(8)
If now (xi, yi), i = 0, 1, ..., n are n + 1 points, and
$q_{i}=(1-t)\,y_{i-1}+t\,y_{i}+t(1-t){\big (}(1-t)\,a_{i}+t\,b_{i}{\big )},$
(9)
where i = 1, 2, ..., n, and $t={\tfrac {x-x_{i-1}}{x_{i}-x_{i-1}}}$ are n third-degree polynomials interpolating y in the interval xi−1 ≤ x ≤ xi for i = 1, ..., n such that q′i (xi) = q′i+1(xi) for i = 1, ..., n − 1, then the n polynomials together define a differentiable function in the interval x0 ≤ x ≤ xn, and
$a_{i}=k_{i-1}(x_{i}-x_{i-1})-(y_{i}-y_{i-1}),$
(10)
$b_{i}=-k_{i}(x_{i}-x_{i-1})+(y_{i}-y_{i-1})$
(11)
for i = 1, ..., n, where
$k_{0}=q_{1}'(x_{0}),$
(12)
$k_{i}=q_{i}'(x_{i})=q_{i+1}'(x_{i}),\qquad i=1,\dots ,n-1,$
(13)
$k_{n}=q_{n}'(x_{n}).$
(14)
If the sequence k0, k1, ..., kn is such that, in addition, q′′i(xi) = q′′i+1(xi) holds for i = 1, ..., n − 1, then the resulting function will even have a continuous second derivative.
From (7), (8), (10) and (11) follows that this is the case if and only if
${\frac {k_{i-1}}{x_{i}-x_{i-1}}}+\left({\frac {1}{x_{i}-x_{i-1}}}+{\frac {1}{x_{i+1}-x_{i}}}\right)2k_{i}+{\frac {k_{i+1}}{x_{i+1}-x_{i}}}=3\left({\frac {y_{i}-y_{i-1}}{{(x_{i}-x_{i-1})}^{2}}}+{\frac {y_{i+1}-y_{i}}{{(x_{i+1}-x_{i})}^{2}}}\right)$
(15)
for i = 1, ..., n − 1. The relations (15) are n − 1 linear equations for the n + 1 values k0, k1, ..., kn.
For the elastic rulers being the model for the spline interpolation, one has that to the left of the left-most "knot" and to the right of the right-most "knot" the ruler can move freely and will therefore take the form of a straight line with q′′ = 0. As q′′ should be a continuous function of x, "natural splines" in addition to the n − 1 linear equations (15) should have
$q''_{1}(x_{0})=2{\frac {3(y_{1}-y_{0})-(k_{1}+2k_{0})(x_{1}-x_{0})}{{(x_{1}-x_{0})}^{2}}}=0,$
$q''_{n}(x_{n})=-2{\frac {3(y_{n}-y_{n-1})-(2k_{n}+k_{n-1})(x_{n}-x_{n-1})}{{(x_{n}-x_{n-1})}^{2}}}=0,$
i.e. that
${\frac {2}{x_{1}-x_{0}}}k_{0}+{\frac {1}{x_{1}-x_{0}}}k_{1}=3{\frac {y_{1}-y_{0}}{(x_{1}-x_{0})^{2}}},$
(16)
${\frac {1}{x_{n}-x_{n-1}}}k_{n-1}+{\frac {2}{x_{n}-x_{n-1}}}k_{n}=3{\frac {y_{n}-y_{n-1}}{(x_{n}-x_{n-1})^{2}}}.$
(17)
Eventually, (15) together with (16) and (17) constitute n + 1 linear equations that uniquely define the n + 1 parameters k0, k1, ..., kn.
There exist other end conditions, "clamped spline", which specifies the slope at the ends of the spline, and the popular "not-a-knot spline", which requires that the third derivative is also continuous at the x1 and xn−1 points. For the "not-a-knot" spline, the additional equations will read:
$q'''_{1}(x_{1})=q'''_{2}(x_{1})\Rightarrow {\frac {1}{\Delta x_{1}^{2}}}k_{0}+\left({\frac {1}{\Delta x_{1}^{2}}}-{\frac {1}{\Delta x_{2}^{2}}}\right)k_{1}-{\frac {1}{\Delta x_{2}^{2}}}k_{2}=2\left({\frac {\Delta y_{1}}{\Delta x_{1}^{3}}}-{\frac {\Delta y_{2}}{\Delta x_{2}^{3}}}\right),$
$q'''_{n-1}(x_{n-1})=q'''_{n}(x_{n-1})\Rightarrow {\frac {1}{\Delta x_{n-1}^{2}}}k_{n-2}+\left({\frac {1}{\Delta x_{n-1}^{2}}}-{\frac {1}{\Delta x_{n}^{2}}}\right)k_{n-1}-{\frac {1}{\Delta x_{n}^{2}}}k_{n}=2\left({\frac {\Delta y_{n-1}}{\Delta x_{n-1}^{3}}}-{\frac {\Delta y_{n}}{\Delta x_{n}^{3}}}\right),$
where $\Delta x_{i}=x_{i}-x_{i-1},\ \Delta y_{i}=y_{i}-y_{i-1}$.
Example
In case of three points the values for $k_{0},k_{1},k_{2}$ are found by solving the tridiagonal linear equation system
${\begin{bmatrix}a_{11}&a_{12}&0\\a_{21}&a_{22}&a_{23}\\0&a_{32}&a_{33}\\\end{bmatrix}}{\begin{bmatrix}k_{0}\\k_{1}\\k_{2}\\\end{bmatrix}}={\begin{bmatrix}b_{1}\\b_{2}\\b_{3}\\\end{bmatrix}}$
with
$a_{11}={\frac {2}{x_{1}-x_{0}}},$
$a_{12}={\frac {1}{x_{1}-x_{0}}},$
$a_{21}={\frac {1}{x_{1}-x_{0}}},$
$a_{22}=2\left({\frac {1}{x_{1}-x_{0}}}+{\frac {1}{x_{2}-x_{1}}}\right),$
$a_{23}={\frac {1}{x_{2}-x_{1}}},$
$a_{32}={\frac {1}{x_{2}-x_{1}}},$
$a_{33}={\frac {2}{x_{2}-x_{1}}},$
$b_{1}=3{\frac {y_{1}-y_{0}}{(x_{1}-x_{0})^{2}}},$
$b_{2}=3\left({\frac {y_{1}-y_{0}}{{(x_{1}-x_{0})}^{2}}}+{\frac {y_{2}-y_{1}}{{(x_{2}-x_{1})}^{2}}}\right),$
$b_{3}=3{\frac {y_{2}-y_{1}}{(x_{2}-x_{1})^{2}}}.$
For the three points
$(-1,0.5),\ (0,0),\ (3,3),$
one gets that
$k_{0}=-0.6875,\ k_{1}=-0.1250,\ k_{2}=1.5625,$
and from (10) and (11) that
$a_{1}=k_{0}(x_{1}-x_{0})-(y_{1}-y_{0})=-0.1875,$
$b_{1}=-k_{1}(x_{1}-x_{0})+(y_{1}-y_{0})=-0.3750,$
$a_{2}=k_{1}(x_{2}-x_{1})-(y_{2}-y_{1})=-3.3750,$
$b_{2}=-k_{2}(x_{2}-x_{1})+(y_{2}-y_{1})=-1.6875.$
In the figure, the spline function consisting of the two cubic polynomials $q_{1}(x)$ and $q_{2}(x)$ given by (9) is displayed.
See also
• Cubic Hermite spline
• Centripetal Catmull–Rom spline
• Discrete spline interpolation
• Monotone cubic interpolation
• Non-uniform rational B-spline
• Multivariate interpolation
• Polynomial interpolation
• Smoothing spline
• Spline wavelet
• Thin plate spline
• Polyharmonic spline
Computer code
TinySpline: Open source C-library for splines which implements cubic spline interpolation
SciPy Spline Interpolation: a Python package that implements interpolation
Cubic Interpolation: Open source C#-library for cubic spline interpolation
References
1. Hall, Charles A.; Meyer, Weston W. (1976). "Optimal Error Bounds for Cubic Spline Interpolation". Journal of Approximation Theory. 16 (2): 105–122. doi:10.1016/0021-9045(76)90040-X.
2. Burden, Richard; Faires, Douglas (2015). Numerical Analysis (10th ed.). Cengage Learning. pp. 142–157. ISBN 9781305253667.
• Schoenberg, Isaac J. (1946). "Contributions to the Problem of Approximation of Equidistant Data by Analytic Functions: Part A.—On the Problem of Smoothing or Graduation. A First Class of Analytic Approximation Formulae". Quarterly of Applied Mathematics. 4 (2): 45–99. doi:10.1090/qam/15914.
• Schoenberg, Isaac J. (1946). "Contributions to the Problem of Approximation of Equidistant Data by Analytic Functions: Part B.—On the Problem of Osculatory Interpolation. A Second Class of Analytic Approximation Formulae". Quarterly of Applied Mathematics. 4 (2): 112–141. doi:10.1090/qam/16705.
External links
• Cubic Spline Interpolation Online Calculation and Visualization Tool (with JavaScript source code)
• "Spline interpolation", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Dynamic cubic splines with JSXGraph
• Lectures on the theory and practice of spline interpolation
• Paper which explains step by step how cubic spline interpolation is done, but only for equidistant knots.
• Numerical Recipes in C, Go to Chapter 3 Section 3-3
• A note on cubic splines
• Information about spline interpolation (including code in Fortran 77)
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Wikipedia
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Split-complex number
In algebra, a split complex number (or hyperbolic number, also perplex number, double number) is based on a hyperbolic unit j satisfying $j^{2}=1.$ A split-complex number has two real number components x and y, and is written $z=x+yj.$ The conjugate of z is $z^{*}=x-yj.$ Since $j^{2}=1,$ the product of a number z with its conjugate is $N(z):=zz^{*}=x^{2}-y^{2},$ an isotropic quadratic form.
The collection D of all split complex numbers $z=x+yj$ for $x,y\in \mathbb {R} $ forms an algebra over the field of real numbers. Two split-complex numbers w and z have a product wz that satisfies $N(wz)=N(w)N(z).$ This composition of N over the algebra product makes (D, +, ×, *) a composition algebra.
A similar algebra based on $\mathbb {R} ^{2}$ and component-wise operations of addition and multiplication, $(\mathbb {R} ^{2},+,\times ,xy),$ where xy is the quadratic form on $\mathbb {R} ^{2},$ also forms a quadratic space. The ring isomorphism
${\begin{aligned}D&\to \mathbb {R} ^{2}\\x+yj&\mapsto (x-y,x+y)\end{aligned}}$
relates proportional quadratic forms, but the mapping is not an isometry since the multiplicative identity (1, 1) of $\mathbb {R} ^{2}$ is at a distance ${\sqrt {2}}$ from 0, which is normalized in D.
Split-complex numbers have many other names; see § Synonyms below. See the article Motor variable for functions of a split-complex number.
Definition
A split-complex number is an ordered pair of real numbers, written in the form
$z=x+jy$
where x and y are real numbers and the hyperbolic unit[1] j satisfies
$j^{2}=+1$
In the field of complex numbers the imaginary unit i satisfies $i^{2}=-1.$ The change of sign distinguishes the split-complex numbers from the ordinary complex ones. The hyperbolic unit j is not a real number but an independent quantity.
The collection of all such z is called the split-complex plane. Addition and multiplication of split-complex numbers are defined by
${\begin{aligned}(x+jy)+(u+jv)&=(x+u)+j(y+v)\\(x+jy)(u+jv)&=(xu+yv)+j(xv+yu).\end{aligned}}$
This multiplication is commutative, associative and distributes over addition.
Conjugate, modulus, and bilinear form
Just as for complex numbers, one can define the notion of a split-complex conjugate. If
$z=x+jy~,$
then the conjugate of z is defined as
$z^{*}=x-jy~.$
The conjugate satisfies similar properties to usual complex conjugate. Namely,
${\begin{aligned}(z+w)^{*}&=z^{*}+w^{*}\\(zw)^{*}&=z^{*}w^{*}\\\left(z^{*}\right)^{*}&=z.\end{aligned}}$
These three properties imply that the split-complex conjugate is an automorphism of order 2.
The squared modulus of a split-complex number $z=x+jy$ is given by the isotropic quadratic form
$\lVert z\rVert ^{2}=zz^{*}=z^{*}z=x^{2}-y^{2}~.$
It has the composition algebra property:
$\lVert zw\rVert =\lVert z\rVert \lVert w\rVert ~.$
However, this quadratic form is not positive-definite but rather has signature (1, −1), so the modulus is not a norm.
The associated bilinear form is given by
$\langle z,w\rangle =\operatorname {\mathcal {R_{e}}} \left(zw^{*}\right)=\operatorname {\mathcal {R_{e}}} \left(z^{*}w\right)=xu-yv~,$
where $z=x+jy$ and $w=u+jv.$ Another expression for the squared modulus is then
$\lVert z\rVert ^{2}=\langle z,z\rangle ~.$
Since it is not positive-definite, this bilinear form is not an inner product; nevertheless the bilinear form is frequently referred to as an indefinite inner product. A similar abuse of language refers to the modulus as a norm.
A split-complex number is invertible if and only if its modulus is nonzero ($\lVert z\rVert \neq 0$), thus numbers of the form x ± j x have no inverse. The multiplicative inverse of an invertible element is given by
$z^{-1}={\frac {z^{*}}{{\lVert z\rVert }^{2}}}~.$
Split-complex numbers which are not invertible are called null vectors. These are all of the form (a ± j a) for some real number a.
The diagonal basis
There are two nontrivial idempotent elements given by $e={\tfrac {1}{2}}(1-j)$ and $e^{*}={\tfrac {1}{2}}(1+j).$ Recall that idempotent means that $ee=e$ and $e^{*}e^{*}=e^{*}.$ Both of these elements are null:
$\lVert e\rVert =\lVert e^{*}\rVert =e^{*}e=0~.$
It is often convenient to use e and e∗ as an alternate basis for the split-complex plane. This basis is called the diagonal basis or null basis. The split-complex number z can be written in the null basis as
$z=x+jy=(x-y)e+(x+y)e^{*}~.$
If we denote the number $z=ae+be^{*}$ for real numbers a and b by (a, b), then split-complex multiplication is given by
$\left(a_{1},b_{1}\right)\left(a_{2},b_{2}\right)=\left(a_{1}a_{2},b_{1}b_{2}\right)~.$
The split-complex conjugate in the diagonal basis is given by
$(a,b)^{*}=(b,a)$
and the modulus by
$\lVert (a,b)\rVert =ab.$
Isomorphism
On the basis {e, e*} it becomes clear that the split-complex numbers are ring-isomorphic to the direct sum $\mathbb {R} \oplus \mathbb {R} $ with addition and multiplication defined pairwise.
The diagonal basis for the split-complex number plane can be invoked by using an ordered pair (x, y) for $z=x+jy$ and making the mapping
$(u,v)=(x,y){\begin{pmatrix}1&1\\1&-1\end{pmatrix}}=(x,y)S~.$
Now the quadratic form is $uv=(x+y)(x-y)=x^{2}-y^{2}~.$ Furthermore,
$(\cosh a,\sinh a){\begin{pmatrix}1&1\\1&-1\end{pmatrix}}=\left(e^{a},e^{-a}\right)$
so the two parametrized hyperbolas are brought into correspondence with S.
The action of hyperbolic versor $e^{bj}\!$ then corresponds under this linear transformation to a squeeze mapping
$\sigma :(u,v)\mapsto \left(ru,{\frac {v}{r}}\right),\quad r=e^{b}~.$ :(u,v)\mapsto \left(ru,{\frac {v}{r}}\right),\quad r=e^{b}~.}
Though lying in the same isomorphism class in the category of rings, the split-complex plane and the direct sum of two real lines differ in their layout in the Cartesian plane. The isomorphism, as a planar mapping, consists of a counter-clockwise rotation by 45° and a dilation by √2. The dilation in particular has sometimes caused confusion in connection with areas of a hyperbolic sector. Indeed, hyperbolic angle corresponds to area of a sector in the $\mathbb {R} \oplus \mathbb {R} $ plane with its "unit circle" given by $\{(a,b)\in \mathbb {R} \oplus \mathbb {R} :ab=1\}.$ The contracted unit hyperbola $\{\cosh a+j\sinh a:a\in \mathbb {R} \}$ of the split-complex plane has only half the area in the span of a corresponding hyperbolic sector. Such confusion may be perpetuated when the geometry of the split-complex plane is not distinguished from that of $\mathbb {R} \oplus \mathbb {R} $.
Geometry
A two-dimensional real vector space with the Minkowski inner product is called (1 + 1)-dimensional Minkowski space, often denoted $\mathbb {R} ^{1,1}.$ Just as much of the geometry of the Euclidean plane $\mathbb {R} ^{2}$ can be described with complex numbers, the geometry of the Minkowski plane $\mathbb {R} ^{1,1}$ can be described with split-complex numbers.
The set of points
$\left\{z:\lVert z\rVert ^{2}=a^{2}\right\}$
is a hyperbola for every nonzero a in $\mathbb {R} .$ The hyperbola consists of a right and left branch passing through (a, 0) and (−a, 0). The case a = 1 is called the unit hyperbola. The conjugate hyperbola is given by
$\left\{z:\lVert z\rVert ^{2}=-a^{2}\right\}$
with an upper and lower branch passing through (0, a) and (0, −a). The hyperbola and conjugate hyperbola are separated by two diagonal asymptotes which form the set of null elements:
$\left\{z:\lVert z\rVert =0\right\}.$
These two lines (sometimes called the null cone) are perpendicular in $\mathbb {R} ^{2}$ and have slopes ±1.
Split-complex numbers z and w are said to be hyperbolic-orthogonal if ⟨z, w⟩ = 0. While analogous to ordinary orthogonality, particularly as it is known with ordinary complex number arithmetic, this condition is more subtle. It forms the basis for the simultaneous hyperplane concept in spacetime.
The analogue of Euler's formula for the split-complex numbers is
$\exp(j\theta )=\cosh(\theta )+j\sinh(\theta ).$
This formula can be derived from a power series expansion using the fact that cosh has only even powers while that for sinh has odd powers.[2] For all real values of the hyperbolic angle θ the split-complex number λ = exp(jθ) has norm 1 and lies on the right branch of the unit hyperbola. Numbers such as λ have been called hyperbolic versors.
Since λ has modulus 1, multiplying any split-complex number z by λ preserves the modulus of z and represents a hyperbolic rotation (also called a Lorentz boost or a squeeze mapping). Multiplying by λ preserves the geometric structure, taking hyperbolas to themselves and the null cone to itself.
The set of all transformations of the split-complex plane which preserve the modulus (or equivalently, the inner product) forms a group called the generalized orthogonal group O(1, 1). This group consists of the hyperbolic rotations, which form a subgroup denoted SO+(1, 1), combined with four discrete reflections given by
$z\mapsto \pm z$
and $z\mapsto \pm z^{*}.$
The exponential map
$\exp \colon (\mathbb {R} ,+)\to \mathrm {SO} ^{+}(1,1)$
sending θ to rotation by exp(jθ) is a group isomorphism since the usual exponential formula applies:
$e^{j(\theta +\phi )}=e^{j\theta }e^{j\phi }.$
If a split-complex number z does not lie on one of the diagonals, then z has a polar decomposition.
Algebraic properties
In abstract algebra terms, the split-complex numbers can be described as the quotient of the polynomial ring $\mathbb {R} [x]$ by the ideal generated by the polynomial $x^{2}-1,$
$\mathbb {R} [x]/(x^{2}-1).$
The image of x in the quotient is the "imaginary" unit j. With this description, it is clear that the split-complex numbers form a commutative algebra over the real numbers. The algebra is not a field since the null elements are not invertible. All of the nonzero null elements are zero divisors.
Since addition and multiplication are continuous operations with respect to the usual topology of the plane, the split-complex numbers form a topological ring.
The algebra of split-complex numbers forms a composition algebra since
$\lVert zw\rVert =\lVert z\rVert \lVert w\rVert ~$
for any numbers z and w.
From the definition it is apparent that the ring of split-complex numbers is isomorphic to the group ring $\mathbb {R} [C_{2}]$ of the cyclic group C2 over the real numbers $\mathbb {R} .$
Matrix representations
One can easily represent split-complex numbers by matrices. The split-complex number $z=x+jy$ can be represented by the matrix $z\mapsto {\begin{pmatrix}x&y\\y&x\end{pmatrix}}.$
Addition and multiplication of split-complex numbers are then given by matrix addition and multiplication. The modulus of z is given by the determinant of the corresponding matrix.
In fact there are many representations of the split-complex plane in the four-dimensional ring of 2x2 real matrices. The real multiples of the identity matrix form a real line in the matrix ring M(2,R). Any hyperbolic unit m provides a basis element with which to extend the real line to the split-complex plane. The matrices
$m={\begin{pmatrix}a&c\\b&-a\end{pmatrix}}$
which square to the identity matrix satisfy $a^{2}+bc=1.$ For example, when a = 0, then (b,c) is a point on the standard hyperbola. More generally, there is a hypersurface in M(2,R) of hyperbolic units, any one of which serves in a basis to represent the split-complex numbers as a subring of M(2,R).[3]
The number $z=x+jy$ can be represented by the matrix $x\ I+y\ m.$
History
The use of split-complex numbers dates back to 1848 when James Cockle revealed his tessarines.[4] William Kingdon Clifford used split-complex numbers to represent sums of spins. Clifford introduced the use of split-complex numbers as coefficients in a quaternion algebra now called split-biquaternions. He called its elements "motors", a term in parallel with the "rotor" action of an ordinary complex number taken from the circle group. Extending the analogy, functions of a motor variable contrast to functions of an ordinary complex variable.
Since the late twentieth century, the split-complex multiplication has commonly been seen as a Lorentz boost of a spacetime plane.[5][6][7][8][9][10] In that model, the number z = x + y j represents an event in a spatio-temporal plane, where x is measured in nanoseconds and y in Mermin's feet. The future corresponds to the quadrant of events {z : |y| < x}, which has the split-complex polar decomposition $z=\rho e^{aj}\!$. The model says that z can be reached from the origin by entering a frame of reference of rapidity a and waiting ρ nanoseconds. The split-complex equation
$e^{aj}\ e^{bj}=e^{(a+b)j}$
expressing products on the unit hyperbola illustrates the additivity of rapidities for collinear velocities. Simultaneity of events depends on rapidity a;
$\{z=\sigma je^{aj}:\sigma \in \mathbb {R} \}$
is the line of events simultaneous with the origin in the frame of reference with rapidity a.
Two events z and w are hyperbolic-orthogonal when $z^{*}w+zw^{*}=0.$ Canonical events exp(aj) and j exp(aj) are hyperbolic orthogonal and lie on the axes of a frame of reference in which the events simultaneous with the origin are proportional to j exp(aj).
In 1933 Max Zorn was using the split-octonions and noted the composition algebra property. He realized that the Cayley–Dickson construction, used to generate division algebras, could be modified (with a factor gamma, γ) to construct other composition algebras including the split-octonions. His innovation was perpetuated by Adrian Albert, Richard D. Schafer, and others.[11] The gamma factor, with R as base field, builds split-complex numbers as a composition algebra. Reviewing Albert for Mathematical Reviews, N. H. McCoy wrote that there was an "introduction of some new algebras of order 2e over F generalizing Cayley–Dickson algebras."[12] Taking F = R and e = 1 corresponds to the algebra of this article.
In 1935 J.C. Vignaux and A. Durañona y Vedia developed the split-complex geometric algebra and function theory in four articles in Contribución a las Ciencias Físicas y Matemáticas, National University of La Plata, República Argentina (in Spanish). These expository and pedagogical essays presented the subject for broad appreciation.[13]
In 1941 E.F. Allen used the split-complex geometric arithmetic to establish the nine-point hyperbola of a triangle inscribed in zz∗ = 1.[14]
In 1956 Mieczyslaw Warmus published "Calculus of Approximations" in Bulletin de l’Académie polonaise des sciences (see link in References). He developed two algebraic systems, each of which he called "approximate numbers", the second of which forms a real algebra.[15] D. H. Lehmer reviewed the article in Mathematical Reviews and observed that this second system was isomorphic to the "hyperbolic complex" numbers, the subject of this article.
In 1961 Warmus continued his exposition, referring to the components of an approximate number as midpoint and radius of the interval denoted.
Synonyms
Different authors have used a great variety of names for the split-complex numbers. Some of these include:
• (real) tessarines, James Cockle (1848)
• (algebraic) motors, W.K. Clifford (1882)
• hyperbolic complex numbers, J.C. Vignaux (1935), G. Cree (1949)[16]
• bireal numbers, U. Bencivenga (1946)
• real hyperbolic numbers, N. Smith (1949)[17]
• approximate numbers, Warmus (1956), for use in interval analysis
• double numbers, I.M. Yaglom (1968), Kantor and Solodovnikov (1989), Hazewinkel (1990), Rooney (2014)
• hyperbolic numbers, W. Miller & R. Boehning (1968),[18] G. Sobczyk (1995)
• anormal-complex numbers, W. Benz (1973)
• perplex numbers, P. Fjelstad (1986) and Poodiack & LeClair (2009)
• countercomplex or hyperbolic, Carmody (1988)
• Lorentz numbers, F.R. Harvey (1990)
• semi-complex numbers, F. Antonuccio (1994)
• paracomplex numbers, Cruceanu, Fortuny & Gadea (1996)
• split-complex numbers, B. Rosenfeld (1997)[19]
• spacetime numbers, N. Borota (2000)
• Study numbers, P. Lounesto (2001)
• twocomplex numbers, S. Olariu (2002)
• split binarions, K. McCrimmon (2004)
See also
The Wikibook Associative Composition Algebra has a page on the topic of: Split binarions
• Minkowski space
• Split-quaternion
• Hypercomplex number
References
1. Vladimir V. Kisil (2012) Geometry of Mobius Transformations: Elliptic, Parabolic, and Hyperbolic actions of SL(2,R), pages 2, 161, Imperial College Press ISBN 978-1-84816-858-9
2. James Cockle (1848) On a New Imaginary in Algebra, Philosophical Magazine 33:438
3. Abstract Algebra/2x2 real matrices at Wikibooks
4. James Cockle (1849) On a New Imaginary in Algebra 34:37–47, London-Edinburgh-Dublin Philosophical Magazine (3) 33:435–9, link from Biodiversity Heritage Library.
5. Francesco Antonuccio (1994) Semi-complex analysis and mathematical physics
6. F. Catoni, D. Boccaletti, R. Cannata, V. Catoni, E. Nichelatti, P. Zampetti. (2008) The Mathematics of Minkowski Space-Time, Birkhäuser Verlag, Basel. Chapter 4: Trigonometry in the Minkowski plane. ISBN 978-3-7643-8613-9.
7. Francesco Catoni; Dino Boccaletti; Roberto Cannata; Vincenzo Catoni; Paolo Zampetti (2011). "Chapter 2: Hyperbolic Numbers". Geometry of Minkowski Space-Time. Springer Science & Business Media. ISBN 978-3-642-17977-8.
8. Fjelstad, Paul (1986), "Extending special relativity via the perplex numbers", American Journal of Physics, 54 (5): 416–422, doi:10.1119/1.14605
9. Louis Kauffman (1985) "Transformations in Special Relativity", International Journal of Theoretical Physics 24:223–36.
10. Sobczyk, G.(1995) Hyperbolic Number Plane, also published in College Mathematics Journal 26:268–80.
11. Robert B. Brown (1967)On Generalized Cayley-Dickson Algebras, Pacific Journal of Mathematics 20(3):415–22, link from Project Euclid.
12. N.H. McCoy (1942) Review of "Quadratic forms permitting composition" by A.A. Albert, Mathematical Reviews #0006140
13. Vignaux, J.(1935) "Sobre el numero complejo hiperbolico y su relacion con la geometria de Borel", Contribucion al Estudio de las Ciencias Fisicas y Matematicas, Universidad Nacional de la Plata, Republica Argentina
14. Allen, E.F. (1941) "On a Triangle Inscribed in a Rectangular Hyperbola", American Mathematical Monthly 48(10): 675–681
15. M. Warmus (1956) "Calculus of Approximations", Bulletin de l'Académie polonaise des sciences, Vol. 4, No. 5, pp. 253–257, MR0081372
16. Cree, George C. (1949). The Number Theory of a System of Hyperbolic Complex Numbers (MA thesis). McGill University.
17. Smith, Norman E. (1949). Introduction to Hyperbolic Number Theory (MA thesis). McGill University.
18. Miller, William; Boehning, Rochelle (1968). "Gaussian, parabolic, and hyperbolic numbers". The Mathematics Teacher. 61 (4): 377–382. doi:10.5951/MT.61.4.0377. JSTOR 27957849.
19. Rosenfeld, B. (1997) Geometry of Lie Groups, page 30, Kluwer Academic Publishers ISBN 0-7923-4390-5
Further reading
• Bencivenga, Uldrico (1946) "Sulla rappresentazione geometrica delle algebre doppie dotate di modulo", Atti della Reale Accademia delle Scienze e Belle-Lettere di Napoli, Ser (3) v.2 No7. MR0021123.
• Walter Benz (1973) Vorlesungen uber Geometrie der Algebren, Springer
• N. A. Borota, E. Flores, and T. J. Osler (2000) "Spacetime numbers the easy way", Mathematics and Computer Education 34: 159–168.
• N. A. Borota and T. J. Osler (2002) "Functions of a spacetime variable", Mathematics and Computer Education 36: 231–239.
• K. Carmody, (1988) "Circular and hyperbolic quaternions, octonions, and sedenions", Appl. Math. Comput. 28:47–72.
• K. Carmody, (1997) "Circular and hyperbolic quaternions, octonions, and sedenions – further results", Appl. Math. Comput. 84:27–48.
• William Kingdon Clifford (1882) Mathematical Works, A. W. Tucker editor, page 392, "Further Notes on Biquaternions"
• V.Cruceanu, P. Fortuny & P.M. Gadea (1996) A Survey on Paracomplex Geometry, Rocky Mountain Journal of Mathematics 26(1): 83–115, link from Project Euclid.
• De Boer, R. (1987) "An also known as list for perplex numbers", American Journal of Physics 55(4):296.
• Anthony A. Harkin & Joseph B. Harkin (2004) Geometry of Generalized Complex Numbers, Mathematics Magazine 77(2):118–29.
• F. Reese Harvey. Spinors and calibrations. Academic Press, San Diego. 1990. ISBN 0-12-329650-1. Contains a description of normed algebras in indefinite signature, including the Lorentz numbers.
• Hazewinkle, M. (1994) "Double and dual numbers", Encyclopaedia of Mathematics, Soviet/AMS/Kluwer, Dordrect.
• Kevin McCrimmon (2004) A Taste of Jordan Algebras, pp 66, 157, Universitext, Springer ISBN 0-387-95447-3 MR2014924
• C. Musès, "Applied hypernumbers: Computational concepts", Appl. Math. Comput. 3 (1977) 211–226.
• C. Musès, "Hypernumbers II—Further concepts and computational applications", Appl. Math. Comput. 4 (1978) 45–66.
• Olariu, Silviu (2002) Complex Numbers in N Dimensions, Chapter 1: Hyperbolic Complex Numbers in Two Dimensions, pages 1–16, North-Holland Mathematics Studies #190, Elsevier ISBN 0-444-51123-7.
• Poodiack, Robert D. & Kevin J. LeClair (2009) "Fundamental theorems of algebra for the perplexes", The College Mathematics Journal 40(5):322–35.
• Isaak Yaglom (1968) Complex Numbers in Geometry, translated by E. Primrose from 1963 Russian original, Academic Press, pp. 18–20.
• J. Rooney (2014). "Generalised Complex Numbers in Mechanics". In Marco Ceccarelli and Victor A. Glazunov (ed.). Advances on Theory and Practice of Robots and Manipulators: Proceedings of Romansy 2014 XX CISM-IFToMM Symposium on Theory and Practice of Robots and Manipulators. Mechanisms and Machine Science. Vol. 22. Springer. pp. 55–62. doi:10.1007/978-3-319-07058-2_7. ISBN 978-3-319-07058-2.
Number systems
Sets of definable numbers
• Natural numbers ($\mathbb {N} $)
• Integers ($\mathbb {Z} $)
• Rational numbers ($\mathbb {Q} $)
• Constructible numbers
• Algebraic numbers ($\mathbb {A} $)
• Closed-form numbers
• Periods
• Computable numbers
• Arithmetical numbers
• Set-theoretically definable numbers
• Gaussian integers
Composition algebras
• Division algebras: Real numbers ($\mathbb {R} $)
• Complex numbers ($\mathbb {C} $)
• Quaternions ($\mathbb {H} $)
• Octonions ($\mathbb {O} $)
Split
types
• Over $\mathbb {R} $:
• Split-complex numbers
• Split-quaternions
• Split-octonions
Over $\mathbb {C} $:
• Bicomplex numbers
• Biquaternions
• Bioctonions
Other hypercomplex
• Dual numbers
• Dual quaternions
• Dual-complex numbers
• Hyperbolic quaternions
• Sedenions ($\mathbb {S} $)
• Split-biquaternions
• Multicomplex numbers
• Geometric algebra/Clifford algebra
• Algebra of physical space
• Spacetime algebra
Other types
• Cardinal numbers
• Extended natural numbers
• Irrational numbers
• Fuzzy numbers
• Hyperreal numbers
• Levi-Civita field
• Surreal numbers
• Transcendental numbers
• Ordinal numbers
• p-adic numbers (p-adic solenoids)
• Supernatural numbers
• Profinite integers
• Superreal numbers
• Normal numbers
• Classification
• List
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Wikipedia
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Epimorphism
In category theory, an epimorphism (also called an epic morphism or, colloquially, an epi) is a morphism f : X → Y that is right-cancellative in the sense that, for all objects Z and all morphisms g1, g2: Y → Z,
$g_{1}\circ f=g_{2}\circ f\implies g_{1}=g_{2}.$
Epimorphisms are categorical analogues of onto or surjective functions (and in the category of sets the concept corresponds exactly to the surjective functions), but they may not exactly coincide in all contexts; for example, the inclusion $\mathbb {Z} \to \mathbb {Q} $ is a ring epimorphism. The dual of an epimorphism is a monomorphism (i.e. an epimorphism in a category C is a monomorphism in the dual category Cop).
Many authors in abstract algebra and universal algebra define an epimorphism simply as an onto or surjective homomorphism. Every epimorphism in this algebraic sense is an epimorphism in the sense of category theory, but the converse is not true in all categories. In this article, the term "epimorphism" will be used in the sense of category theory given above. For more on this, see § Terminology below.
Examples
Every morphism in a concrete category whose underlying function is surjective is an epimorphism. In many concrete categories of interest the converse is also true. For example, in the following categories, the epimorphisms are exactly those morphisms that are surjective on the underlying sets:
• Set: sets and functions. To prove that every epimorphism f: X → Y in Set is surjective, we compose it with both the characteristic function g1: Y → {0,1} of the image f(X) and the map g2: Y → {0,1} that is constant 1.
• Rel: sets with binary relations and relation-preserving functions. Here we can use the same proof as for Set, equipping {0,1} with the full relation {0,1}×{0,1}.
• Pos: partially ordered sets and monotone functions. If f : (X, ≤) → (Y, ≤) is not surjective, pick y0 in Y \ f(X) and let g1 : Y → {0,1} be the characteristic function of {y | y0 ≤ y} and g2 : Y → {0,1} the characteristic function of {y | y0 < y}. These maps are monotone if {0,1} is given the standard ordering 0 < 1.
• Grp: groups and group homomorphisms. The result that every epimorphism in Grp is surjective is due to Otto Schreier (he actually proved more, showing that every subgroup is an equalizer using the free product with one amalgamated subgroup); an elementary proof can be found in (Linderholm 1970).
• FinGrp: finite groups and group homomorphisms. Also due to Schreier; the proof given in (Linderholm 1970) establishes this case as well.
• Ab: abelian groups and group homomorphisms.
• K-Vect: vector spaces over a field K and K-linear transformations.
• Mod-R: right modules over a ring R and module homomorphisms. This generalizes the two previous examples; to prove that every epimorphism f: X → Y in Mod-R is surjective, we compose it with both the canonical quotient map g 1: Y → Y/f(X) and the zero map g2: Y → Y/f(X).
• Top: topological spaces and continuous functions. To prove that every epimorphism in Top is surjective, we proceed exactly as in Set, giving {0,1} the indiscrete topology, which ensures that all considered maps are continuous.
• HComp: compact Hausdorff spaces and continuous functions. If f: X → Y is not surjective, let y ∈ Y − fX. Since fX is closed, by Urysohn's Lemma there is a continuous function g1:Y → [0,1] such that g1 is 0 on fX and 1 on y. We compose f with both g1 and the zero function g2: Y → [0,1].
However, there are also many concrete categories of interest where epimorphisms fail to be surjective. A few examples are:
• In the category of monoids, Mon, the inclusion map N → Z is a non-surjective epimorphism. To see this, suppose that g1 and g2 are two distinct maps from Z to some monoid M. Then for some n in Z, g1(n) ≠ g2(n), so g1(−n) ≠ g2(−n). Either n or −n is in N, so the restrictions of g1 and g2 to N are unequal.
• In the category of algebras over commutative ring R, take R[N] → R[Z], where R[G] is the group ring of the group G and the morphism is induced by the inclusion N → Z as in the previous example. This follows from the observation that 1 generates the algebra R[Z] (note that the unit in R[Z] is given by 0 of Z), and the inverse of the element represented by n in Z is just the element represented by −n. Thus any homomorphism from R[Z] is uniquely determined by its value on the element represented by 1 of Z.
• In the category of rings, Ring, the inclusion map Z → Q is a non-surjective epimorphism; to see this, note that any ring homomorphism on Q is determined entirely by its action on Z, similar to the previous example. A similar argument shows that the natural ring homomorphism from any commutative ring R to any one of its localizations is an epimorphism.
• In the category of commutative rings, a finitely generated homomorphism of rings f : R → S is an epimorphism if and only if for all prime ideals P of R, the ideal Q generated by f(P) is either S or is prime, and if Q is not S, the induced map Frac(R/P) → Frac(S/Q) is an isomorphism (EGA IV 17.2.6).
• In the category of Hausdorff spaces, Haus, the epimorphisms are precisely the continuous functions with dense images. For example, the inclusion map Q → R, is a non-surjective epimorphism.
The above differs from the case of monomorphisms where it is more frequently true that monomorphisms are precisely those whose underlying functions are injective.
As for examples of epimorphisms in non-concrete categories:
• If a monoid or ring is considered as a category with a single object (composition of morphisms given by multiplication), then the epimorphisms are precisely the right-cancellable elements.
• If a directed graph is considered as a category (objects are the vertices, morphisms are the paths, composition of morphisms is the concatenation of paths), then every morphism is an epimorphism.
Properties
Every isomorphism is an epimorphism; indeed only a right-sided inverse is needed: if there exists a morphism j : Y → X such that fj = idY, then f: X → Y is easily seen to be an epimorphism. A map with such a right-sided inverse is called a split epi. In a topos, a map that is both a monic morphism and an epimorphism is an isomorphism.
The composition of two epimorphisms is again an epimorphism. If the composition fg of two morphisms is an epimorphism, then f must be an epimorphism.
As some of the above examples show, the property of being an epimorphism is not determined by the morphism alone, but also by the category of context. If D is a subcategory of C, then every morphism in D that is an epimorphism when considered as a morphism in C is also an epimorphism in D. However the converse need not hold; the smaller category can (and often will) have more epimorphisms.
As for most concepts in category theory, epimorphisms are preserved under equivalences of categories: given an equivalence F : C → D, a morphism f is an epimorphism in the category C if and only if F(f) is an epimorphism in D. A duality between two categories turns epimorphisms into monomorphisms, and vice versa.
The definition of epimorphism may be reformulated to state that f : X → Y is an epimorphism if and only if the induced maps
${\begin{matrix}\operatorname {Hom} (Y,Z)&\rightarrow &\operatorname {Hom} (X,Z)\\g&\mapsto &gf\end{matrix}}$
are injective for every choice of Z. This in turn is equivalent to the induced natural transformation
${\begin{matrix}\operatorname {Hom} (Y,-)&\rightarrow &\operatorname {Hom} (X,-)\end{matrix}}$
being a monomorphism in the functor category SetC.
Every coequalizer is an epimorphism, a consequence of the uniqueness requirement in the definition of coequalizers. It follows in particular that every cokernel is an epimorphism. The converse, namely that every epimorphism be a coequalizer, is not true in all categories.
In many categories it is possible to write every morphism as the composition of an epimorphism followed by a monomorphism. For instance, given a group homomorphism f : G → H, we can define the group K = im(f) and then write f as the composition of the surjective homomorphism G → K that is defined like f, followed by the injective homomorphism K → H that sends each element to itself. Such a factorization of an arbitrary morphism into an epimorphism followed by a monomorphism can be carried out in all abelian categories and also in all the concrete categories mentioned above in § Examples (though not in all concrete categories).
Related concepts
Among other useful concepts are regular epimorphism, extremal epimorphism, immediate epimorphism, strong epimorphism, and split epimorphism.
• An epimorphism is said to be regular if it is a coequalizer of some pair of parallel morphisms.
• An epimorphism $\varepsilon $ is said to be extremal[1] if in each representation $\varepsilon =\mu \circ \varphi $, where $\mu $ is a monomorphism, the morphism $\mu $ is automatically an isomorphism.
• An epimorphism $\varepsilon $ is said to be immediate if in each representation $\varepsilon =\mu \circ \varepsilon '$, where $\mu $ is a monomorphism and $\varepsilon '$ is an epimorphism, the morphism $\mu $ is automatically an isomorphism.
• An epimorphism $\varepsilon :A\to B$ is said to be strong[1][2] if for any monomorphism $\mu :C\to D$ and any morphisms $\alpha :A\to C$ and $\beta :B\to D$ such that $\beta \circ \varepsilon =\mu \circ \alpha $, there exists a morphism $\delta :B\to C$ such that $\delta \circ \varepsilon =\alpha $ and $\mu \circ \delta =\beta $.
• An epimorphism $\varepsilon $ is said to be split if there exists a morphism $\mu $ such that $\varepsilon \circ \mu =1$ (in this case $\mu $ is called a right-sided inverse for $\varepsilon $).
There is also the notion of homological epimorphism in ring theory. A morphism f: A → B of rings is a homological epimorphism if it is an epimorphism and it induces a full and faithful functor on derived categories: D(f) : D(B) → D(A).
A morphism that is both a monomorphism and an epimorphism is called a bimorphism. Every isomorphism is a bimorphism but the converse is not true in general. For example, the map from the half-open interval [0,1) to the unit circle S1 (thought of as a subspace of the complex plane) that sends x to exp(2πix) (see Euler's formula) is continuous and bijective but not a homeomorphism since the inverse map is not continuous at 1, so it is an instance of a bimorphism that is not an isomorphism in the category Top. Another example is the embedding Q → R in the category Haus; as noted above, it is a bimorphism, but it is not bijective and therefore not an isomorphism. Similarly, in the category of rings, the map Z → Q is a bimorphism but not an isomorphism.
Epimorphisms are used to define abstract quotient objects in general categories: two epimorphisms f1 : X → Y1 and f2 : X → Y2 are said to be equivalent if there exists an isomorphism j : Y1 → Y2 with j f1 = f2. This is an equivalence relation, and the equivalence classes are defined to be the quotient objects of X.
Terminology
The companion terms epimorphism and monomorphism were first introduced by Bourbaki. Bourbaki uses epimorphism as shorthand for a surjective function. Early category theorists believed that epimorphisms were the correct analogue of surjections in an arbitrary category, similar to how monomorphisms are very nearly an exact analogue of injections. Unfortunately this is incorrect; strong or regular epimorphisms behave much more closely to surjections than ordinary epimorphisms. Saunders Mac Lane attempted to create a distinction between epimorphisms, which were maps in a concrete category whose underlying set maps were surjective, and epic morphisms, which are epimorphisms in the modern sense. However, this distinction never caught on.
It is a common mistake to believe that epimorphisms are either identical to surjections or that they are a better concept. Unfortunately this is rarely the case; epimorphisms can be very mysterious and have unexpected behavior. It is very difficult, for example, to classify all the epimorphisms of rings. In general, epimorphisms are their own unique concept, related to surjections but fundamentally different.
See also
• List of category theory topics
• Monomorphism
Notes
1. Borceux 1994.
2. Tsalenko & Shulgeifer 1974.
References
• Adámek, Jiří; Herrlich, Horst; Strecker, George E. (1990). Abstract and Concrete Categories (PDF). John Wiley & Sons. ISBN 0-471-60922-6.
• Bergman, George (2015). An Invitation to General Algebra and Universal Constructions. Springer. ISBN 978-3-319-11478-1.
• Borceux, Francis (1994). Handbook of Categorical Algebra. Volume 1: Basic Category Theory. Cambridge University Press. ISBN 978-0521061193.
• Riehl, Emily (2016). Category Theory in Context. Dover Publications, Inc Mineola, New York. ISBN 9780486809038.
• Tsalenko, M.S.; Shulgeifer, E.G. (1974). Foundations of category theory. Nauka. ISBN 5-02-014427-4.
• "Epimorphism", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Lawvere, F. William; Rosebrugh, Robert (2015). Sets for Mathematics. Cambridge university press. ISBN 978-0-521-80444-8.
• Linderholm, Carl (1970). "A Group Epimorphism is Surjective". American Mathematical Monthly. 77 (2): 176–177. doi:10.1080/00029890.1970.11992448.
External links
• epimorphism at the nLab
• Strong epimorphism at the nLab
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Section (category theory)
In category theory, a branch of mathematics, a section is a right inverse of some morphism. Dually, a retraction is a left inverse of some morphism. In other words, if $f:X\to Y$ and $g:Y\to X$ are morphisms whose composition $f\circ g:Y\to Y$ is the identity morphism on $Y$, then $g$ is a section of $f$, and $f$ is a retraction of $g$.[1]
Every section is a monomorphism (every morphism with a left inverse is left-cancellative), and every retraction is an epimorphism (every morphism with a right inverse is right-cancellative).
In algebra, sections are also called split monomorphisms and retractions are also called split epimorphisms. In an abelian category, if $f:X\to Y$ is a split epimorphism with split monomorphism $g:Y\to X$, then $X$ is isomorphic to the direct sum of $Y$ and the kernel of $f$. The synonym coretraction for section is sometimes seen in the literature, although rarely in recent work.
Properties
• A section that is also an epimorphism is an isomorphism. Dually a retraction that is also a monomorphism is an isomorphism.
Terminology
The concept of a retraction in category theory comes from the essentially similar notion of a retraction in topology: $f:X\to Y$ where $Y$ is a subspace of $X$ is a retraction in the topological sense, if it's a retraction of the inclusion map $i:Y\hookrightarrow X$ in the category theory sense. The concept in topology was defined by Karol Borsuk in 1931.[2]
Borsuk's student, Samuel Eilenberg, was with Saunders Mac Lane the founder of category theory, and (as the earliest publications on category theory concerned various topological spaces) one might have expected this term to have initially be used. In fact, their earlier publications, up to, e.g., Mac Lane (1963)'s Homology, used the term right inverse. It was not until 1965 when Eilenberg and John Coleman Moore coined the dual term 'coretraction' that Borsuk's term was lifted to category theory in general.[3] The term coretraction gave way to the term section by the end of the 1960s.
Both use of left/right inverse and section/retraction are commonly seen in the literature: the former use has the advantage that it is familiar from the theory of semigroups and monoids; the latter is considered less confusing by some because one does not have to think about 'which way around' composition goes, an issue that has become greater with the increasing popularity of the synonym f;g for g∘f.[4]
Examples
In the category of sets, every monomorphism (injective function) with a non-empty domain is a section, and every epimorphism (surjective function) is a retraction; the latter statement is equivalent to the axiom of choice.
In the category of vector spaces over a field K, every monomorphism and every epimorphism splits; this follows from the fact that linear maps can be uniquely defined by specifying their values on a basis.
In the category of abelian groups, the epimorphism Z → Z/2Z which sends every integer to its remainder modulo 2 does not split; in fact the only morphism Z/2Z → Z is the zero map. Similarly, the natural monomorphism Z/2Z → Z/4Z doesn't split even though there is a non-trivial morphism Z/4Z → Z/2Z.
The categorical concept of a section is important in homological algebra, and is also closely related to the notion of a section of a fiber bundle in topology: in the latter case, a section of a fiber bundle is a section of the bundle projection map of the fiber bundle.
Given a quotient space ${\bar {X}}$ with quotient map $\pi \colon X\to {\bar {X}}$, a section of $\pi $ is called a transversal.
Bibliography
• Mac Lane, Saunders (1978). Categories for the working mathematician (2nd ed.). Springer Verlag.
• Barry, Mitchell (1965). Theory of categories. Academic Press.
See also
• Splitting lemma
• Inverse function#Left and right inverses
• Transversal (combinatorics)
Notes
1. Mac Lane (1978, p.19).
2. Borsuk, Karol (1931), "Sur les rétractes", Fundamenta Mathematicae, 17: 152–170, doi:10.4064/fm-17-1-152-170, Zbl 0003.02701
3. Eilenberg, S., & Moore, J. C. (1965). Foundations of relative homological algebra. Memoirs of the American Mathematical Society number 55. American Mathematical Society, Providence: RI, OCLC 1361982. The term was popularised by Barry Mitchell (1965)'s influential Theory of categories.
4. Cf. e.g., https://blog.juliosong.com/linguistics/mathematics/category-theory-notes-9/
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Split exact sequence
In mathematics, a split exact sequence is a short exact sequence in which the middle term is built out of the two outer terms in the simplest possible way.
Equivalent characterizations
A short exact sequence of abelian groups or of modules over a fixed ring, or more generally of objects in an abelian category
$0\to A\mathrel {\stackrel {a}{\to }} B\mathrel {\stackrel {b}{\to }} C\to 0$
is called split exact if it is isomorphic to the exact sequence where the middle term is the direct sum of the outer ones:
$0\to A\mathrel {\stackrel {i}{\to }} A\oplus C\mathrel {\stackrel {p}{\to }} C\to 0$
The requirement that the sequence is isomorphic means that there is an isomorphism $f:B\to A\oplus C$ such that the composite $f\circ a$ is the natural inclusion $i:A\to A\oplus C$ and such that the composite $p\circ f$ equals b. This can be summarized by a commutative diagram as:
The splitting lemma provides further equivalent characterizations of split exact sequences.
Examples
A trivial example of a split short exact sequence is
$0\to M_{1}\mathrel {\stackrel {q}{\to }} M_{1}\oplus M_{2}\mathrel {\stackrel {p}{\to }} M_{2}\to 0$
where $M_{1},M_{2}$ are R-modules, $q$ is the canonical injection and $p$ is the canonical projection.
Any short exact sequence of vector spaces is split exact. This is a rephrasing of the fact that any set of linearly independent vectors in a vector space can be extended to a basis.
The exact sequence $0\to \mathbf {Z} \mathrel {\stackrel {2}{\to }} \mathbf {Z} \to \mathbf {Z} /2\to 0$ (where the first map is multiplication by 2) is not split exact.
Related notions
Pure exact sequences can be characterized as the filtered colimits of split exact sequences.[1]
References
1. Fuchs (2015, Ch. 5, Thm. 3.4)
Sources
• Fuchs, László (2015), Abelian Groups, Springer Monographs in Mathematics, Springer, ISBN 9783319194226
• Sharp, R. Y., Rodney (2001), Steps in Commutative Algebra, 2nd ed., London Mathematical Society Student Texts, Cambridge University Press, ISBN 0521646235
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Group extension
In mathematics, a group extension is a general means of describing a group in terms of a particular normal subgroup and quotient group. If $Q$ and $N$ are two groups, then $G$ is an extension of $Q$ by $N$ if there is a short exact sequence
$1\to N\;{\overset {\iota }{\to }}\;G\;{\overset {\pi }{\to }}\;Q\to 1.$
If $G$ is an extension of $Q$ by $N$, then $G$ is a group, $\iota (N)$ is a normal subgroup of $G$ and the quotient group $G/\iota (N)$ is isomorphic to the group $Q$. Group extensions arise in the context of the extension problem, where the groups $Q$ and $N$ are known and the properties of $G$ are to be determined. Note that the phrasing "$G$ is an extension of $N$ by $Q$" is also used by some.[1]
Since any finite group $G$ possesses a maximal normal subgroup $N$ with simple factor group $G/N$, all finite groups may be constructed as a series of extensions with finite simple groups. This fact was a motivation for completing the classification of finite simple groups.
An extension is called a central extension if the subgroup $N$ lies in the center of $G$.
Extensions in general
One extension, the direct product, is immediately obvious. If one requires $G$ and $Q$ to be abelian groups, then the set of isomorphism classes of extensions of $Q$ by a given (abelian) group $N$ is in fact a group, which is isomorphic to
$\operatorname {Ext} _{\mathbb {Z} }^{1}(Q,N);$
cf. the Ext functor. Several other general classes of extensions are known but no theory exists that treats all the possible extensions at one time. Group extension is usually described as a hard problem; it is termed the extension problem.
To consider some examples, if $G=K\times H$, then $G$ is an extension of both $H$ and $K$. More generally, if $G$ is a semidirect product of $K$ and $H$, written as $G=K\rtimes H$, then $G$ is an extension of $H$ by $K$, so such products as the wreath product provide further examples of extensions.
Extension problem
The question of what groups $G$ are extensions of $H$ by $N$ is called the extension problem, and has been studied heavily since the late nineteenth century. As to its motivation, consider that the composition series of a finite group is a finite sequence of subgroups $\{A_{i}\}$, where each $\{A_{i+1}\}$ is an extension of $\{A_{i}\}$ by some simple group. The classification of finite simple groups gives us a complete list of finite simple groups; so the solution to the extension problem would give us enough information to construct and classify all finite groups in general.
Classifying extensions
Solving the extension problem amounts to classifying all extensions of H by K; or more practically, by expressing all such extensions in terms of mathematical objects that are easier to understand and compute. In general, this problem is very hard, and all the most useful results classify extensions that satisfy some additional condition.
It is important to know when two extensions are equivalent or congruent. We say that the extensions
$1\to K{\stackrel {i}{{}\to {}}}G{\stackrel {\pi }{{}\to {}}}H\to 1$
and
$1\to K{\stackrel {i'}{{}\to {}}}G'{\stackrel {\pi '}{{}\to {}}}H\to 1$
are equivalent (or congruent) if there exists a group isomorphism $T:G\to G'$ making commutative the diagram of Figure 1. In fact it is sufficient to have a group homomorphism; due to the assumed commutativity of the diagram, the map $T$ is forced to be an isomorphism by the short five lemma.
Warning
It may happen that the extensions $1\to K\to G\to H\to 1$ and $1\to K\to G^{\prime }\to H\to 1$ are inequivalent but G and G' are isomorphic as groups. For instance, there are $8$ inequivalent extensions of the Klein four-group by $\mathbb {Z} /2\mathbb {Z} $,[2] but there are, up to group isomorphism, only four groups of order $8$ containing a normal subgroup of order $2$ with quotient group isomorphic to the Klein four-group.
Trivial extensions
A trivial extension is an extension
$1\to K\to G\to H\to 1$
that is equivalent to the extension
$1\to K\to K\times H\to H\to 1$
where the left and right arrows are respectively the inclusion and the projection of each factor of $K\times H$.
Classifying split extensions
A split extension is an extension
$1\to K\to G\to H\to 1$
with a homomorphism $s\colon H\to G$ such that going from H to G by s and then back to H by the quotient map of the short exact sequence induces the identity map on H i.e., $\pi \circ s=\mathrm {id} _{H}$. In this situation, it is usually said that s splits the above exact sequence.
Split extensions are very easy to classify, because an extension is split if and only if the group G is a semidirect product of K and H. Semidirect products themselves are easy to classify, because they are in one-to-one correspondence with homomorphisms from $H\to \operatorname {Aut} (K)$, where Aut(K) is the automorphism group of K. For a full discussion of why this is true, see semidirect product.
Warning on terminology
In general in mathematics, an extension of a structure K is usually regarded as a structure L of which K is a substructure. See for example field extension. However, in group theory the opposite terminology has crept in, partly because of the notation $\operatorname {Ext} (Q,N)$, which reads easily as extensions of Q by N, and the focus is on the group Q.
A paper of Ronald Brown and Timothy Porter on Otto Schreier's theory of nonabelian extensions uses the terminology that an extension of K gives a larger structure.[3]
Central extension
A central extension of a group G is a short exact sequence of groups
$1\to A\to E\to G\to 1$
such that A is included in $Z(E)$, the center of the group E. The set of isomorphism classes of central extensions of G by A is in one-to-one correspondence with the cohomology group $H^{2}(G,A)$.
Examples of central extensions can be constructed by taking any group G and any abelian group A, and setting E to be $A\times G$. This kind of split example corresponds to the element 0 in $H^{2}(G,A)$ under the above correspondence. Another split example is given for a normal subgroup A with E set to the semidirect product $A\rtimes G$. More serious examples are found in the theory of projective representations, in cases where the projective representation cannot be lifted to an ordinary linear representation.
In the case of finite perfect groups, there is a universal perfect central extension.
Similarly, the central extension of a Lie algebra ${\mathfrak {g}}$ is an exact sequence
$0\rightarrow {\mathfrak {a}}\rightarrow {\mathfrak {e}}\rightarrow {\mathfrak {g}}\rightarrow 0$
such that ${\mathfrak {a}}$ is in the center of ${\mathfrak {e}}$.
There is a general theory of central extensions in Maltsev varieties.[4]
Generalization to general extensions
There is a similar classification of all extensions of G by A in terms of homomorphisms from $G\to \operatorname {Out} (A)$, a tedious but explicitly checkable existence condition involving $H^{3}(G,Z(A))$ and the cohomology group $H^{2}(G,Z(A))$.[5]
Lie groups
In Lie group theory, central extensions arise in connection with algebraic topology. Roughly speaking, central extensions of Lie groups by discrete groups are the same as covering groups. More precisely, a connected covering space G∗ of a connected Lie group G is naturally a central extension of G, in such a way that the projection
$\pi \colon G^{*}\to G$
is a group homomorphism, and surjective. (The group structure on G∗ depends on the choice of an identity element mapping to the identity in G.) For example, when G∗ is the universal cover of G, the kernel of π is the fundamental group of G, which is known to be abelian (see H-space). Conversely, given a Lie group G and a discrete central subgroup Z, the quotient G/Z is a Lie group and G is a covering space of it.
More generally, when the groups A, E and G occurring in a central extension are Lie groups, and the maps between them are homomorphisms of Lie groups, then if the Lie algebra of G is g, that of A is a, and that of E is e, then e is a central Lie algebra extension of g by a. In the terminology of theoretical physics, generators of a are called central charges. These generators are in the center of e; by Noether's theorem, generators of symmetry groups correspond to conserved quantities, referred to as charges.
The basic examples of central extensions as covering groups are:
• the spin groups, which double cover the special orthogonal groups, which (in even dimension) doubly cover the projective orthogonal group.
• the metaplectic groups, which double cover the symplectic groups.
The case of SL2(R) involves a fundamental group that is infinite cyclic. Here the central extension involved is well known in modular form theory, in the case of forms of weight ½. A projective representation that corresponds is the Weil representation, constructed from the Fourier transform, in this case on the real line. Metaplectic groups also occur in quantum mechanics.
See also
• Lie algebra extension
• Virasoro algebra
• HNN extension
• Group contraction
• Extension of a topological group
References
1. group+extension#Definition at the nLab Remark 2.2.
2. page no. 830, Dummit, David S., Foote, Richard M., Abstract algebra (Third edition), John Wiley & Sons, Inc., Hoboken, NJ (2004).
3. Brown, Ronald; Porter, Timothy (1996). "On the Schreier theory of non-abelian extensions: generalisations and computations". Proceedings of the Royal Irish Academy Sect A. 96 (2): 213–227. MR 1641218.
4. Janelidze, George; Kelly, Gregory Maxwell (2000). "Central extensions in Malt'sev varieties". Theory and Applications of Categories. 7 (10): 219–226. MR 1774075.
5. P. J. Morandi, Group Extensions and H3 Archived 2018-05-17 at the Wayback Machine. From his collection of short mathematical notes.
• Mac Lane, Saunders (1975), Homology, Classics in Mathematics, Springer Verlag, ISBN 3-540-58662-8
• R.L. Taylor, Covering groups of non connected topological groups, Proceedings of the American Mathematical Society, vol. 5 (1954), 753–768.
• R. Brown and O. Mucuk, Covering groups of non-connected topological groups revisited, Mathematical Proceedings of the Cambridge Philosophical Society, vol. 115 (1994), 97–110.
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Real form (Lie theory)
In mathematics, the notion of a real form relates objects defined over the field of real and complex numbers. A real Lie algebra g0 is called a real form of a complex Lie algebra g if g is the complexification of g0:
${\mathfrak {g}}\simeq {\mathfrak {g}}_{0}\otimes _{\mathbb {R} }\mathbb {C} .$
Lie groups and Lie algebras
Classical groups
• General linear GL(n)
• Special linear SL(n)
• Orthogonal O(n)
• Special orthogonal SO(n)
• Unitary U(n)
• Special unitary SU(n)
• Symplectic Sp(n)
Simple Lie groups
Classical
• An
• Bn
• Cn
• Dn
Exceptional
• G2
• F4
• E6
• E7
• E8
Other Lie groups
• Circle
• Lorentz
• Poincaré
• Conformal group
• Diffeomorphism
• Loop
• Euclidean
Lie algebras
• Lie group–Lie algebra correspondence
• Exponential map
• Adjoint representation
• Killing form
• Index
• Simple Lie algebra
• Loop algebra
• Affine Lie algebra
Semisimple Lie algebra
• Dynkin diagrams
• Cartan subalgebra
• Root system
• Weyl group
• Real form
• Complexification
• Split Lie algebra
• Compact Lie algebra
Representation theory
• Lie group representation
• Lie algebra representation
• Representation theory of semisimple Lie algebras
• Representations of classical Lie groups
• Theorem of the highest weight
• Borel–Weil–Bott theorem
Lie groups in physics
• Particle physics and representation theory
• Lorentz group representations
• Poincaré group representations
• Galilean group representations
Scientists
• Sophus Lie
• Henri Poincaré
• Wilhelm Killing
• Élie Cartan
• Hermann Weyl
• Claude Chevalley
• Harish-Chandra
• Armand Borel
• Glossary
• Table of Lie groups
The notion of a real form can also be defined for complex Lie groups. Real forms of complex semisimple Lie groups and Lie algebras have been completely classified by Élie Cartan.
Real forms for Lie groups and algebraic groups
Using the Lie correspondence between Lie groups and Lie algebras, the notion of a real form can be defined for Lie groups. In the case of linear algebraic groups, the notions of complexification and real form have a natural description in the language of algebraic geometry.
Classification
Main article: List of simple Lie groups
Just as complex semisimple Lie algebras are classified by Dynkin diagrams, the real forms of a semisimple Lie algebra are classified by Satake diagrams, which are obtained from the Dynkin diagram of the complex form by labeling some vertices black (filled), and connecting some other vertices in pairs by arrows, according to certain rules.
It is a basic fact in the structure theory of complex semisimple Lie algebras that every such algebra has two special real forms: one is the compact real form and corresponds to a compact Lie group under the Lie correspondence (its Satake diagram has all vertices blackened), and the other is the split real form and corresponds to a Lie group that is as far as possible from being compact (its Satake diagram has no vertices blackened and no arrows). In the case of the complex special linear group SL(n,C), the compact real form is the special unitary group SU(n) and the split real form is the real special linear group SL(n,R). The classification of real forms of semisimple Lie algebras was accomplished by Élie Cartan in the context of Riemannian symmetric spaces. In general, there may be more than two real forms.
Suppose that g0 is a semisimple Lie algebra over the field of real numbers. By Cartan's criterion, the Killing form is nondegenerate, and can be diagonalized in a suitable basis with the diagonal entries +1 or −1. By Sylvester's law of inertia, the number of positive entries, or the positive index of inertia, is an invariant of the bilinear form, i.e. it does not depend on the choice of the diagonalizing basis. This is a number between 0 and the dimension of g which is an important invariant of the real Lie algebra, called its index.
Split real form
See also: Split Lie algebra
A real form g0 of a finite-dimensional complex semisimple Lie algebra g is said to be split, or normal, if in each Cartan decomposition g0 = k0 ⊕ p0, the space p0 contains a maximal abelian subalgebra of g0, i.e. its Cartan subalgebra. Élie Cartan proved that every complex semisimple Lie algebra g has a split real form, which is unique up to isomorphism.[1] It has maximal index among all real forms.
The split form corresponds to the Satake diagram with no vertices blackened and no arrows.
Compact real form
See also: Compact Lie algebra
A real Lie algebra g0 is called compact if the Killing form is negative definite, i.e. the index of g0 is zero. In this case g0 = k0 is a compact Lie algebra. It is known that under the Lie correspondence, compact Lie algebras correspond to compact Lie groups.
The compact form corresponds to the Satake diagram with all vertices blackened.
Construction of the compact real form
In general, the construction of the compact real form uses structure theory of semisimple Lie algebras. For classical Lie algebras there is a more explicit construction.
Let g0 be a real Lie algebra of matrices over R that is closed under the transpose map,
$X\to {X}^{t}.$
Then g0 decomposes into the direct sum of its skew-symmetric part k0 and its symmetric part p0. This is the Cartan decomposition:
${\mathfrak {g}}_{0}={\mathfrak {k}}_{0}\oplus {\mathfrak {p}}_{0}.$
The complexification g of g0 decomposes into the direct sum of g0 and ig0. The real vector space of matrices
${\mathfrak {u}}_{0}={\mathfrak {k}}_{0}\oplus i{\mathfrak {p}}_{0}$
is a subspace of the complex Lie algebra g that is closed under the commutators and consists of skew-hermitian matrices. It follows that u0 is a real Lie subalgebra of g, that its Killing form is negative definite (making it a compact Lie algebra), and that the complexification of u0 is g. Therefore, u0 is a compact form of g.
See also
• Complexification (Lie group)
Notes
1. Helgason 1978, p. 426
References
• Helgason, Sigurdur (1978), Differential geometry, Lie groups and symmetric spaces, Academic Press, ISBN 0-12-338460-5
• Knapp, Anthony (2004), Lie Groups: Beyond an Introduction, Progress in Mathematics, vol. 140, Birkhäuser, ISBN 0-8176-4259-5
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Split link
In the mathematical field of knot theory, a split link is a link that has a (topological) 2-sphere in its complement separating one or more link components from the others.[1] A split link is said to be splittable, and a link that is not split is called a non-split link or not splittable. Whether a link is split or non-split corresponds to whether the link complement is reducible or irreducible as a 3-manifold.
A link with an alternating diagram, i.e. an alternating link, will be non-split if and only if this diagram is connected. This is a result of the work of William Menasco.[2] A split link has many connected, non-alternating link diagrams.
References
1. Cromwell, Peter R. (2004), Knots and Links, Cambridge University Press, Definition 4.1.1, p. 78, ISBN 9780521548311.
2. Lickorish, W. B. Raymond (1997), An Introduction to Knot Theory, Graduate Texts in Mathematics, vol. 175, Springer, p. 32, ISBN 9780387982540.
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Split-octonion
In mathematics, the split-octonions are an 8-dimensional nonassociative algebra over the real numbers. Unlike the standard octonions, they contain non-zero elements which are non-invertible. Also the signatures of their quadratic forms differ: the split-octonions have a split signature (4,4) whereas the octonions have a positive-definite signature (8,0).
Up to isomorphism, the octonions and the split-octonions are the only two 8-dimensional composition algebras over the real numbers. They are also the only two octonion algebras over the real numbers. Split-octonion algebras analogous to the split-octonions can be defined over any field.
Definition
Cayley–Dickson construction
The octonions and the split-octonions can be obtained from the Cayley–Dickson construction by defining a multiplication on pairs of quaternions. We introduce a new imaginary unit ℓ and write a pair of quaternions (a, b) in the form a + ℓb. The product is defined by the rule:[1]
$(a+\ell b)(c+\ell d)=(ac+\lambda {\bar {d}}b)+\ell (da+b{\bar {c}})$
where
$\lambda =\ell ^{2}.$
If λ is chosen to be −1, we get the octonions. If, instead, it is taken to be +1 we get the split-octonions. One can also obtain the split-octonions via a Cayley–Dickson doubling of the split-quaternions. Here either choice of λ (±1) gives the split-octonions.
Multiplication table
A basis for the split-octonions is given by the set $\{\ 1,\ i,\ j,\ k,\ \ell ,\ \ell i,\ \ell j,\ \ell k\ \}$.
Every split-octonion $x$ can be written as a linear combination of the basis elements,
$x=x_{0}+x_{1}\,i+x_{2}\,j+x_{3}\,k+x_{4}\,\ell +x_{5}\,\ell i+x_{6}\,\ell j+x_{7}\,\ell k,$
with real coefficients $x_{a}$.
By linearity, multiplication of split-octonions is completely determined by the following multiplication table:
multiplier
$1$ $i$ $j$ $k$ $\ell $ $\ell i$ $\ell j$ $\ell k$
multiplicand $1$ $1$ $i$ $j$ $k$ $\ell $ $\ell i$ $\ell j$ $\ell k$
$i$ $i$ $-1$ $k$ $-j$ $-\ell i$ $\ell $ $-\ell k$ $\ell j$
$j$ $j$ $-k$ $-1$ $i$ $-\ell j$ $\ell k$ $\ell $ $-\ell i$
$k$ $k$ $j$ $-i$ $-1$ $-\ell k$ $-\ell j$ $\ell i$ $\ell $
$\ell $ $\ell $ $\ell i$ $\ell j$ $\ell k$ $1$ $i$ $j$ $k$
$\ell i$ $\ell i$ $-\ell $ $-\ell k$ $\ell j$ $-i$ $1$ $k$ $-j$
$\ell j$ $\ell j$ $\ell k$ $-\ell $ $-\ell i$ $-j$ $-k$ $1$ $i$
$\ell k$ $\ell k$ $-\ell j$ $\ell i$ $-\ell $ $-k$ $j$ $-i$ $1$
A convenient mnemonic is given by the diagram at the right, which represents the multiplication table for the split-octonions. This one is derived from its parent octonion (one of 480 possible), which is defined by:
$e_{i}e_{j}=-\delta _{ij}e_{0}+\varepsilon _{ijk}e_{k},\,$
where $\delta _{ij}$ is the Kronecker delta and $\varepsilon _{ijk}$ is the Levi-Civita symbol with value $+1$ when $ijk=123,154,176,264,257,374,365,$ and:
$e_{i}e_{0}=e_{0}e_{i}=e_{i};\,\,\,\,e_{0}e_{0}=e_{0},$
with $e_{0}$ the scalar element, and $i,j,k=1...7.$
The red arrows indicate possible direction reversals imposed by negating the lower right quadrant of the parent creating a split octonion with this multiplication table.
Conjugate, norm and inverse
The conjugate of a split-octonion x is given by
${\bar {x}}=x_{0}-x_{1}\,i-x_{2}\,j-x_{3}\,k-x_{4}\,\ell -x_{5}\,\ell i-x_{6}\,\ell j-x_{7}\,\ell k,$
just as for the octonions.
The quadratic form on x is given by
$N(x)={\bar {x}}x=(x_{0}^{2}+x_{1}^{2}+x_{2}^{2}+x_{3}^{2})-(x_{4}^{2}+x_{5}^{2}+x_{6}^{2}+x_{7}^{2}).$
This quadratic form N(x) is an isotropic quadratic form since there are non-zero split-octonions x with N(x) = 0. With N, the split-octonions form a pseudo-Euclidean space of eight dimensions over R, sometimes written R4,4 to denote the signature of the quadratic form.
If N(x) ≠ 0, then x has a (two-sided) multiplicative inverse x−1 given by
$x^{-1}=N(x)^{-1}{\bar {x}}.$
Properties
The split-octonions, like the octonions, are noncommutative and nonassociative. Also like the octonions, they form a composition algebra since the quadratic form N is multiplicative. That is,
$N(xy)=N(x)N(y).$
The split-octonions satisfy the Moufang identities and so form an alternative algebra. Therefore, by Artin's theorem, the subalgebra generated by any two elements is associative. The set of all invertible elements (i.e. those elements for which N(x) ≠ 0) form a Moufang loop.
The automorphism group of the split-octonions is a 14-dimensional Lie group, the split real form of the exceptional simple Lie group G2.
Zorn's vector-matrix algebra
Since the split-octonions are nonassociative they cannot be represented by ordinary matrices (matrix multiplication is always associative). Zorn found a way to represent them as "matrices" containing both scalars and vectors using a modified version of matrix multiplication.[2] Specifically, define a vector-matrix to be a 2×2 matrix of the form[3][4][5][6]
${\begin{bmatrix}a&\mathbf {v} \\\mathbf {w} &b\end{bmatrix}},$
where a and b are real numbers and v and w are vectors in R3. Define multiplication of these matrices by the rule
${\begin{bmatrix}a&\mathbf {v} \\\mathbf {w} &b\end{bmatrix}}{\begin{bmatrix}a'&\mathbf {v} '\\\mathbf {w} '&b'\end{bmatrix}}={\begin{bmatrix}aa'+\mathbf {v} \cdot \mathbf {w} '&a\mathbf {v} '+b'\mathbf {v} +\mathbf {w} \times \mathbf {w} '\\a'\mathbf {w} +b\mathbf {w} '-\mathbf {v} \times \mathbf {v} '&bb'+\mathbf {v} '\cdot \mathbf {w} \end{bmatrix}}$
where · and × are the ordinary dot product and cross product of 3-vectors. With addition and scalar multiplication defined as usual the set of all such matrices forms a nonassociative unital 8-dimensional algebra over the reals, called Zorn's vector-matrix algebra.
Define the "determinant" of a vector-matrix by the rule
$\det {\begin{bmatrix}a&\mathbf {v} \\\mathbf {w} &b\end{bmatrix}}=ab-\mathbf {v} \cdot \mathbf {w} $.
This determinant is a quadratic form on Zorn's algebra which satisfies the composition rule:
$\det(AB)=\det(A)\det(B).\,$
Zorn's vector-matrix algebra is, in fact, isomorphic to the algebra of split-octonions. Write an octonion $x$ in the form
$x=(a+\mathbf {v} )+\ell (b+\mathbf {w} )$
where $a$ and $b$ are real numbers and v and w are pure imaginary quaternions regarded as vectors in R3. The isomorphism from the split-octonions to Zorn's algebra is given by
$x\mapsto \phi (x)={\begin{bmatrix}a+b&\mathbf {v} +\mathbf {w} \\-\mathbf {v} +\mathbf {w} &a-b\end{bmatrix}}.$
This isomorphism preserves the norm since $N(x)=\det(\phi (x))$.
Applications
Split-octonions are used in the description of physical law. For example:
• The Dirac equation in physics (the equation of motion of a free spin 1/2 particle, like e.g. an electron or a proton) can be expressed on native split-octonion arithmetic.[7]
• Supersymmetric quantum mechanics has an octonionic extension.[8]
• The Zorn-based split-octonion algebra can be used in modeling local gauge symmetric SU(3) quantum chromodynamics.[9]
• The problem of a ball rolling without slipping on a ball of radius 3 times as large has the split real form of the exceptional group G2 as its symmetry group, owing to the fact that this problem can be described using split-octonions.[10]
References
1. Kevin McCrimmon (2004) A Taste of Jordan Algebras, page 158, Universitext, Springer ISBN 0-387-95447-3 MR2014924
2. Max Zorn (1931) "Alternativekörper und quadratische Systeme", Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 9(3/4): 395–402
3. Nathan Jacobson (1962) Lie Algebras, page 142, Interscience Publishers.
4. Schafer, Richard D. (1966). An Introduction to Nonassociative Algebras. Academic Press. pp. 52–6. ISBN 0-486-68813-5.
5. Lowell J. Page (1963) "Jordan Algebras", pages 144–186 in Studies in Modern Algebra edited by A.A. Albert, Mathematics Association of America : Zorn’s vector-matrix algebra on page 180
6. Arthur A. Sagle & Ralph E. Walde (1973) Introduction to Lie Groups and Lie Algebras, page 199, Academic Press
7. M. Gogberashvili (2006) "Octonionic Electrodynamics", Journal of Physics A 39: 7099-7104. doi:10.1088/0305-4470/39/22/020
8. V. Dzhunushaliev (2008) "Non-associativity, supersymmetry and hidden variables", Journal of Mathematical Physics 49: 042108 doi:10.1063/1.2907868; arXiv:0712.1647
9. B. Wolk, Adv. Appl. Clifford Algebras 27(4), 3225 (2017).
10. J. Baez and J. Huerta, G2 and the rolling ball, Trans. Amer. Math. Soc. 366, 5257-5293 (2014); arXiv:1205.2447.
• Harvey, F. Reese (1990). Spinors and Calibrations. San Diego: Academic Press. ISBN 0-12-329650-1.
• Nash, Patrick L (1990) "On the structure of the split octonion algebra", Il Nuovo Cimento B 105(1): 31–41. doi:10.1007/BF02723550
• Springer, T. A.; F. D. Veldkamp (2000). Octonions, Jordan Algebras and Exceptional Groups. Springer-Verlag. ISBN 3-540-66337-1.
Number systems
Sets of definable numbers
• Natural numbers ($\mathbb {N} $)
• Integers ($\mathbb {Z} $)
• Rational numbers ($\mathbb {Q} $)
• Constructible numbers
• Algebraic numbers ($\mathbb {A} $)
• Closed-form numbers
• Periods
• Computable numbers
• Arithmetical numbers
• Set-theoretically definable numbers
• Gaussian integers
Composition algebras
• Division algebras: Real numbers ($\mathbb {R} $)
• Complex numbers ($\mathbb {C} $)
• Quaternions ($\mathbb {H} $)
• Octonions ($\mathbb {O} $)
Split
types
• Over $\mathbb {R} $:
• Split-complex numbers
• Split-quaternions
• Split-octonions
Over $\mathbb {C} $:
• Bicomplex numbers
• Biquaternions
• Bioctonions
Other hypercomplex
• Dual numbers
• Dual quaternions
• Dual-complex numbers
• Hyperbolic quaternions
• Sedenions ($\mathbb {S} $)
• Split-biquaternions
• Multicomplex numbers
• Geometric algebra/Clifford algebra
• Algebra of physical space
• Spacetime algebra
Other types
• Cardinal numbers
• Extended natural numbers
• Irrational numbers
• Fuzzy numbers
• Hyperreal numbers
• Levi-Civita field
• Surreal numbers
• Transcendental numbers
• Ordinal numbers
• p-adic numbers (p-adic solenoids)
• Supernatural numbers
• Profinite integers
• Superreal numbers
• Normal numbers
• Classification
• List
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Wikipedia
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Split Lie algebra
In the mathematical field of Lie theory, a split Lie algebra is a pair $({\mathfrak {g}},{\mathfrak {h}})$ where ${\mathfrak {g}}$ is a Lie algebra and ${\mathfrak {h}}<{\mathfrak {g}}$ is a splitting Cartan subalgebra, where "splitting" means that for all $x\in {\mathfrak {h}}$, $\operatorname {ad} _{\mathfrak {g}}x$ is triangularizable. If a Lie algebra admits a splitting, it is called a splittable Lie algebra.[1] Note that for reductive Lie algebras, the Cartan subalgebra is required to contain the center.
Lie groups and Lie algebras
Classical groups
• General linear GL(n)
• Special linear SL(n)
• Orthogonal O(n)
• Special orthogonal SO(n)
• Unitary U(n)
• Special unitary SU(n)
• Symplectic Sp(n)
Simple Lie groups
Classical
• An
• Bn
• Cn
• Dn
Exceptional
• G2
• F4
• E6
• E7
• E8
Other Lie groups
• Circle
• Lorentz
• Poincaré
• Conformal group
• Diffeomorphism
• Loop
• Euclidean
Lie algebras
• Lie group–Lie algebra correspondence
• Exponential map
• Adjoint representation
• Killing form
• Index
• Simple Lie algebra
• Loop algebra
• Affine Lie algebra
Semisimple Lie algebra
• Dynkin diagrams
• Cartan subalgebra
• Root system
• Weyl group
• Real form
• Complexification
• Split Lie algebra
• Compact Lie algebra
Representation theory
• Lie group representation
• Lie algebra representation
• Representation theory of semisimple Lie algebras
• Representations of classical Lie groups
• Theorem of the highest weight
• Borel–Weil–Bott theorem
Lie groups in physics
• Particle physics and representation theory
• Lorentz group representations
• Poincaré group representations
• Galilean group representations
Scientists
• Sophus Lie
• Henri Poincaré
• Wilhelm Killing
• Élie Cartan
• Hermann Weyl
• Claude Chevalley
• Harish-Chandra
• Armand Borel
• Glossary
• Table of Lie groups
Over an algebraically closed field such as the complex numbers, all semisimple Lie algebras are splittable (indeed, not only does the Cartan subalgebra act by triangularizable matrices, but even stronger, it acts by diagonalizable ones) and all splittings are conjugate; thus split Lie algebras are of most interest for non-algebraically closed fields.
Split Lie algebras are of interest both because they formalize the split real form of a complex Lie algebra, and because split semisimple Lie algebras (more generally, split reductive Lie algebras) over any field share many properties with semisimple Lie algebras over algebraically closed fields – having essentially the same representation theory, for instance – the splitting Cartan subalgebra playing the same role as the Cartan subalgebra plays over algebraically closed fields. This is the approach followed in (Bourbaki 2005), for instance.
Properties
• Over an algebraically closed field, all Cartan subalgebras are conjugate. Over a non-algebraically closed field, not all Cartan subalgebras are conjugate in general; however, in a splittable semisimple Lie algebra all splitting Cartan algebras are conjugate.
• Over an algebraically closed field, all semisimple Lie algebras are splittable.
• Over a non-algebraically closed field, there exist non-splittable semisimple Lie algebras.[2]
• In a splittable Lie algebra, there may exist Cartan subalgebras that are not splitting.[3]
• Direct sums of splittable Lie algebras and ideals in splittable Lie algebras are splittable.
Split real Lie algebras
See also: Real form
For a real Lie algebra, splittable is equivalent to either of these conditions:[4]
• The real rank equals the complex rank.
• The Satake diagram has neither black vertices nor arrows.
Every complex semisimple Lie algebra has a unique (up to isomorphism) split real Lie algebra, which is also semisimple, and is simple if and only if the complex Lie algebra is.[5]
For real semisimple Lie algebras, split Lie algebras are opposite to compact Lie algebras – the corresponding Lie group is "as far as possible" from being compact.
Examples
The split real forms for the complex semisimple Lie algebras are:[6]
• $A_{n},{\mathfrak {sl}}_{n+1}(\mathbf {C} ):{\mathfrak {sl}}_{n+1}(\mathbf {R} )$
• $B_{n},{\mathfrak {so}}_{2n+1}(\mathbf {C} ):{\mathfrak {so}}_{n,n+1}(\mathbf {R} )$
• $C_{n},{\mathfrak {sp}}_{n}(\mathbf {C} ):{\mathfrak {sp}}_{n}(\mathbf {R} )$
• $D_{n},{\mathfrak {so}}_{2n}(\mathbf {C} ):{\mathfrak {so}}_{n,n}(\mathbf {R} )$
• Exceptional Lie algebras: $E_{6},E_{7},E_{8},F_{4},G_{2}$ have split real forms EI, EV, EVIII, FI, G.
These are the Lie algebras of the split real groups of the complex Lie groups.
Note that for ${\mathfrak {sl}}$ and ${\mathfrak {sp}}$, the real form is the real points of (the Lie algebra of) the same algebraic group, while for ${\mathfrak {so}}$ one must use the split forms (of maximally indefinite index), as the group SO is compact.
See also
• Compact Lie algebra
• Real form
• Split-complex number
• Split orthogonal group
References
1. (Bourbaki 2005, Chapter VIII, Section 2: Root System of a Split Semi-Simple Lie Algebra, p. 77)
2. (Bourbaki 2005, Chapter VIII, Section 2: Root System of a Split Semi-Simple Lie Algebra, Exercise 2 a p. 77)
3. (Bourbaki 2005, Chapter VIII, Section 2: Root System of a Split Semi-Simple Lie Algebra, Exercise 2 b p. 77)
4. (Onishchik & Vinberg 1994, p. 157)
5. (Onishchik & Vinberg 1994, Theorem 4.4, p. 158)
6. (Onishchik & Vinberg 1994, p. 158)
• Bourbaki, Nicolas (2005), "VIII: Split Semi-simple Lie Algebras", Elements of Mathematics: Lie Groups and Lie Algebras: Chapters 7–9
• Onishchik, A. L.; Vinberg, Ėrnest Borisovich (1994), "4.4: Split Real Semisimple Lie Algebras", Lie groups and Lie algebras III: structure of Lie groups and Lie algebras, pp. 157–158
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Splitting lemma
In mathematics, and more specifically in homological algebra, the splitting lemma states that in any abelian category, the following statements are equivalent for a short exact sequence
$0\longrightarrow A\mathrel {\overset {q}{\longrightarrow }} B\mathrel {\overset {r}{\longrightarrow }} C\longrightarrow 0.$
1. Left split
There exists a morphism t: B → A such that tq is the identity on A, idA,
2. Right split
There exists a morphism u: C → B such that ru is the identity on C, idC,
3. Direct sum
There is an isomorphism h from B to the direct sum of A and C, such that hq is the natural injection of A into the direct sum, and $rh^{-1}$ is the natural projection of the direct sum onto C.
Not to be confused with the splitting lemma in singularity theory.
If any of these statements holds, the sequence is called a split exact sequence, and the sequence is said to split.
In the above short exact sequence, where the sequence splits, it allows one to refine the first isomorphism theorem, which states that:
C ≅ B/ker r ≅ B/q(A) (i.e., C isomorphic to the coimage of r or cokernel of q)
to:
B = q(A) ⊕ u(C) ≅ A ⊕ C
where the first isomorphism theorem is then just the projection onto C.
It is a categorical generalization of the rank–nullity theorem (in the form V ≅ ker T ⊕ im T) in linear algebra.
Proof for the category of abelian groups
3. ⇒ 1. and 3. ⇒ 2.
First, to show that 3. implies both 1. and 2., we assume 3. and take as t the natural projection of the direct sum onto A, and take as u the natural injection of C into the direct sum.
1. ⇒ 3.
To prove that 1. implies 3., first note that any member of B is in the set (ker t + im q). This follows since for all b in B, b = (b − qt(b)) + qt(b); qt(b) is in im q, and b − qt(b) is in ker t, since
t(b − qt(b)) = t(b) − tqt(b) = t(b) − (tq)t(b) = t(b) − t(b) = 0.
Next, the intersection of im q and ker t is 0, since if there exists a in A such that q(a) = b, and t(b) = 0, then 0 = tq(a) = a; and therefore, b = 0.
This proves that B is the direct sum of im q and ker t. So, for all b in B, b can be uniquely identified by some a in A, k in ker t, such that b = q(a) + k.
By exactness ker r = im q. The subsequence B ⟶ C ⟶ 0 implies that r is onto; therefore for any c in C there exists some b = q(a) + k such that c = r(b) = r(q(a) + k) = r(k). Therefore, for any c in C, exists k in ker t such that c = r(k), and r(ker t) = C.
If r(k) = 0, then k is in im q; since the intersection of im q and ker t = 0, then k = 0. Therefore, the restriction r: ker t → C is an isomorphism; and ker t is isomorphic to C.
Finally, im q is isomorphic to A due to the exactness of 0 ⟶ A ⟶ B; so B is isomorphic to the direct sum of A and C, which proves (3).
2. ⇒ 3.
To show that 2. implies 3., we follow a similar argument. Any member of B is in the set ker r + im u; since for all b in B, b = (b − ur(b)) + ur(b), which is in ker r + im u. The intersection of ker r and im u is 0, since if r(b) = 0 and u(c) = b, then 0 = ru(c) = c.
By exactness, im q = ker r, and since q is an injection, im q is isomorphic to A, so A is isomorphic to ker r. Since ru is a bijection, u is an injection, and thus im u is isomorphic to C. So B is again the direct sum of A and C.
An alternative "abstract nonsense" proof of the splitting lemma may be formulated entirely in category theoretic terms.
Non-abelian groups
In the form stated here, the splitting lemma does not hold in the full category of groups, which is not an abelian category.
Partially true
It is partially true: if a short exact sequence of groups is left split or a direct sum (1. or 3.), then all of the conditions hold. For a direct sum this is clear, as one can inject from or project to the summands. For a left split sequence, the map t × r: B → A × C gives an isomorphism, so B is a direct sum (3.), and thus inverting the isomorphism and composing with the natural injection C → A × C gives an injection C → B splitting r (2.).
However, if a short exact sequence of groups is right split (2.), then it need not be left split or a direct sum (neither 1. nor 3. follows): the problem is that the image of the right splitting need not be normal. What is true in this case is that B is a semidirect product, though not in general a direct product.
Counterexample
To form a counterexample, take the smallest non-abelian group B ≅ S3, the symmetric group on three letters. Let A denote the alternating subgroup, and let C = B/A ≅ {±1}. Let q and r denote the inclusion map and the sign map respectively, so that
$0\longrightarrow A\mathrel {\stackrel {q}{\longrightarrow }} B\mathrel {\stackrel {r}{\longrightarrow }} C\longrightarrow 0$
is a short exact sequence. 3. fails, because S3 is not abelian, but 2. holds: we may define u: C → B by mapping the generator to any two-cycle. Note for completeness that 1. fails: any map t: B → A must map every two-cycle to the identity because the map has to be a group homomorphism, while the order of a two-cycle is 2 which can not be divided by the order of the elements in A other than the identity element, which is 3 as A is the alternating subgroup of S3, or namely the cyclic group of order 3. But every permutation is a product of two-cycles, so t is the trivial map, whence tq: A → A is the trivial map, not the identity.
References
• Saunders Mac Lane: Homology. Reprint of the 1975 edition, Springer Classics in Mathematics, ISBN 3-540-58662-8, p. 16
• Allen Hatcher: Algebraic Topology. 2002, Cambridge University Press, ISBN 0-521-79540-0, p. 147
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Splitting lemma (functions)
In mathematics, especially in singularity theory, the splitting lemma is a useful result due to René Thom which provides a way of simplifying the local expression of a function usually applied in a neighbourhood of a degenerate critical point.
Not to be confused with the splitting lemma in homological algebra.
Formal statement
Let $f:(\mathbb {R} ^{n},0)\to (\mathbb {R} ,0)$ be a smooth function germ, with a critical point at 0 (so $(\partial f/\partial x_{i})(0)=0$ for $i=1,\dots ,n$). Let V be a subspace of $\mathbb {R} ^{n}$ such that the restriction f |V is non-degenerate, and write B for the Hessian matrix of this restriction. Let W be any complementary subspace to V. Then there is a change of coordinates $\Phi (x,y)$ of the form $\Phi (x,y)=(\phi (x,y),y)$ with $x\in V,y\in W$, and a smooth function h on W such that
$f\circ \Phi (x,y)={\frac {1}{2}}x^{T}Bx+h(y).$
This result is often referred to as the parametrized Morse lemma, which can be seen by viewing y as the parameter. It is the gradient version of the implicit function theorem.
Extensions
There are extensions to infinite dimensions, to complex analytic functions, to functions invariant under the action of a compact group, ...
References
• Poston, Tim; Stewart, Ian (1979), Catastrophe Theory and Its Applications, Pitman, ISBN 978-0-273-08429-7.
• Brocker, Th (1975), Differentiable Germs and Catastrophes, Cambridge University Press, ISBN 978-0-521-20681-5.
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Wikipedia
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Splitting of prime ideals in Galois extensions
In mathematics, the interplay between the Galois group G of a Galois extension L of a number field K, and the way the prime ideals P of the ring of integers OK factorise as products of prime ideals of OL, provides one of the richest parts of algebraic number theory. The splitting of prime ideals in Galois extensions is sometimes attributed to David Hilbert by calling it Hilbert theory. There is a geometric analogue, for ramified coverings of Riemann surfaces, which is simpler in that only one kind of subgroup of G need be considered, rather than two. This was certainly familiar before Hilbert.
Definitions
Let L/K be a finite extension of number fields, and let OK and OL be the corresponding ring of integers of K and L, respectively, which are defined to be the integral closure of the integers Z in the field in question.
${\begin{array}{ccc}O_{K}&\hookrightarrow &O_{L}\\\downarrow &&\downarrow \\K&\hookrightarrow &L\end{array}}$
Finally, let p be a non-zero prime ideal in OK, or equivalently, a maximal ideal, so that the residue OK/p is a field.
From the basic theory of one-dimensional rings follows the existence of a unique decomposition
$pO_{L}=\prod _{j=1}^{g}P_{j}^{e_{j}}$
of the ideal pOL generated in OL by p into a product of distinct maximal ideals Pj, with multiplicities ej.
The field F = OK/p naturally embeds into Fj = OL/Pj for every j, the degree fj = [OL/Pj : OK/p] of this residue field extension is called inertia degree of Pj over p.
The multiplicity ej is called ramification index of Pj over p. If it is bigger than 1 for some j, the field extension L/K is called ramified at p (or we say that p ramifies in L, or that it is ramified in L). Otherwise, L/K is called unramified at p. If this is the case then by the Chinese remainder theorem the quotient OL/pOL is a product of fields Fj. The extension L/K is ramified in exactly those primes that divide the relative discriminant, hence the extension is unramified in all but finitely many prime ideals.
Multiplicativity of ideal norm implies
$[L:K]=\sum _{j=1}^{g}e_{j}f_{j}.$
If fj = ej = 1 for every j (and thus g = [L : K]), we say that p splits completely in L. If g = 1 and f1 = 1 (and so e1 = [L : K]), we say that p ramifies completely in L. Finally, if g = 1 and e1 = 1 (and so f1 = [L : K]), we say that p is inert in L.
The Galois situation
In the following, the extension L/K is assumed to be a Galois extension. Then the prime avoidance lemma can be used to show the Galois group $G=\operatorname {Gal} (L/K)$ acts transitively on the Pj. That is, the prime ideal factors of p in L form a single orbit under the automorphisms of L over K. From this and the unique factorisation theorem, it follows that f = fj and e = ej are independent of j; something that certainly need not be the case for extensions that are not Galois. The basic relations then read
$pO_{L}=\left(\prod _{j=1}^{g}P_{j}\right)^{e}$.
and
$[L:K]=efg.$
The relation above shows that [L : K]/ef equals the number g of prime factors of p in OL. By the orbit-stabilizer formula this number is also equal to |G|/|DPj| for every j, where DPj, the decomposition group of Pj, is the subgroup of elements of G sending a given Pj to itself. Since the degree of L/K and the order of G are equal by basic Galois theory, it follows that the order of the decomposition group DPj is ef for every j.
This decomposition group contains a subgroup IPj, called inertia group of Pj, consisting of automorphisms of L/K that induce the identity automorphism on Fj. In other words, IPj is the kernel of reduction map $D_{P_{j}}\to \operatorname {Gal} (F_{j}/F)$. It can be shown that this map is surjective, and it follows that $\operatorname {Gal} (F_{j}/F)$ is isomorphic to DPj/IPj and the order of the inertia group IPj is e.
The theory of the Frobenius element goes further, to identify an element of DPj/IPj for given j which corresponds to the Frobenius automorphism in the Galois group of the finite field extension Fj /F. In the unramified case the order of DPj is f and IPj is trivial, so the Frobenius element is in this case an element of DPj, and thus also an element of G. For varying j, the groups DPj are conjugate subgroups inside G: Recalling that G acts transitively on the Pj, one checks that if $\sigma $ maps Pj to Pj', $\sigma D_{P_{j}}\sigma ^{-1}=D_{P_{j'}}$. Therefore, if G is an abelian group, the Frobenius element of an unramified prime P does not depend on which Pj we take. Furthermore, in the abelian case, associating an unramified prime of K to its Frobenius and extending multiplicatively defines a homomorphism from the group of unramified ideals of K into G. This map, known as the Artin map, is a crucial ingredient of class field theory, which studies the finite abelian extensions of a given number field K.[1]
In the geometric analogue, for complex manifolds or algebraic geometry over an algebraically closed field, the concepts of decomposition group and inertia group coincide. There, given a Galois ramified cover, all but finitely many points have the same number of preimages.
The splitting of primes in extensions that are not Galois may be studied by using a splitting field initially, i.e. a Galois extension that is somewhat larger. For example, cubic fields usually are 'regulated' by a degree 6 field containing them.
Example — the Gaussian integers
This section describes the splitting of prime ideals in the field extension Q(i)/Q. That is, we take K = Q and L = Q(i), so OK is simply Z, and OL = Z[i] is the ring of Gaussian integers. Although this case is far from representative — after all, Z[i] has unique factorisation, and there aren't many quadratic fields with unique factorization — it exhibits many of the features of the theory.
Writing G for the Galois group of Q(i)/Q, and σ for the complex conjugation automorphism in G, there are three cases to consider.
The prime p = 2
The prime 2 of Z ramifies in Z[i]:
$(2)=(1+i)^{2}$
The ramification index here is therefore e = 2. The residue field is
$O_{L}/(1+i)O_{L}$
which is the finite field with two elements. The decomposition group must be equal to all of G, since there is only one prime of Z[i] above 2. The inertia group is also all of G, since
$a+bi\equiv a-bi{\bmod {1}}+i$
for any integers a and b, as $a+bi=2bi+a-bi=(1+i)\cdot (1-i)bi+a-bi\equiv a-bi{\bmod {1}}+i$ .
In fact, 2 is the only prime that ramifies in Z[i], since every prime that ramifies must divide the discriminant of Z[i], which is −4.
Primes p ≡ 1 mod 4
Any prime p ≡ 1 mod 4 splits into two distinct prime ideals in Z[i]; this is a manifestation of Fermat's theorem on sums of two squares. For example:
$13=(2+3i)(2-3i)$
The decomposition groups in this case are both the trivial group {1}; indeed the automorphism σ switches the two primes (2 + 3i) and (2 − 3i), so it cannot be in the decomposition group of either prime. The inertia group, being a subgroup of the decomposition group, is also the trivial group. There are two residue fields, one for each prime,
$O_{L}/(2\pm 3i)O_{L}\ ,$
which are both isomorphic to the finite field with 13 elements. The Frobenius element is the trivial automorphism; this means that
$(a+bi)^{13}\equiv a+bi{\bmod {2}}\pm 3i$
for any integers a and b.
Primes p ≡ 3 mod 4
Any prime p ≡ 3 mod 4 remains inert in Z[i]; that is, it does not split. For example, (7) remains prime in Z[i]. In this situation, the decomposition group is all of G, again because there is only one prime factor. However, this situation differs from the p = 2 case, because now σ does not act trivially on the residue field
$O_{L}/(7)O_{L}\ ,$
which is the finite field with 72 = 49 elements. For example, the difference between 1 + i and σ(1 + i) = 1 − i is 2i, which is certainly not divisible by 7. Therefore, the inertia group is the trivial group {1}. The Galois group of this residue field over the subfield Z/7Z has order 2, and is generated by the image of the Frobenius element. The Frobenius element is none other than σ; this means that
$(a+bi)^{7}\equiv a-bi{\bmod {7}}$
for any integers a and b.
Summary
Prime in Z How it splits in Z[i] Inertia group Decomposition group
2 Ramifies with index 2 G G
p ≡ 1 mod 4 Splits into two distinct factors 1 1
p ≡ 3 mod 4 Remains inert 1 G
Computing the factorisation
Suppose that we wish to determine the factorisation of a prime ideal P of OK into primes of OL. The following procedure (Neukirch, p. 47) solves this problem in many cases. The strategy is to select an integer θ in OL so that L is generated over K by θ (such a θ is guaranteed to exist by the primitive element theorem), and then to examine the minimal polynomial H(X) of θ over K; it is a monic polynomial with coefficients in OK. Reducing the coefficients of H(X) modulo P, we obtain a monic polynomial h(X) with coefficients in F, the (finite) residue field OK/P. Suppose that h(X) factorises in the polynomial ring F[X] as
$h(X)=h_{1}(X)^{e_{1}}\cdots h_{n}(X)^{e_{n}},$
where the hj are distinct monic irreducible polynomials in F[X]. Then, as long as P is not one of finitely many exceptional primes (the precise condition is described below), the factorisation of P has the following form:
$PO_{L}=Q_{1}^{e_{1}}\cdots Q_{n}^{e_{n}},$
where the Qj are distinct prime ideals of OL. Furthermore, the inertia degree of each Qj is equal to the degree of the corresponding polynomial hj, and there is an explicit formula for the Qj:
$Q_{j}=PO_{L}+h_{j}(\theta )O_{L},$
where hj denotes here a lifting of the polynomial hj to K[X].
In the Galois case, the inertia degrees are all equal, and the ramification indices e1 = ... = en are all equal.
The exceptional primes, for which the above result does not necessarily hold, are the ones not relatively prime to the conductor of the ring OK[θ]. The conductor is defined to be the ideal
$\{y\in O_{L}:yO_{L}\subseteq O_{K}[\theta ]\};$
it measures how far the order OK[θ] is from being the whole ring of integers (maximal order) OL.
A significant caveat is that there exist examples of L/K and P such that there is no available θ that satisfies the above hypotheses (see for example [2]). Therefore, the algorithm given above cannot be used to factor such P, and more sophisticated approaches must be used, such as that described in.[3]
An example
Consider again the case of the Gaussian integers. We take θ to be the imaginary unit i, with minimal polynomial H(X) = X2 + 1. Since Z[$i$] is the whole ring of integers of Q($i$), the conductor is the unit ideal, so there are no exceptional primes.
For P = (2), we need to work in the field Z/(2)Z, which amounts to factorising the polynomial X2 + 1 modulo 2:
$X^{2}+1=(X+1)^{2}{\pmod {2}}.$
Therefore, there is only one prime factor, with inertia degree 1 and ramification index 2, and it is given by
$Q=(2)\mathbf {Z} [i]+(i+1)\mathbf {Z} [i]=(1+i)\mathbf {Z} [i].$
The next case is for P = (p) for a prime p ≡ 3 mod 4. For concreteness we will take P = (7). The polynomial X2 + 1 is irreducible modulo 7. Therefore, there is only one prime factor, with inertia degree 2 and ramification index 1, and it is given by
$Q=(7)\mathbf {Z} [i]+(i^{2}+1)\mathbf {Z} [i]=7\mathbf {Z} [i].$
The last case is P = (p) for a prime p ≡ 1 mod 4; we will again take P = (13). This time we have the factorisation
$X^{2}+1=(X+5)(X-5){\pmod {13}}.$
Therefore, there are two prime factors, both with inertia degree and ramification index 1. They are given by
$Q_{1}=(13)\mathbf {Z} [i]+(i+5)\mathbf {Z} [i]=\cdots =(2+3i)\mathbf {Z} [i]$
and
$Q_{2}=(13)\mathbf {Z} [i]+(i-5)\mathbf {Z} [i]=\cdots =(2-3i)\mathbf {Z} [i].$
See also
• Chebotarev's density theorem
References
1. Milne, J.S. (2020). Class Field Theory.
2. Stein, William A. (2002). "Essential Discriminant Divisors". Factoring Primes in Rings of Integers.
3. Stein 2002, Method that Always Works
External links
• "Splitting and ramification in number fields and Galois extensions". PlanetMath.
• Stein, William (2004), A brief introduction to classical and adelic algebraic number theory
• Neukirch, Jürgen (1999). Algebraische Zahlentheorie. Grundlehren der mathematischen Wissenschaften. Vol. 322. Berlin: Springer-Verlag. ISBN 978-3-540-65399-8. MR 1697859. Zbl 0956.11021.
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Wikipedia
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Splitting principle
In mathematics, the splitting principle is a technique used to reduce questions about vector bundles to the case of line bundles.
In the theory of vector bundles, one often wishes to simplify computations, say of Chern classes. Often computations are well understood for line bundles and for direct sums of line bundles. In this case the splitting principle can be quite useful.
Theorem — Let $\xi \colon E\rightarrow X$ be a vector bundle of rank $n$ over a paracompact space $X$. There exists a space $Y=Fl(E)$, called the flag bundle associated to $E$, and a map $p\colon Y\rightarrow X$ such that
1. the induced cohomology homomorphism $p^{*}\colon H^{*}(X)\rightarrow H^{*}(Y)$ is injective, and
2. the pullback bundle $p^{*}\xi \colon p^{*}E\rightarrow Y$ breaks up as a direct sum of line bundles: $p^{*}(E)=L_{1}\oplus L_{2}\oplus \cdots \oplus L_{n}.$
The theorem above holds for complex vector bundles and integer coefficients or for real vector bundles with $\mathbb {Z} _{2}$ coefficients. In the complex case, the line bundles $L_{i}$ or their first characteristic classes are called Chern roots.
The fact that $p^{*}\colon H^{*}(X)\rightarrow H^{*}(Y)$ is injective means that any equation which holds in $H^{*}(Y)$ (say between various Chern classes) also holds in $H^{*}(X)$.
The point is that these equations are easier to understand for direct sums of line bundles than for arbitrary vector bundles, so equations should be understood in $Y$ and then pushed down to $X$.
Since vector bundles on $X$ are used to define the K-theory group $K(X)$, it is important to note that $p^{*}\colon K(X)\rightarrow K(Y)$ is also injective for the map $p$ in the above theorem.[1]
The splitting principle admits many variations. The following, in particular, concerns real vector bundles and their complexifications: [2]
Theorem — Let $\xi \colon E\rightarrow X$ be a real vector bundle of rank $2n$ over a paracompact space $X$. There exists a space $Y$ and a map $p\colon Y\rightarrow X$ such that
1. the induced cohomology homomorphism $p^{*}\colon H^{*}(X)\rightarrow H^{*}(Y)$ is injective, and
2. the pullback bundle $p^{*}\xi \colon p^{*}E\rightarrow Y$ breaks up as a direct sum of line bundles and their conjugates: $p^{*}(E\otimes \mathbb {C} )=L_{1}\oplus {\overline {L_{1}}}\oplus \cdots \oplus L_{n}\oplus {\overline {L_{n}}}.$
Symmetric polynomial
Under the splitting principle, characteristic classes for complex vector bundles correspond to symmetric polynomials in the first Chern classes of complex line bundles; these are the Chern classes.
See also
• K-theory
• Grothendieck splitting principle for holomorphic vector bundles on the complex projective line
References
1. Oscar Randal-Williams, Characteristic classes and K-theory, Corollary 4.3.4, https://www.dpmms.cam.ac.uk/~or257/teaching/notes/Kthy.pdf
2. H. Blane Lawson and Marie-Louise Michelsohn, Spin Geometry, Proposition 11.2.
• Hatcher, Allen (2003), Vector Bundles & K-Theory (2.0 ed.) section 3.1
• Raoul Bott and Loring Tu. Differential Forms in Algebraic Topology, section 21.
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Wikipedia
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Splitting theorem
In the mathematical field of differential geometry, there are various splitting theorems on when a pseudo-Riemannian manifold can be given as a metric product. The best-known is the Cheeger–Gromoll splitting theorem for Riemannian manifolds, although there has also been research into splitting of Lorentzian manifolds.
Cheeger and Gromoll's Riemannian splitting theorem
Any connected Riemannian manifold M has an underlying metric space structure, and this allows the definition of a geodesic line as a map c: ℝ → M such that the distance from c(s) to c(t) equals | t − s | for arbitrary s and t. This is to say that the restriction of c to any bounded interval is a curve of minimal length which connects its endpoints.[1]
In 1971, Jeff Cheeger and Detlef Gromoll proved that, if a geodesically complete and connected Riemannian manifold of nonnegative Ricci curvature contains any geodesic line, then it must split isometrically as the product of a complete Riemannian manifold with ℝ. The proof was later simplified by Jost Eschenburg and Ernst Heintze. In 1936, Stefan Cohn-Vossen had originally formulated and proved the theorem in the case of two-dimensional manifolds, and Victor Toponogov had extended Cohn-Vossen's work to higher dimensions, under the special condition of nonnegative sectional curvature.[2]
The proof can be summarized as follows.[3] The condition of a geodesic line allows for two Busemann functions to be defined. These can be thought of as a normalized Riemannian distance function to the two endpoints of the line. From the fundamental Laplacian comparison theorem proved earlier by Eugenio Calabi, these functions are both superharmonic under the Ricci curvature assumption. Either of these functions could be negative at some points, but the triangle inequality implies that their sum is nonnegative. The strong maximum principle implies that the sum is identically zero and hence that each Busemann function is in fact (weakly) a harmonic function. Weyl's lemma implies the infinite differentiability of the Busemann functions. Then, the proof can be finished by using Bochner's formula to construct parallel vector fields, setting up the de Rham decomposition theorem.[4] Alternatively, the theory of Riemannian submersions may be invoked.[5]
As a consequence of their splitting theorem, Cheeger and Gromoll were able to prove that the universal cover of any closed manifold of nonnegative Ricci curvature must split isometrically as the product of a closed manifold with a Euclidean space. If the universal cover is topologically contractible, then it follows that all metrics involved must be flat.[6]
Lorentzian splitting theorem
In 1982, Shing-Tung Yau conjectured that a particular Lorentzian version of Cheeger and Gromoll's theorem should hold.[7] Proofs in various levels of generality were found by Jost Eschenburg, Gregory Galloway, and Richard Newman. In these results, the role of geodesic completeness is replaced by either the condition of global hyperbolicity or of timelike geodesic completeness. The nonnegativity of Ricci curvature is replaced by the timelike convergence condition that the Ricci curvature is nonnegative in all timelike directions. The geodesic line is required to be timelike.[8]
References
Notes.
1. Besse 1987, Definition 6.64; Petersen 2016, p. 298; Schoen & Yau 1994, p. 12.
2. Besse 1987, Section 6E; Petersen 2016, Theorem 7.3.5; Schoen & Yau 1994, Section 1.2.
3. Besse 1987, Section 6G; Petersen 2016, Section 7.3; Schoen & Yau 1994, Section 1.2.
4. Schoen & Yau 1994, Section 1.2.
5. Besse 1987, p. 176.
6. Petersen 2016, Section 7.3.3.
7. Yau 1982, Problem 115.
8. Beem, Ehrlich & Easley 1996, Chapter 14.
Historical articles.
• Cheeger, Jeff; Gromoll, Detlef (1971). "The splitting theorem for manifolds of nonnegative Ricci curvature". Journal of Differential Geometry. 6 (1): 119–128. doi:10.4310/jdg/1214430220. MR 0303460. Zbl 0223.53033.
• Cohn-Vossen, S. (1936). "Totalkrümmung und geodätische Linien auf einfachzusammenhängenden offenen vollständigen Flächenstücken". Matematicheskii Sbornik. 43 (2): 139–163. JFM 62.0862.01. Zbl 0014.27601.
• Toponogov, V. A. (1964). Translated by Robinson, A. "Riemannian spaces which contain straight lines". American Mathematical Society Translations. Second Series. 37 (Twenty-two papers on algebra, number theory and differential geometry): 287–290. doi:10.1090/trans2/037. Zbl 0138.42902.
• Toponogov, V. A. (1968). Translated by West, A. "The metric structure of Riemannian spaces with nonnegative curvature which contain straight lines". American Mathematical Society Translations. Second Series. 70 (Thirty-one invited addresses (eight in abstract) at the International Congress of Mathematicians in Moscow, 1966): 225–239. doi:10.1090/trans2/070. Zbl 0187.43801.
• Yau, Shing Tung (1982). "Problem section". In Yau, Shing-Tung (ed.). Seminar on Differential Geometry. Annals of Mathematics Studies. Vol. 102. Princeton, NJ: Princeton University Press. pp. 669–706. doi:10.1515/9781400881918-035. MR 0645762. Zbl 0479.53001. Reprinted in Schoen & Yau (1994).
Textbooks.
• Beem, John K.; Ehrlich, Paul E.; Easley, Kevin L. (1996). Global Lorentzian geometry. Monographs and Textbooks in Pure and Applied Mathematics. Vol. 202 (Second edition of 1981 original ed.). New York: Marcel Dekker, Inc. doi:10.1201/9780203753125. ISBN 0-8247-9324-2. MR 1384756. Zbl 0846.53001.
• Besse, Arthur L. (1987). Einstein manifolds. Ergebnisse der Mathematik und ihrer Grenzgebiete (3). Vol. 10. Reprinted in 2008. Berlin: Springer-Verlag. doi:10.1007/978-3-540-74311-8. ISBN 3-540-15279-2. MR 0867684. Zbl 0613.53001.
• Petersen, Peter (2016). Riemannian geometry. Graduate Texts in Mathematics. Vol. 171 (Third edition of 1998 original ed.). Springer, Cham. doi:10.1007/978-3-319-26654-1. ISBN 978-3-319-26652-7. MR 3469435. Zbl 1417.53001.
• Schoen, R.; Yau, S.-T. (1994). Lectures on differential geometry. Conference Proceedings and Lecture Notes in Geometry and Topology. Vol. 1. Translated by Ding, Wei Yue; Cheng, S. Y. Cambridge, MA: International Press. ISBN 1-57146-012-8. MR 1333601. Zbl 0830.53001.
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Wikipedia
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Descartes number
In number theory, a Descartes number is an odd number which would have been an odd perfect number, if one of its composite factors were prime. They are named after René Descartes who observed that the number D = 32⋅72⋅112⋅132⋅22021 = (3⋅1001)2 ⋅ (22⋅1001 − 1) = 198585576189 would be an odd perfect number if only 22021 were a prime number, since the sum-of-divisors function for D would satisfy, if 22021 were prime,
${\begin{aligned}\sigma (D)&=(3^{2}+3+1)\cdot (7^{2}+7+1)\cdot (11^{2}+11+1)\cdot (13^{2}+13+1)\cdot (22021+1)=(13)\cdot (3\cdot 19)\cdot (7\cdot 19)\cdot (3\cdot 61)\cdot (22\cdot 1001)\\&=3^{2}\cdot 7\cdot 13\cdot 19^{2}\cdot 61\cdot (22\cdot 7\cdot 11\cdot 13)=2\cdot (3^{2}\cdot 7^{2}\cdot 11^{2}\cdot 13^{2})\cdot (19^{2}\cdot 61)=2\cdot (3^{2}\cdot 7^{2}\cdot 11^{2}\cdot 13^{2})\cdot 22021=2D,\end{aligned}}$
where we ignore the fact that 22021 is composite (22021 = 192 ⋅ 61).
A Descartes number is defined as an odd number n = m ⋅ p where m and p are coprime and 2n = σ(m) ⋅ (p + 1), whence p is taken as a 'spoof' prime. The example given is the only one currently known.
If m is an odd almost perfect number,[1] that is, σ(m) = 2m − 1 and 2m − 1 is taken as a 'spoof' prime, then n = m ⋅ (2m − 1) is a Descartes number, since σ(n) = σ(m ⋅ (2m − 1)) = σ(m) ⋅ 2m = (2m − 1) ⋅ 2m = 2n. If 2m − 1 were prime, n would be an odd perfect number.
Properties
Banks et al. showed in 2008 that if n is a cube-free Descartes number not divisible by $3$, then n has over a million distinct prime divisors.
Generalizations
John Voight generalized Descartes numbers to allow negative bases. He found the example $3^{4}7^{2}11^{2}19^{2}(-127)^{1}$.[2] Subsequent work by a group at Brigham Young University found more examples similar to Voight's example, [2] and also allowed a new class of spoofs where one is allowed to also not notice that a prime is the same as another prime in the factorization.[3]
See also
• Erdős–Nicolas number, another type of almost-perfect number
Notes
1. Currently, the only known almost perfect numbers are the non-negative powers of 2, whence the only known odd almost perfect number is 20 = 1.
2. Nadis, Steve (September 10, 2020). "Mathematicians Open a New Front on an Ancient Number Problem". Quanta Magazine. Retrieved 3 October 2021.
3. Andersen, Nickolas; Durham, Spencer ; Griffin, Michael J. ; Hales, Jonathan ; Jenkins, Paul ; Keck, Ryan ; Ko, Hankun ; Molnar, Grant; Moss, Eric ; Nielsen, Pace P. ; Niendorf, Kyle ; Tombs, Vandy; Warnick, Merrill ; Wu, Dongsheng (2020). "Odd, spoof perfect factorizations". J. Number Theory (234): 31-47. arXiv:2006.10697.{{cite journal}}: CS1 maint: multiple names: authors list (link) arXiv version
References
• Banks, William D.; Güloğlu, Ahmet M.; Nevans, C. Wesley; Saidak, Filip (2008). "Descartes numbers". In De Koninck, Jean-Marie; Granville, Andrew; Luca, Florian (eds.). Anatomy of integers. Based on the CRM workshop, Montreal, Canada, March 13--17, 2006. CRM Proceedings and Lecture Notes. Vol. 46. Providence, RI: American Mathematical Society. pp. 167–173. ISBN 978-0-8218-4406-9. Zbl 1186.11004.
• Klee, Victor; Wagon, Stan (1991). Old and new unsolved problems in plane geometry and number theory. The Dolciani Mathematical Expositions. Vol. 11. Washington, DC: Mathematical Association of America. ISBN 0-88385-315-9. Zbl 0784.51002.
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Sporadic group
In mathematics, a sporadic group is one of the 26 exceptional groups found in the classification of finite simple groups.
Algebraic structure → Group theory
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• Cyclic group Zn
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• Cauchy's theorem
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• cyclic
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A simple group is a group G that does not have any normal subgroups except for the trivial group and G itself. The classification theorem states that the list of finite simple groups consists of 18 countably infinite families[lower-alpha 1] plus 26 exceptions that do not follow such a systematic pattern. These 26 exceptions are the sporadic groups. They are also known as the sporadic simple groups, or the sporadic finite groups. Because it is not strictly a group of Lie type, the Tits group is sometimes regarded as a sporadic group,[1] in which case there would be 27 sporadic groups.
The monster group, or friendly giant, is the largest of the sporadic groups, and all but six of the other sporadic groups are subquotients of it.[2]
Names
Five of the sporadic groups were discovered by Mathieu in the 1860s and the other 21 were found between 1965 and 1975. Several of these groups were predicted to exist before they were constructed. Most of the groups are named after the mathematician(s) who first predicted their existence. The full list is:[1][3][4]
• Mathieu groups M11 (M11), M12 (M12), M22 (M22), M23 (M23), M24 (M24)
• Janko groups J1 (J1), J2 or HJ (J2), J3 or HJM (J3), J4 (J4)
• Conway groups Co1 (Co1), Co2 (Co2), Co3 (Co3)
• Fischer groups Fi22 (Fi22), Fi23 (Fi23), Fi24′ or F3+ (Fi24)
• Higman–Sims group HS
• McLaughlin group McL
• Held group He or F7+ or F7
• Rudvalis group Ru
• Suzuki group Suz or F3−
• O'Nan group O'N (ON)
• Harada–Norton group HN or F5+ or F5
• Lyons group Ly
• Thompson group Th or F3|3 or F3
• Baby Monster group B or F2+ or F2
• Fischer–Griess Monster group M or F1
Various constructions for these groups were first compiled in Conway et al. (1985), including character tables, individual conjugacy classes and lists of maximal subgroup, as well as Schur multipliers and orders of their outer automorphisms. These are also listed online at Wilson et al. (1999), updated with their group presentations and semi-presentations. The degrees of minimal faithful representation or Brauer characters over fields of characteristic p ≥ 0 for all sporadic groups have also been calculated, and for some of their covering groups. These are detailed in Jansen (2005).
An exception found in the classification of sporadic groups within finite simple groups is the Tits group T, that is sometimes also considered as being sporadic — it is almost but not strictly a group of Lie type — which is why in some sources the number of sporadic groups is given as 27, instead of 26.[1] In some other sources, the Tits group is regarded as neither sporadic nor of Lie type.[lower-alpha 2] The Tits group is the (n = 0)-member 2F4(2)′ of the infinite family of commutator groups 2F4(22n+1)′; thus by definition not sporadic. For n > 0 these finite simple groups coincide with the groups of Lie type 2F4(22n+1), also known as Ree groups of type 2F4.
The earliest use of the term sporadic group may be Burnside (1911, p. 504) where he comments about the Mathieu groups: "These apparently sporadic simple groups would probably repay a closer examination than they have yet received."
The diagram at right is based on Ronan (2006, p. 247). It does not show the numerous non-sporadic simple subquotients of the sporadic groups.
Organization
Happy Family
Of the 26 sporadic groups, 20 can be seen inside the monster group as subgroups or quotients of subgroups (sections). These twenty have been called the happy family by Robert Griess, and can be organized into three generations.[5][lower-alpha 3]
First generation (5 groups): the Mathieu groups
Main article: Mathieu groups
Mn for n = 11, 12, 22, 23 and 24 are multiply transitive permutation groups on n points. They are all subgroups of M24, which is a permutation group on 24 points.[6]
Second generation (7 groups): the Leech lattice
See also: Leech lattice and Conway groups
All the subquotients of the automorphism group of a lattice in 24 dimensions called the Leech lattice:[7]
• Co1 is the quotient of the automorphism group by its center {±1}
• Co2 is the stabilizer of a type 2 (i.e., length 2) vector
• Co3 is the stabilizer of a type 3 (i.e., length √6) vector
• Suz is the group of automorphisms preserving a complex structure (modulo its center)
• McL is the stabilizer of a type 2-2-3 triangle
• HS is the stabilizer of a type 2-3-3 triangle
• J2 is the group of automorphisms preserving a quaternionic structure (modulo its center).
Third generation (8 groups): other subgroups of the Monster
Consists of subgroups which are closely related to the Monster group M:[8]
• B or F2 has a double cover which is the centralizer of an element of order 2 in M
• Fi24′ has a triple cover which is the centralizer of an element of order 3 in M (in conjugacy class "3A")
• Fi23 is a subgroup of Fi24′
• Fi22 has a double cover which is a subgroup of Fi23
• The product of Th = F3 and a group of order 3 is the centralizer of an element of order 3 in M (in conjugacy class "3C")
• The product of HN = F5 and a group of order 5 is the centralizer of an element of order 5 in M
• The product of He = F7 and a group of order 7 is the centralizer of an element of order 7 in M.
• Finally, the Monster group itself is considered to be in this generation.
(This series continues further: the product of M12 and a group of order 11 is the centralizer of an element of order 11 in M.)
The Tits group, if regarded as a sporadic group, would belong in this generation: there is a subgroup S4 ×2F4(2)′ normalising a 2C2 subgroup of B, giving rise to a subgroup 2·S4 ×2F4(2)′ normalising a certain Q8 subgroup of the Monster. 2F4(2)′ is also a subquotient of the Fischer group Fi22, and thus also of Fi23 and Fi24′, and of the Baby Monster B. 2F4(2)′ is also a subquotient of the (pariah) Rudvalis group Ru, and has no involvements in sporadic simple groups except the ones already mentioned.
Pariahs
Main article: Pariah group
The six exceptions are J1, J3, J4, O'N, Ru and Ly, sometimes known as the pariahs.[9][10]
Table of the sporadic group orders (w/ Tits group)
Group Discoverer [11]
Year
Generation
[4][12]
Order
[1][4]
Factorized order
[13]
Minimal faithful Brauer character degree
[14][15]
$(a,b,ab)$
Generators
[15][lower-alpha 4]
$\langle \langle a,b\mid o(z)\rangle \rangle $
Semi-presentation
M or F1Fischer, Griess1973 3rd808017424794512875886459904961710757005754368000000000≈ 8×1053246 · 320 · 59 · 76 · 112 · 133 · 17 · 19 · 23 · 29 · 31 · 41 · 47 · 59 · 711968832A, 3B, 29$o{\bigl (}(ab)^{4}(ab^{2})^{2}{\bigr )}=50$
B or F2Fischer1973 3rd4154781481226426191177580544000000≈ 4×1033241 · 313 · 56 · 72 · 11 · 13 · 17 · 19 · 23 · 31 · 4743712C, 3A, 55$o{\bigl (}(ab)^{2}(abab^{2})^{2}ab^{2}{\bigr )}=23$
Fi24 or F3+Fischer1971 3rd1255205709190661721292800≈ 1×1024221 · 316 · 52 · 73 · 11 · 13 · 17 · 23 · 2986712A, 3E, 29$o{\bigl (}(ab)^{3}b{\bigr )}=33$
Fi23Fischer1971 3rd4089470473293004800≈ 4×1018218 · 313 · 52 · 7 · 11 · 13 · 17 · 237822B, 3D, 28$o{\bigl (}a^{bb}(ab)^{14}{\bigr )}=5$
Fi22Fischer1971 3rd64561751654400≈ 6×1013217 · 39 · 52 · 7 · 11 · 13782A, 13, 11$o{\bigl (}(ab)^{2}(abab^{2})^{2}ab^{2}{\bigr )}=12$
Th or F3Thompson1976 3rd90745943887872000≈ 9×1016215 · 310 · 53 · 72 · 13 · 19 · 312482, 3A, 19$o{\bigl (}(ab)^{3}b{\bigr )}=21$
LyLyons1972 Pariah51765179004000000≈ 5×101628 · 37 · 56 · 7 · 11 · 31 · 37 · 6724802, 5A, 14$o{\bigl (}ababab^{2}{\bigr )}=67$
HN or F5Harada, Norton1976 3rd273030912000000≈ 3×1014214 · 36 · 56 · 7 · 11 · 191332A, 3B, 22$o{\bigl (}[a,b]{\bigr )}=5$
Co1Conway1969 2nd4157776806543360000≈ 4×1018221 · 39 · 54 · 72 · 11 · 13 · 232762B, 3C, 40$o{\bigl (}ab(abab^{2})^{2}{\bigr )}=42$
Co2Conway1969 2nd42305421312000≈ 4×1013218 · 36 · 53 · 7 · 11 · 23232A, 5A, 28$o{\bigl (}[a,b]{\bigr )}=4$
Co3Conway1969 2nd495766656000≈ 5×1011210 · 37 · 53 · 7 · 11 · 23232A, 7C, 17$o{\bigl (}(uvv)^{3}(uv)^{6}{\bigr )}=5$[lower-alpha 5]
ON or O'NO'Nan1976 Pariah460815505920≈ 5×101129 · 34 · 5 · 73 · 11 · 19 · 31109442A, 4A, 11$o{\bigl (}abab(b^{2}(b^{2})^{abab})^{5}{\bigr )}=5$
SuzSuzuki1969 2nd448345497600≈ 4×1011213 · 37 · 52 · 7 · 11 · 131432B, 3B, 13$o{\bigl (}[a,b]{\bigr )}=15$
RuRudvalis1972 Pariah145926144000≈ 1×1011214 · 33 · 53 · 7 · 13 · 293782B, 4A, 13$o(abab^{2})=29$
He or F7Held1969 3rd4030387200≈ 4×109210 · 33 · 52 · 73 · 17512A, 7C, 17$o{\bigl (}ab^{2}abab^{2}ab^{2}{\bigr )}=10$
McLMcLaughlin1969 2nd898128000≈ 9×10827 · 36 · 53 · 7 · 11222A, 5A, 11$o{\bigl (}(ab)^{2}(abab^{2})^{2}ab^{2}{\bigr )}=7$
HSHigman, Sims1967 2nd44352000≈ 4×10729 · 32 · 53 · 7 · 11222A, 5A, 11$o(abab^{2})=15$
J4Janko1976 Pariah86775571046077562880≈ 9×1019221 · 33 · 5 · 7 · 113 · 23 · 29 · 31 · 37 · 4313332A, 4A, 37$o{\bigl (}abab^{2}{\bigr )}=10$
J3 or HJMJanko1968 Pariah50232960≈ 5×10727 · 35 · 5 · 17 · 19852A, 3A, 19$o{\bigl (}[a,b]{\bigr )}=9$
J2 or HJJanko1968 2nd604800≈ 6×10527 · 33 · 52 · 7142B, 3B, 7$o{\bigl (}[a,b]{\bigr )}=12$
J1Janko1965 Pariah175560≈ 2×10523 · 3 · 5 · 7 · 11 · 19562, 3, 7$o{\bigl (}abab^{2}{\bigr )}=19$
T (or 2F4(2)′)Tits1964 3rd17971200≈ 2×107211 · 33 · 52 · 13104[16]2A, 3, 13$o{\bigl (}[a,b]{\bigr )}=5$
M24Mathieu1861 1st244823040≈ 2×108210 · 33 · 5 · 7 · 11 · 23232B, 3A, 23$o{\bigl (}ab(abab^{2})^{2}ab^{2}{\bigr )}=4$
M23Mathieu1861 1st10200960≈ 1×10727 · 32 · 5 · 7 · 11 · 23222, 4, 23$o{\bigl (}(ab)^{2}(abab^{2})^{2}ab^{2}{\bigr )}=8$
M22Mathieu1861 1st443520≈ 4×10527 · 32 · 5 · 7 · 11212A, 4A, 11$o{\bigl (}abab^{2}{\bigr )}=11$
M12Mathieu1861 1st95040≈ 1×10526 · 33 · 5 · 11112B, 3B, 11$o{\bigl (}[a,b]{\bigr )}=o{\bigl (}ababab^{2}{\bigr )}=6$
M11Mathieu1861 1st7920≈ 8×10324 · 32 · 5 · 11102, 4, 11$o{\bigl (}(ab)^{2}(abab^{2})^{2}ab^{2}{\bigr )}=4$
Notes
1. The groups of prime order, the alternating groups of degree at least 5, the infinite family of commutator groups 2F4(22n+1)′ of groups of Lie type (containing the Tits group), and 15 families of groups of Lie type.
2. For example, in Eric W. Weisstein, "Tits Group", MathWorld there is a link from the Tits group to "Sporadic Group", as opposed to in Eric W. Weisstein, "Sporadic Group", MathWorld, where the Tits group is not listed among the 26 sporadic groups. Both sources checked on 2018-05-26.
3. Conway et al. (1985, p. viii) organizes the 26 sporadic groups in likeness:
"The sporadic simple groups may be roughly sorted as the Mathieu groups, the Leech lattice groups, Fischer's 3-transposition groups, the further Monster centralizers, and the half-dozen oddments."
4. Here listed are semi-presentations from standard generators of each sporadic group. Most sporadic groups have multiple presentations & semi-presentations; the more prominent examples are listed.
5. Where $u=(b^{2}(b^{2})abb)^{3}$ and $v=t(b^{2}(b^{2})t)^{2}$ with $t=abab^{3}a^{2}$.
References
1. Conway et al. (1985, p. viii)
2. Griess, Jr. (1998, p. 146)
3. Gorenstein, Lyons & Solomon (1998, pp. 262–302)
4. Ronan (2006, pp. 244–246)
5. Griess, Jr. (1982, p. 91)
6. Griess, Jr. (1998, pp. 54–79)
7. Griess, Jr. (1998, pp. 104–145)
8. Griess, Jr. (1998, pp. 146−150)
9. Griess, Jr. (1982, pp. 91−96)
10. Griess, Jr. (1998, pp. 146, 150−152)
11. Hiss (2003, p. 172)
Tabelle 2. Die Entdeckung der sporadischen Gruppen (Table 2. The discovery of the sporadic groups)
12. Sloane (2015)
13. Jansen (2005, pp. 122–123)
14. Nickerson & Wilson (2011, p. 365)
15. Wilson et al. (1999)
16. Lubeck (2001, p. 2151)
Works cited
• Burnside, William (1911). Theory of groups of finite order (2nd ed.). Cambridge: Cambridge University Press. pp. xxiv, 1–512. hdl:2027/uc1.b4062919. ISBN 0-486-49575-2. MR 0069818. OCLC 54407807. S2CID 117347785.
• Conway, J. H. (1968). "A perfect group of order 8,315,553,613,086,720,000 and the sporadic simple groups". Proc. Natl. Acad. Sci. U.S.A. 61 (2): 398–400. Bibcode:1968PNAS...61..398C. doi:10.1073/pnas.61.2.398. MR 0237634. PMID 16591697. S2CID 29358882. Zbl 0186.32401.
• Conway, J. H.; Curtis, R. T.; Norton, S. P.; Parker, R. A.; Wilson, R. A. (1985). ATLAS of Finite Groups: Maximal Subgroups and Ordinary Characters for Simple Groups. Oxford: Clarendon Press. pp. xxxiii, 1–252. ISBN 978-0-19-853199-9. MR 0827219. OCLC 12106933. S2CID 117473588. Zbl 0568.20001.
• Gorenstein, D.; Lyons, Richard; Solomon, Ronald (1998). The classification of the finite simple groups, Number 3. Mathematical Surveys and Monographs. Vol. 40. Providence, R.I.: American Mathematical Society. pp. xiii, 1–362. doi:10.1112/S0024609398255457. ISBN 978-0-8218-0391-2. MR 1490581. OCLC 6907721813. S2CID 209854856.
• Griess, Jr., Robert L. (1982). "The Friendly Giant". Inventiones Mathematicae. 69: 1−102. Bibcode:1982InMat..69....1G. doi:10.1007/BF01389186. hdl:2027.42/46608. MR 0671653. S2CID 123597150. Zbl 0498.20013.
• Griess, Jr., Robert L. (1998). Twelve Sporadic Groups. Springer Monographs in Mathematics. Berlin: Springer-Verlag. pp. 1−169. ISBN 9783540627784. MR 1707296. OCLC 38910263. Zbl 0908.20007.
• Hiss, Gerhard (2003). "Die Sporadischen Gruppen (The Sporadic Groups)" (PDF). Jahresber. Deutsch. Math.-Verein. (Annual Report of the German Mathematicians Association). 105 (4): 169−193. ISSN 0012-0456. MR 2033760. Zbl 1042.20007. (German)
• Jansen, Christoph (2005). "The Minimal Degrees of Faithful Representations of the Sporadic Simple Groups and their Covering Groups". LMS Journal of Computation and Mathematics. London Mathematical Society. 8: 122−144. doi:10.1112/S1461157000000930. MR 2153793. S2CID 121362819. Zbl 1089.20006.
• Lubeck, Frank (2001). "Smallest degrees of representations of exceptional groups of Lie type". Communications in Algebra. Philadelphia, PA: Taylor & Francis. 29 (5): 2147−2169. doi:10.1081/AGB-100002175. MR 1837968. S2CID 122060727. Zbl 1004.20003.
• Nickerson, S.J.; Wilson, R.A. (2011). "Semi-Presentations for the Sporadic Simple Groups". Experimental Mathematics. Oxfordshire: Taylor & Francis. 14 (3): 359−371. doi:10.1080/10586458.2005.10128927. MR 2172713. S2CID 13100616. Zbl 1087.20025.
• Ronan, Mark (2006). Symmetry and the Monster: One of the Greatest Quests of Mathematics. New York: Oxford University Press. pp. vii, 1–255. doi:10.1007/s00283-008-9007-9. ISBN 978-0-19-280722-9. MR 2215662. OCLC 180766312. Zbl 1113.00002.
• Sloane, N.J.A., ed. (2015). "Orders of sporadic simple groups (A001228)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
• Wilson, R.A (1998). "Chapter: An Atlas of Sporadic Group Representations" (PDF). The Atlas of Finite Groups - Ten Years On (LMS Lecture Note Series 249). Cambridge, U.K: Cambridge University Press. pp. 261–273. ISBN 9780511565830. OCLC 726827806. S2CID 59394831. Zbl 0914.20016.
• Wilson, R.A.; Parker, R.A.; Nickerson, S.J.; Bray, J.N. (1999). "ATLAS: Sporadic Groups". ATLAS of Finite Group Representations.
External links
• Weisstein, Eric W. "Sporadic Group". MathWorld.
• Atlas of Finite Group Representations: Sporadic groups
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Wikipedia
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Spouge's approximation
In mathematics, Spouge's approximation is a formula for computing an approximation of the gamma function. It was named after John L. Spouge, who defined the formula in a 1994 paper.[1] The formula is a modification of Stirling's approximation, and has the form
$\Gamma (z+1)=(z+a)^{z+{\frac {1}{2}}}e^{-z-a}\left(c_{0}+\sum _{k=1}^{a-1}{\frac {c_{k}}{z+k}}+\varepsilon _{a}(z)\right)$
where a is an arbitrary positive integer and the coefficients are given by
${\begin{aligned}c_{0}&={\sqrt {2\pi }}\\c_{k}&={\frac {(-1)^{k-1}}{(k-1)!}}(-k+a)^{k-{\frac {1}{2}}}e^{-k+a}\qquad k\in \{1,2,\dots ,a-1\}.\end{aligned}}$
Spouge has proved that, if Re(z) > 0 and a > 2, the relative error in discarding εa(z) is bounded by
$a^{-{\frac {1}{2}}}(2\pi )^{-a-{\frac {1}{2}}}.$
The formula is similar to the Lanczos approximation, but has some distinct features.[2] Whereas the Lanczos formula exhibits faster convergence, Spouge's coefficients are much easier to calculate and the error can be set arbitrarily low. The formula is therefore feasible for arbitrary-precision evaluation of the gamma function. However, special care must be taken to use sufficient precision when computing the sum due to the large size of the coefficients ck, as well as their alternating sign. For example, for a = 49, one must compute the sum using about 65 decimal digits of precision in order to obtain the promised 40 decimal digits of accuracy.
See also
• Stirling's approximation
• Lanczos approximation
References
1. Spouge, John L. (1994). "Computation of the Gamma, Digamma, and Trigamma Functions" (PDF). SIAM Journal on Numerical Analysis. 31 (3): 931–000. doi:10.1137/0731050. JSTOR 2158038.
• Pugh, Glendon (2004). An analysis of the Lanczos Gamma approximation (PDF) (PhD thesis).
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Wikipedia
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SPQR tree
In graph theory, a branch of mathematics, the triconnected components of a biconnected graph are a system of smaller graphs that describe all of the 2-vertex cuts in the graph. An SPQR tree is a tree data structure used in computer science, and more specifically graph algorithms, to represent the triconnected components of a graph. The SPQR tree of a graph may be constructed in linear time[1] and has several applications in dynamic graph algorithms and graph drawing.
The basic structures underlying the SPQR tree, the triconnected components of a graph, and the connection between this decomposition and the planar embeddings of a planar graph, were first investigated by Saunders Mac Lane (1937); these structures were used in efficient algorithms by several other researchers[2] prior to their formalization as the SPQR tree by Di Battista and Tamassia (1989, 1990, 1996).
Structure
An SPQR tree takes the form of an unrooted tree in which for each node x there is associated an undirected graph or multigraph Gx. The node, and the graph associated with it, may have one of four types, given the initials SPQR:
• In an S node, the associated graph is a cycle graph with three or more vertices and edges. This case is analogous to series composition in series–parallel graphs; the S stands for "series".[3]
• In a P node, the associated graph is a dipole graph, a multigraph with two vertices and three or more edges, the planar dual to a cycle graph. This case is analogous to parallel composition in series–parallel graphs; the P stands for "parallel".[3]
• In a Q node, the associated graph has a single real edge. This trivial case is necessary to handle the graph that has only one edge. In some works on SPQR trees, this type of node does not appear in the SPQR trees of graphs with more than one edge; in other works, all non-virtual edges are required to be represented by Q nodes with one real and one virtual edge, and the edges in the other node types must all be virtual.
• In an R node, the associated graph is a 3-connected graph that is not a cycle or dipole. The R stands for "rigid": in the application of SPQR trees in planar graph embedding, the associated graph of an R node has a unique planar embedding.[3]
Each edge xy between two nodes of the SPQR tree is associated with two directed virtual edges, one of which is an edge in Gx and the other of which is an edge in Gy. Each edge in a graph Gx may be a virtual edge for at most one SPQR tree edge.
An SPQR tree T represents a 2-connected graph GT, formed as follows. Whenever SPQR tree edge xy associates the virtual edge ab of Gx with the virtual edge cd of Gy, form a single larger graph by merging a and c into a single supervertex, merging b and d into another single supervertex, and deleting the two virtual edges. That is, the larger graph is the 2-clique-sum of Gx and Gy. Performing this gluing step on each edge of the SPQR tree produces the graph GT; the order of performing the gluing steps does not affect the result. Each vertex in one of the graphs Gx may be associated in this way with a unique vertex in GT, the supervertex into which it was merged.
Typically, it is not allowed within an SPQR tree for two S nodes to be adjacent, nor for two P nodes to be adjacent, because if such an adjacency occurred the two nodes could be merged into a single larger node. With this assumption, the SPQR tree is uniquely determined from its graph. When a graph G is represented by an SPQR tree with no adjacent P nodes and no adjacent S nodes, then the graphs Gx associated with the nodes of the SPQR tree are known as the triconnected components of G.
Construction
The SPQR tree of a given 2-vertex-connected graph can be constructed in linear time.[1]
The problem of constructing the triconnected components of a graph was first solved in linear time by Hopcroft & Tarjan (1973). Based on this algorithm, Di Battista & Tamassia (1996) suggested that the full SPQR tree structure, and not just the list of components, should be constructible in linear time. After an implementation of a slower algorithm for SPQR trees was provided as part of the GDToolkit library, Gutwenger & Mutzel (2001) provided the first linear-time implementation. As part of this process of implementing this algorithm, they also corrected some errors in the earlier work of Hopcroft & Tarjan (1973).
The algorithm of Gutwenger & Mutzel (2001) includes the following overall steps.
1. Sort the edges of the graph by the pairs of numerical indices of their endpoints, using a variant of radix sort that makes two passes of bucket sort, one for each endpoint. After this sorting step, parallel edges between the same two vertices will be adjacent to each other in the sorted list and can be split off into a P-node of the eventual SPQR tree, leaving the remaining graph simple.
2. Partition the graph into split components; these are graphs that can be formed by finding a pair of separating vertices, splitting the graph at these two vertices into two smaller graphs (with a linked pair of virtual edges having the separating vertices as endpoints), and repeating this splitting process until no more separating pairs exist. The partition found in this way is not uniquely defined, because the parts of the graph that should become S-nodes of the SPQR tree will be subdivided into multiple triangles.
3. Label each split component with a P (a two-vertex split component with multiple edges), an S (a split component in the form of a triangle), or an R (any other split component). While there exist two split components that share a linked pair of virtual edges, and both components have type S or both have type P, merge them into a single larger component of the same type.
To find the split components, Gutwenger & Mutzel (2001) use depth-first search to find a structure that they call a palm tree; this is a depth-first search tree with its edges oriented away from the root of the tree, for the edges belonging to the tree, and towards the root for all other edges. They then find a special preorder numbering of the nodes in the tree, and use certain patterns in this numbering to identify pairs of vertices that can separate the graph into smaller components. When a component is found in this way, a stack data structure is used to identify the edges that should be part of the new component.
Usage
Finding 2-vertex cuts
With the SPQR tree of a graph G (without Q nodes) it is straightforward to find every pair of vertices u and v in G such that removing u and v from G leaves a disconnected graph, and the connected components of the remaining graphs:
• The two vertices u and v may be the two endpoints of a virtual edge in the graph associated with an R node, in which case the two components are represented by the two subtrees of the SPQR tree formed by removing the corresponding SPQR tree edge.
• The two vertices u and v may be the two vertices in the graph associated with a P node that has two or more virtual edges. In this case the components formed by the removal of u and v are represented by subtrees of the SPQR tree, one for each virtual edge in the node.
• The two vertices u and v may be two vertices in the graph associated with an S node such that either u and v are not adjacent, or the edge uv is virtual. If the edge is virtual, then the pair (u,v) also belongs to a node of type P and R and the components are as described above. If the two vertices are not adjacent then the two components are represented by two paths of the cycle graph associated with the S node and with the SPQR tree nodes attached to those two paths.
Representing all embeddings of planar graphs
If a planar graph is 3-connected, it has a unique planar embedding up to the choice of which face is the outer face and of orientation of the embedding: the faces of the embedding are exactly the nonseparating cycles of the graph. However, for a planar graph (with labeled vertices and edges) that is 2-connected but not 3-connected, there may be greater freedom in finding a planar embedding. Specifically, whenever two nodes in the SPQR tree of the graph are connected by a pair of virtual edges, it is possible to flip the orientation of one of the nodes (replacing it by its mirror image) relative to the other one. Additionally, in a P node of the SPQR tree, the different parts of the graph connected to virtual edges of the P node may be arbitrarily permuted. All planar representations may be described in this way.[4]
See also
• Block-cut tree, a similar tree structure for the 2-vertex-connected components
• Gomory–Hu tree, a different tree structure that characterizes the edge connectivity of a graph
• Tree decomposition, a generalization (no longer unique) to larger cuts
Notes
1. Hopcroft & Tarjan (1973); Gutwenger & Mutzel (2001).
2. E.g., Hopcroft & Tarjan (1973) and Bienstock & Monma (1988), both of which are cited as precedents by Di Battista and Tamassia.
3. Di Battista & Tamassia (1989).
4. Mac Lane (1937).
References
• Bienstock, Daniel; Monma, Clyde L. (1988), "On the complexity of covering vertices by faces in a planar graph", SIAM Journal on Computing, 17 (1): 53–76, CiteSeerX 10.1.1.542.2314, doi:10.1137/0217004.
• Di Battista, Giuseppe; Tamassia, Roberto (1989), "Incremental planarity testing", Proc. 30th Annual Symposium on Foundations of Computer Science, pp. 436–441, doi:10.1109/SFCS.1989.63515.
• Di Battista, Giuseppe; Tamassia, Roberto (1990), "On-line graph algorithms with SPQR-trees", Proc. 17th International Colloquium on Automata, Languages and Programming, Lecture Notes in Computer Science, vol. 443, Springer-Verlag, pp. 598–611, doi:10.1007/BFb0032061.
• Di Battista, Giuseppe; Tamassia, Roberto (1996), "On-line planarity testing" (PDF), SIAM Journal on Computing, 25 (5): 956–997, doi:10.1137/S0097539794280736.
• Gutwenger, Carsten; Mutzel, Petra (2001), "A linear time implementation of SPQR-trees", Proc. 8th International Symposium on Graph Drawing (GD 2000), Lecture Notes in Computer Science, vol. 1984, Springer-Verlag, pp. 77–90, doi:10.1007/3-540-44541-2_8.
• Hopcroft, John; Tarjan, Robert (1973), "Dividing a graph into triconnected components", SIAM Journal on Computing, 2 (3): 135–158, doi:10.1137/0202012, hdl:1813/6037.
• Mac Lane, Saunders (1937), "A structural characterization of planar combinatorial graphs", Duke Mathematical Journal, 3 (3): 460–472, doi:10.1215/S0012-7094-37-00336-3.
External links
• SPQR tree implementation in the Open Graph Drawing Framework.
• The tree of the triconnected components Java implementation in the jBPT library (see TCTree class).
Tree data structures
Search trees
(dynamic sets/associative arrays)
• 2–3
• 2–3–4
• AA
• (a,b)
• AVL
• B
• B+
• B*
• Bx
• (Optimal) Binary search
• Dancing
• HTree
• Interval
• Order statistic
• (Left-leaning) Red–black
• Scapegoat
• Splay
• T
• Treap
• UB
• Weight-balanced
Heaps
• Binary
• Binomial
• Brodal
• Fibonacci
• Leftist
• Pairing
• Skew
• van Emde Boas
• Weak
Tries
• Ctrie
• C-trie (compressed ADT)
• Hash
• Radix
• Suffix
• Ternary search
• X-fast
• Y-fast
Spatial data partitioning trees
• Ball
• BK
• BSP
• Cartesian
• Hilbert R
• k-d (implicit k-d)
• M
• Metric
• MVP
• Octree
• PH
• Priority R
• Quad
• R
• R+
• R*
• Segment
• VP
• X
Other trees
• Cover
• Exponential
• Fenwick
• Finger
• Fractal tree index
• Fusion
• Hash calendar
• iDistance
• K-ary
• Left-child right-sibling
• Link/cut
• Log-structured merge
• Merkle
• PQ
• Range
• SPQR
• Top
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Wikipedia
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Sprague–Grundy theorem
In combinatorial game theory, the Sprague–Grundy theorem states that every impartial game under the normal play convention is equivalent to a one-heap game of nim, or to an infinite generalization of nim. It can therefore be represented as a natural number, the size of the heap in its equivalent game of nim, as an ordinal number in the infinite generalization, or alternatively as a nimber, the value of that one-heap game in an algebraic system whose addition operation combines multiple heaps to form a single equivalent heap in nim.
The Grundy value or nim-value of any impartial game is the unique nimber that the game is equivalent to. In the case of a game whose positions are indexed by the natural numbers (like nim itself, which is indexed by its heap sizes), the sequence of nimbers for successive positions of the game is called the nim-sequence of the game.
The Sprague–Grundy theorem and its proof encapsulate the main results of a theory discovered independently by R. P. Sprague (1936)[1] and P. M. Grundy (1939).[2]
Definitions
For the purposes of the Sprague–Grundy theorem, a game is a two-player sequential game of perfect information satisfying the ending condition (all games come to an end: there are no infinite lines of play) and the normal play condition (a player who cannot move loses).
At any given point in the game, a player's position is the set of moves they are allowed to make. As an example, we can define the zero game to be the two-player game where neither player has any legal moves. Referring to the two players as $A$ (for Alice) and $B$ (for Bob), we would denote their positions as $(A,B)=(\{\},\{\})$, since the set of moves each player can make is empty.
An impartial game is one in which at any given point in the game, each player is allowed exactly the same set of moves. Normal-play nim is an example of an impartial game. In nim, there are one or more heaps of objects, and two players (we'll call them Alice and Bob), take turns choosing a heap and removing 1 or more objects from it. The winner is the player who removes the final object from the final heap. The game is impartial because for any given configuration of pile sizes, the moves Alice can make on her turn are exactly the same moves Bob would be allowed to make if it were his turn. In contrast, a game such as checkers is not impartial because, supposing Alice were playing red and Bob were playing black, for any given arrangement of pieces on the board, if it were Alice's turn, she would only be allowed to move the red pieces, and if it were Bob's turn, he would only be allowed to move the black pieces.
Note that any configuration of an impartial game can therefore be written as a single position, because the moves will be the same no matter whose turn it is. For example, the position of the zero game can simply be written $\{\}$, because if it's Alice's turn, she has no moves to make, and if it's Bob's turn, he has no moves to make either. A move can be associated with the position it leaves the next player in.
Doing so allows positions to be defined recursively. For example, consider the following game of Nim played by Alice and Bob.
Example Nim Game
Sizes of heaps Moves
A B C
1 2 2 Alice takes 1 from A
0 2 2 Bob takes 1 from B
0 1 2 Alice takes 1 from C
0 1 1 Bob takes 1 from B
0 0 1 Alice takes 1 from C
0 0 0 Bob has no moves, so Alice wins
• At step 6 of the game (when all of the heaps are empty) the position is $\{\}$, because Bob has no valid moves to make. We name this position $*0$.
• At step 5, Alice had exactly one option: to remove one object from heap C, leaving Bob with no moves. Since her move leaves Bob in position $*0$, her position is written $\{*0\}$. We name this position $*1$.
• At step 4, Bob had two options: remove one from B or remove one from C. Note, however, that it didn't really matter which heap Bob removed the object from: Either way, Alice would be left with exactly one object in exactly one pile. So, using our recursive definition, Bob really only has one move: $*1$. Thus, Bob's position is $\{*1\}$.
• At step 3, Alice had 3 options: remove two from C, remove one from C, or remove one from B. Removing two from C leaves Bob in position $*1$. Removing one from C leaves Bob with two piles, each of size one, i.e., position $\{*1\}$, as described in step 4. However, removing 1 from B would leave Bob with two objects in a single pile. His moves would then be $*0$ and $*1$, so her move would result in the position $\{*0,*1\}$. We call this position $*2$. Alice's position is then the set of all her moves: ${\big \{}*1,\{*1\},*2{\big \}}$.
• Following the same recursive logic, at step 2, Bob's position is
${\big \{}\{*1,\{*1\},*2\},*2{\big \}}.$
• Finally, at step 1, Alice's position is
${\Big \{}{\big \{}*1,\{*1\},*2{\big \}},{\big \{}*2,\{*1,\{*1\},*2\}{\big \}},{\big \{}\{*1\},\{\{*1\}\},\{*1,\{*1\},*2\}{\big \}}{\Big \}}.$
Nimbers
The special names $*0$, $*1$, and $*2$ referenced in our example game are called nimbers. In general, the nimber $*n$ corresponds to the position in a game of nim where there are exactly $n$ objects in exactly one heap. Formally, nimbers are defined inductively as follows: $*0$ is $\{\}$, $*1=\{*0\}$, $*2=\{*0,*1\}$ and for all $n\geq 0$, $*(n+1)=*n\cup \{*n\}$.
While the word nimber comes from the game nim, nimbers can be used to describe the positions of any finite, impartial game, and in fact, the Sprague–Grundy theorem states that every instance of a finite, impartial game can be associated with a single nimber.
Combining Games
Two games can be combined by adding their positions together. For example, consider another game of nim with heaps $A'$, $B'$, and $C'$.
Example Game 2
Sizes of heaps Moves
A' B' C'
1 1 1 Alice takes 1 from A'
0 1 1 Bob takes one from B'
0 0 1 Alice takes one from C'
0 0 0 Bob has no moves, so Alice wins.
We can combine it with our first example to get a combined game with six heaps: $A$, $B$, $C$, $A'$, $B'$, and $C'$:
Combined Game
Sizes of heaps Moves
A B C A' B' C'
1 2 2 1 1 1 Alice takes 1 from A
0 2 2 1 1 1 Bob takes 1 from A'
0 2 2 0 1 1 Alice takes 1 from B'
0 2 2 0 0 1 Bob takes 1 from C'
0 2 2 0 0 0 Alice takes 2 from B
0 0 2 0 0 0 Bob takes 2 from C
0 0 0 0 0 0 Alice has no moves, so Bob wins.
To differentiate between the two games, for the first example game, we'll label its starting position $\color {blue}S$, and color it blue:
$\color {blue}S={\Big \{}{\big \{}*1,\{*1\},*2{\big \}},{\big \{}*2,\{*1,\{*1\},*2\}{\big \}},{\big \{}\{*1\},\{\{*1\}\},\{*1,\{*1\},*2\}{\big \}}{\Big \}}$
For the second example game, we'll label the starting position $\color {red}S'$ and color it red:
$\color {red}S'={\Big \{}\{*1\}{\Big \}}.$
To compute the starting position of the combined game, remember that a player can either make a move in the first game, leaving the second game untouched, or make a move in the second game, leaving the first game untouched. So the combined game's starting position is:
$\color {blue}S\color {black}+\color {red}S'\color {black}={\Big \{}\color {blue}S\color {black}+\color {red}\{*1\}\color {black}{\Big \}}\cup {\Big \{}\color {red}S'\color {black}+\color {blue}\{*1,\{*1\},*2\}\color {black},\color {red}S'\color {black}+\color {blue}\{*2,\{*1,\{*1\},*2\}\}\color {black},\color {red}S'\color {black}+\color {blue}\{\{*1\},\{\{*1\}\},\{*1,\{*1\},*2\}\}\color {black}{\Big \}}$
The explicit formula for adding positions is: $S+S'=\{S+s'\mid s'\in S'\}\cup \{s+S'\mid s\in S\}$, which means that addition is both commutative and associative.
Equivalence
Positions in impartial games fall into two outcome classes: either the next player (the one whose turn it is) wins (an ${\boldsymbol {\mathcal {N}}}$- position), or the previous player wins (a ${\boldsymbol {\mathcal {P}}}$- position). So, for example, $*0$ is a ${\mathcal {P}}$-position, while $*1$ is an ${\mathcal {N}}$-position.
Two positions $G$ and $G'$ are equivalent if, no matter what position $H$ is added to them, they are always in the same outcome class. Formally, $G\approx G'$ if and only if $\forall H$, $G+H$ is in the same outcome class as $G'+H$.
To use our running examples, notice that in both the first and second games above, we can show that on every turn, Alice has a move that forces Bob into a ${\mathcal {P}}$-position. Thus, both $\color {blue}S$ and $\color {red}S'$ are ${\mathcal {N}}$-positions. (Notice that in the combined game, Bob is the player with the ${\mathcal {N}}$-positions. In fact, $\color {blue}S\color {black}+\color {red}S'$ is a ${\mathcal {P}}$-position, which as we will see in Lemma 2, means $\color {blue}S\color {black}\approx \color {red}S'$.)
First Lemma
As an intermediate step to proving the main theorem, we show that for every position $G$ and every ${\mathcal {P}}$-position $A$, the equivalence $G\approx A+G$ holds. By the above definition of equivalence, this amounts to showing that $G+H$ and $A+G+H$ share an outcome class for all $H$.
Suppose that $G+H$ is a ${\mathcal {P}}$-position. Then the previous player has a winning strategy for $A+G+H$: respond to moves in $A$ according to their winning strategy for $A$ (which exists by virtue of $A$ being a ${\mathcal {P}}$-position), and respond to moves in $G+H$ according to their winning strategy for $G+H$ (which exists for the analogous reason). So $A+G+H$ must also be a ${\mathcal {P}}$-position.
On the other hand, if $G+H$ is an ${\mathcal {N}}$-position, then $A+G+H$ is also an ${\mathcal {N}}$-position, because the next player has a winning strategy: choose a ${\mathcal {P}}$-position from among the $G+H$ options, and we conclude from the previous paragraph that adding $A$ to that position is still a ${\mathcal {P}}$-position. Thus, in this case, $A+G+H$ must be a ${\mathcal {N}}$-position, just like $G+H$.
As these are the only two cases, the lemma holds.
Second Lemma
As a further step, we show that $G\approx G'$ if and only if $G+G'$ is a ${\mathcal {P}}$-position.
In the forward direction, suppose that $G\approx G'$. Applying the definition of equivalence with $H=G$, we find that $G'+G$ (which is equal to $G+G'$ by commutativity of addition) is in the same outcome class as $G+G$. But $G+G$ must be a ${\mathcal {P}}$-position: for every move made in one copy of $G$, the previous player can respond with the same move in the other copy, and so always make the last move.
In the reverse direction, since $A=G+G'$ is a ${\mathcal {P}}$-position by hypothesis, it follows from the first lemma, $G\approx G+A$, that $G\approx G+(G+G')$. Similarly, since $B=G+G$ is also a ${\mathcal {P}}$-position, it follows from the first lemma in the form $G'\approx G'+B$ that $G'\approx G'+(G+G)$. By associativity and commutativity, the right-hand sides of these results are equal. Furthermore, $\approx $ is an equivalence relation because equality is an equivalence relation on outcome classes. Via the transitivity of $\approx $, we can conclude that $G\approx G'$.
Proof
We prove that all positions are equivalent to a nimber by structural induction. The more specific result, that the given game's initial position must be equivalent to a nimber, shows that the game is itself equivalent to a nimber.
Consider a position $G=\{G_{1},G_{2},\ldots ,G_{k}\}$. By the induction hypothesis, all of the options are equivalent to nimbers, say $G_{i}\approx *n_{i}$. So let $G'=\{*n_{1},*n_{2},\ldots ,*n_{k}\}$. We will show that $G\approx *m$, where $m$ is the mex (minimum exclusion) of the numbers $n_{1},n_{2},\ldots ,n_{k}$, that is, the smallest non-negative integer not equal to some $n_{i}$.
The first thing we need to note is that $G\approx G'$, by way of the second lemma. If $k$ is zero, the claim is trivially true. Otherwise, consider $G+G'$. If the next player makes a move to $G_{i}$ in $G$, then the previous player can move to $*n_{i}$ in $G'$, and conversely if the next player makes a move in $G'$. After this, the position is a ${\mathcal {P}}$-position by the lemma's forward implication. Therefore, $G+G'$ is a ${\mathcal {P}}$-position, and, citing the lemma's reverse implication, $G\approx G'$.
Now let us show that $G'+*m$ is a ${\mathcal {P}}$-position, which, using the second lemma once again, means that $G'\approx *m$. We do so by giving an explicit strategy for the previous player.
Suppose that $G'$ and $*m$ are empty. Then $G'+*m$ is the null set, clearly a ${\mathcal {P}}$-position.
Or consider the case that the next player moves in the component $*m$ to the option $*m'$ where $m'<m$. Because $m$ was the minimum excluded number, the previous player can move in $G'$ to $*m'$. And, as shown before, any position plus itself is a ${\mathcal {P}}$-position.
Finally, suppose instead that the next player moves in the component $G'$ to the option $*n_{i}$. If $n_{i}<m$ then the previous player moves in $*m$ to $*n_{i}$; otherwise, if $n_{i}>m$, the previous player moves in $*n_{i}$ to $*m$; in either case the result is a position plus itself. (It is not possible that $n_{i}=m$ because $m$ was defined to be different from all the $n_{i}$.)
In summary, we have $G\approx G'$ and $G'\approx *m$. By transitivity, we conclude that $G\approx *m$, as desired.
Development
If $G$ is a position of an impartial game, the unique integer $m$ such that $G\approx *m$ is called its Grundy value, or Grundy number, and the function that assigns this value to each such position is called the Sprague–Grundy function. R. L. Sprague and P. M. Grundy independently gave an explicit definition of this function, not based on any concept of equivalence to nim positions, and showed that it had the following properties:
• The Grundy value of a single nim pile of size $m$ (i.e. of the position $*m$) is $m$;
• A position is a loss for the next player to move (i.e. a ${\mathcal {P}}$-position) if and only if its Grundy value is zero; and
• The Grundy value of the sum of a finite set of positions is just the nim-sum of the Grundy values of its summands.
It follows straightforwardly from these results that if a position $G$ has a Grundy value of $m$, then $G+H$ has the same Grundy value as $*m+H$, and therefore belongs to the same outcome class, for any position $H$. Thus, although Sprague and Grundy never explicitly stated the theorem described in this article, it follows directly from their results and is credited to them.[3][4] These results have subsequently been developed into the field of combinatorial game theory, notably by Richard Guy, Elwyn Berlekamp, John Horton Conway and others, where they are now encapsulated in the Sprague–Grundy theorem and its proof in the form described here. The field is presented in the books Winning Ways for your Mathematical Plays and On Numbers and Games.
See also
• Genus theory
• Indistinguishability quotient
References
1. Sprague, R. P. (1936). "Über mathematische Kampfspiele". Tohoku Mathematical Journal (in German). 41: 438–444. JFM 62.1070.03. Zbl 0013.29004.
2. Grundy, P. M. (1939). "Mathematics and games". Eureka. 2: 6–8. Archived from the original on 2007-09-27. Reprinted, 1964, 27: 9–11.
3. Smith, Cedric A.B. (1960), "Patrick Michael Grundy, 1917–1959", Journal of the Royal Statistical Society, Series A, 123 (2): 221–22
4. Schleicher, Dierk; Stoll, Michael (2006). "An introduction to Conway's games and numbers". Moscow Mathematical Journal. 6 (2): 359–388. arXiv:math.CO/0410026. doi:10.17323/1609-4514-2006-6-2-359-388. S2CID 7175146.
External links
• Grundy's game at cut-the-knot
• Easily readable, introductory account from the UCLA Math Department
• The Game of Nim at sputsoft.com
• Milvang-Jensen, Brit C. A. (2000), Combinatorial Games, Theory and Applications (PDF), CiteSeerX 10.1.1.89.805
Topics in game theory
Definitions
• Congestion game
• Cooperative game
• Determinacy
• Escalation of commitment
• Extensive-form game
• First-player and second-player win
• Game complexity
• Graphical game
• Hierarchy of beliefs
• Information set
• Normal-form game
• Preference
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• Simultaneous game
• Simultaneous action selection
• Solved game
• Succinct game
Equilibrium
concepts
• Bayesian Nash equilibrium
• Berge equilibrium
• Core
• Correlated equilibrium
• Epsilon-equilibrium
• Evolutionarily stable strategy
• Gibbs equilibrium
• Mertens-stable equilibrium
• Markov perfect equilibrium
• Nash equilibrium
• Pareto efficiency
• Perfect Bayesian equilibrium
• Proper equilibrium
• Quantal response equilibrium
• Quasi-perfect equilibrium
• Risk dominance
• Satisfaction equilibrium
• Self-confirming equilibrium
• Sequential equilibrium
• Shapley value
• Strong Nash equilibrium
• Subgame perfection
• Trembling hand
Strategies
• Backward induction
• Bid shading
• Collusion
• Forward induction
• Grim trigger
• Markov strategy
• Dominant strategies
• Pure strategy
• Mixed strategy
• Strategy-stealing argument
• Tit for tat
Classes
of games
• Bargaining problem
• Cheap talk
• Global game
• Intransitive game
• Mean-field game
• Mechanism design
• n-player game
• Perfect information
• Large Poisson game
• Potential game
• Repeated game
• Screening game
• Signaling game
• Stackelberg competition
• Strictly determined game
• Stochastic game
• Symmetric game
• Zero-sum game
Games
• Go
• Chess
• Infinite chess
• Checkers
• Tic-tac-toe
• Prisoner's dilemma
• Gift-exchange game
• Optional prisoner's dilemma
• Traveler's dilemma
• Coordination game
• Chicken
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• Lewis signaling game
• Volunteer's dilemma
• Dollar auction
• Battle of the sexes
• Stag hunt
• Matching pennies
• Ultimatum game
• Rock paper scissors
• Pirate game
• Dictator game
• Public goods game
• Blotto game
• War of attrition
• El Farol Bar problem
• Fair division
• Fair cake-cutting
• Cournot game
• Deadlock
• Diner's dilemma
• Guess 2/3 of the average
• Kuhn poker
• Nash bargaining game
• Induction puzzles
• Trust game
• Princess and monster game
• Rendezvous problem
Theorems
• Arrow's impossibility theorem
• Aumann's agreement theorem
• Folk theorem
• Minimax theorem
• Nash's theorem
• Negamax theorem
• Purification theorem
• Revelation principle
• Sprague–Grundy theorem
• Zermelo's theorem
Key
figures
• Albert W. Tucker
• Amos Tversky
• Antoine Augustin Cournot
• Ariel Rubinstein
• Claude Shannon
• Daniel Kahneman
• David K. Levine
• David M. Kreps
• Donald B. Gillies
• Drew Fudenberg
• Eric Maskin
• Harold W. Kuhn
• Herbert Simon
• Hervé Moulin
• John Conway
• Jean Tirole
• Jean-François Mertens
• Jennifer Tour Chayes
• John Harsanyi
• John Maynard Smith
• John Nash
• John von Neumann
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• Melvin Dresher
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• Olga Bondareva
• Oskar Morgenstern
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• Peyton Young
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• Robert Axelrod
• Robert Aumann
• Robert B. Wilson
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• Suzanne Scotchmer
• Thomas Schelling
• William Vickrey
Miscellaneous
• All-pay auction
• Alpha–beta pruning
• Bertrand paradox
• Bounded rationality
• Combinatorial game theory
• Confrontation analysis
• Coopetition
• Evolutionary game theory
• First-move advantage in chess
• Game Description Language
• Game mechanics
• Glossary of game theory
• List of game theorists
• List of games in game theory
• No-win situation
• Solving chess
• Topological game
• Tragedy of the commons
• Tyranny of small decisions
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Wikipedia
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Spray (mathematics)
In differential geometry, a spray is a vector field H on the tangent bundle TM that encodes a quasilinear second order system of ordinary differential equations on the base manifold M. Usually a spray is required to be homogeneous in the sense that its integral curves t→ΦHt(ξ)∈TM obey the rule ΦHt(λξ)=ΦHλt(ξ) in positive reparameterizations. If this requirement is dropped, H is called a semispray.
Sprays arise naturally in Riemannian and Finsler geometry as the geodesic sprays whose integral curves are precisely the tangent curves of locally length minimizing curves. Semisprays arise naturally as the extremal curves of action integrals in Lagrangian mechanics. Generalizing all these examples, any (possibly nonlinear) connection on M induces a semispray H, and conversely, any semispray H induces a torsion-free nonlinear connection on M. If the original connection is torsion-free it coincides with the connection induced by H, and homogeneous torsion-free connections are in one-to-one correspondence with full sprays.[1]
Formal definitions
Let M be a differentiable manifold and (TM,πTM,M) its tangent bundle. Then a vector field H on TM (that is, a section of the double tangent bundle TTM) is a semispray on M, if any of the three following equivalent conditions holds:
• (πTM)*Hξ = ξ.
• JH=V, where J is the tangent structure on TM and V is the canonical vector field on TM\0.
• j∘H=H, where j:TTM→TTM is the canonical flip and H is seen as a mapping TM→TTM.
A semispray H on M is a (full) spray if any of the following equivalent conditions hold:
• Hλξ = λ*(λHξ), where λ*:TTM→TTM is the push-forward of the multiplication λ:TM→TM by a positive scalar λ>0.
• The Lie-derivative of H along the canonical vector field V satisfies [V,H]=H.
• The integral curves t→ΦHt(ξ)∈TM\0 of H satisfy ΦHt(λξ)=λΦHλt(ξ) for any λ>0.
Let $(x^{i},\xi ^{i})$ be the local coordinates on $TM$ associated with the local coordinates $(x^{i}$) on $M$ using the coordinate basis on each tangent space. Then $H$ is a semispray on $M$ if it has a local representation of the form
$H_{\xi }=\xi ^{i}{\frac {\partial }{\partial x^{i}}}{\Big |}_{(x,\xi )}-2G^{i}(x,\xi ){\frac {\partial }{\partial \xi ^{i}}}{\Big |}_{(x,\xi )}.$
on each associated coordinate system on TM. The semispray H is a (full) spray, if and only if the spray coefficients Gi satisfy
$G^{i}(x,\lambda \xi )=\lambda ^{2}G^{i}(x,\xi ),\quad \lambda >0.\,$
Semisprays in Lagrangian mechanics
A physical system is modeled in Lagrangian mechanics by a Lagrangian function L:TM→R on the tangent bundle of some configuration space M. The dynamical law is obtained from the Hamiltonian principle, which states that the time evolution γ:[a,b]→M of the state of the system is stationary for the action integral
${\mathcal {S}}(\gamma ):=\int _{a}^{b}L(\gamma (t),{\dot {\gamma }}(t))dt$.
In the associated coordinates on TM the first variation of the action integral reads as
${\frac {d}{ds}}{\Big |}_{s=0}{\mathcal {S}}(\gamma _{s})={\Big |}_{a}^{b}{\frac {\partial L}{\partial \xi ^{i}}}X^{i}-\int _{a}^{b}{\Big (}{\frac {\partial ^{2}L}{\partial \xi ^{j}\partial \xi ^{i}}}{\ddot {\gamma }}^{j}+{\frac {\partial ^{2}L}{\partial x^{j}\partial \xi ^{i}}}{\dot {\gamma }}^{j}-{\frac {\partial L}{\partial x^{i}}}{\Big )}X^{i}dt,$
where X:[a,b]→R is the variation vector field associated with the variation γs:[a,b]→M around γ(t) = γ0(t). This first variation formula can be recast in a more informative form by introducing the following concepts:
• The covector $\alpha _{\xi }=\alpha _{i}(x,\xi )dx^{i}|_{x}\in T_{x}^{*}M$ with $\alpha _{i}(x,\xi )={\tfrac {\partial L}{\partial \xi ^{i}}}(x,\xi )$ is the conjugate momentum of $\xi \in T_{x}M$.
• The corresponding one-form $\alpha \in \Omega ^{1}(TM)$ with $\alpha _{\xi }=\alpha _{i}(x,\xi )dx^{i}|_{(x,\xi )}\in T_{\xi }^{*}TM$ is the Hilbert-form associated with the Lagrangian.
• The bilinear form $g_{\xi }=g_{ij}(x,\xi )(dx^{i}\otimes dx^{j})|_{x}$ with $g_{ij}(x,\xi )={\tfrac {\partial ^{2}L}{\partial \xi ^{i}\partial \xi ^{j}}}(x,\xi )$ is the fundamental tensor of the Lagrangian at $\xi \in T_{x}M$.
• The Lagrangian satisfies the Legendre condition if the fundamental tensor $\displaystyle g_{\xi }$ is non-degenerate at every $\xi \in T_{x}M$. Then the inverse matrix of $\displaystyle g_{ij}(x,\xi )$ is denoted by $\displaystyle g^{ij}(x,\xi )$.
• The Energy associated with the Lagrangian is $\displaystyle E(\xi )=\alpha _{\xi }(\xi )-L(\xi )$.
If the Legendre condition is satisfied, then dα∈Ω2(TM) is a symplectic form, and there exists a unique Hamiltonian vector field H on TM corresponding to the Hamiltonian function E such that
$\displaystyle dE=-\iota _{H}d\alpha $.
Let (Xi,Yi) be the components of the Hamiltonian vector field H in the associated coordinates on TM. Then
$\iota _{H}d\alpha =Y^{i}{\frac {\partial ^{2}L}{\partial \xi ^{i}\partial x^{j}}}dx^{j}-X^{i}{\frac {\partial ^{2}L}{\partial \xi ^{i}\partial x^{j}}}d\xi ^{j}$
and
$dE={\Big (}{\frac {\partial ^{2}L}{\partial x^{i}\partial \xi ^{j}}}\xi ^{j}-{\frac {\partial L}{\partial x^{i}}}{\Big )}dx^{i}+\xi ^{j}{\frac {\partial ^{2}L}{\partial \xi ^{i}\partial x^{j}}}d\xi ^{i}$
so we see that the Hamiltonian vector field H is a semispray on the configuration space M with the spray coefficients
$G^{k}(x,\xi )={\frac {g^{ki}}{2}}{\Big (}{\frac {\partial ^{2}L}{\partial \xi ^{i}\partial x^{j}}}\xi ^{j}-{\frac {\partial L}{\partial x^{i}}}{\Big )}.$
Now the first variational formula can be rewritten as
${\frac {d}{ds}}{\Big |}_{s=0}{\mathcal {S}}(\gamma _{s})={\Big |}_{a}^{b}\alpha _{i}X^{i}-\int _{a}^{b}g_{ik}({\ddot {\gamma }}^{k}+2G^{k})X^{i}dt,$
and we see γ[a,b]→M is stationary for the action integral with fixed end points if and only if its tangent curve γ':[a,b]→TM is an integral curve for the Hamiltonian vector field H. Hence the dynamics of mechanical systems are described by semisprays arising from action integrals.
Geodesic spray
Main article: Geodesic spray
Further information: Geodesic flow
The locally length minimizing curves of Riemannian and Finsler manifolds are called geodesics. Using the framework of Lagrangian mechanics one can describe these curves with spray structures. Define a Lagrangian function on TM by
$L(x,\xi )={\tfrac {1}{2}}F^{2}(x,\xi ),$
where F:TM→R is the Finsler function. In the Riemannian case one uses F2(x,ξ) = gij(x)ξiξj. Now introduce the concepts from the section above. In the Riemannian case it turns out that the fundamental tensor gij(x,ξ) is simply the Riemannian metric gij(x). In the general case the homogeneity condition
$F(x,\lambda \xi )=\lambda F(x,\xi ),\quad \lambda >0$
of the Finsler-function implies the following formulae:
$\alpha _{i}=g_{ij}\xi ^{i},\quad F^{2}=g_{ij}\xi ^{i}\xi ^{j},\quad E=\alpha _{i}\xi ^{i}-L={\tfrac {1}{2}}F^{2}.$
In terms of classical mechanical the last equation states that all the energy in the system (M,L) is in the kinetic form. Furthermore, one obtains the homogeneity properties
$g_{ij}(\lambda \xi )=g_{ij}(\xi ),\quad \alpha _{i}(x,\lambda \xi )=\lambda \alpha _{i}(x,\xi ),\quad G^{i}(x,\lambda \xi )=\lambda ^{2}G^{i}(x,\xi ),$
of which the last one says that the Hamiltonian vector field H for this mechanical system is a full spray. The constant speed geodesics of the underlying Finsler (or Riemannian) manifold are described by this spray for the following reasons:
• Since gξ is positive definite for Finsler spaces, every short enough stationary curve for the length functional is length minimizing.
• Every stationary curve for the action integral is of constant speed $F(\gamma (t),{\dot {\gamma }}(t))=\lambda $, since the energy is automatically a constant of motion.
• For any curve $\gamma :[a,b]\to M$ :[a,b]\to M} of constant speed the action integral and the length functional are related by
${\mathcal {S}}(\gamma )={\frac {(b-a)\lambda ^{2}}{2}}={\frac {\ell (\gamma )^{2}}{2(b-a)}}.$
Therefore, a curve $\gamma :[a,b]\to M$ :[a,b]\to M} is stationary to the action integral if and only if it is of constant speed and stationary to the length functional. The Hamiltonian vector field H is called the geodesic spray of the Finsler manifold (M,F) and the corresponding flow ΦHt(ξ) is called the geodesic flow.
Correspondence with nonlinear connections
A semispray $H$ on a smooth manifold $M$ defines an Ehresmann-connection $T(TM\setminus 0)=H(TM\setminus 0)\oplus V(TM\setminus 0)$ on the slit tangent bundle through its horizontal and vertical projections
$h:T(TM\setminus 0)\to T(TM\setminus 0)\quad ;\quad h={\tfrac {1}{2}}{\big (}I-{\mathcal {L}}_{H}J{\big )},$ ;\quad h={\tfrac {1}{2}}{\big (}I-{\mathcal {L}}_{H}J{\big )},}
$v:T(TM\setminus 0)\to T(TM\setminus 0)\quad ;\quad v={\tfrac {1}{2}}{\big (}I+{\mathcal {L}}_{H}J{\big )}.$ ;\quad v={\tfrac {1}{2}}{\big (}I+{\mathcal {L}}_{H}J{\big )}.}
This connection on TM\0 always has a vanishing torsion tensor, which is defined as the Frölicher-Nijenhuis bracket T=[J,v]. In more elementary terms the torsion can be defined as
$\displaystyle T(X,Y)=J[hX,hY]-v[JX,hY)-v[hX,JY].$
Introducing the canonical vector field V on TM\0 and the adjoint structure Θ of the induced connection the horizontal part of the semispray can be written as hH=ΘV. The vertical part ε=vH of the semispray is known as the first spray invariant, and the semispray H itself decomposes into
$\displaystyle H=\Theta V+\epsilon .$
The first spray invariant is related to the tension
$\tau ={\mathcal {L}}_{V}v={\tfrac {1}{2}}{\mathcal {L}}_{[V,H]-H}J$
of the induced non-linear connection through the ordinary differential equation
${\mathcal {L}}_{V}\epsilon +\epsilon =\tau \Theta V.$
Therefore, the first spray invariant ε (and hence the whole semi-spray H) can be recovered from the non-linear connection by
$\epsilon |_{\xi }=\int \limits _{-\infty }^{0}e^{-s}(\Phi _{V}^{-s})_{*}(\tau \Theta V)|_{\Phi _{V}^{s}(\xi )}ds.$
From this relation one also sees that the induced connection is homogeneous if and only if H is a full spray.
Jacobi fields of sprays and semisprays
A good source for Jacobi fields of semisprays is Section 4.4, Jacobi equations of a semispray of the publicly available book Finsler-Lagrange Geometry by Bucătaru and Miron. Of particular note is their concept of a dynamical covariant derivative. In another paper Bucătaru, Constantinescu and Dahl relate this concept to that of the Kosambi biderivative operator.
For a good introduction to Kosambi's methods, see the article, What is Kosambi-Cartan-Chern theory?.
References
1. I. Bucataru, R. Miron, Finsler-Lagrange Geometry, Editura Academiei Române, 2007.
• Sternberg, Shlomo (1964), Lectures on Differential Geometry, Prentice-Hall.
• Lang, Serge (1999), Fundamentals of Differential Geometry, Springer-Verlag.
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Wikipedia
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Spread (projective geometry)
A frequently studied problem in discrete geometry is to identify ways in which an object can be covered by other simpler objects such as points, lines, and planes. In projective geometry, a specific instance of this problem that has numerous applications is determining whether, and how, a projective space can be covered by pairwise disjoint subspaces which have the same dimension; such a partition is called a spread. Specifically, a spread of a projective space $PG(d,K)$, where $d\geq 1$ is an integer and $K$ a division ring, is a set of $r$-dimensional subspaces, for some $0<r<d$ such that every point of the space lies in exactly one of the elements of the spread.
Spreads are particularly well-studied in projective geometries over finite fields, though some notable results apply to infinite projective geometries as well. In the finite case, the foundational work on spreads appears in André[1] and independently in Bruck-Bose[2] in connection with the theory of translation planes. In these papers, it is shown that a spread of $r$-dimensional subspaces of the finite projective space $PG(d,q)$ exists if and only if $r+1\mid d+1$.[3]
Spreads and translation planes
For all integers $n\geq 1$, the projective space $PG(2n+1,q)$ always has a spread of $n$-dimensional subspaces, and in this section the term spread refers to this specific type of spread; spreads of this form may (and frequently do) occur in infinite projective geometries as well. These spreads are the most widely studied in the literature, due to the fact that every such spread can be used to create a translation plane using the André/Bruck-Bose construction.[1][2]
Reguli and regular spreads
Let $\Sigma $ be the projective space $PG(2n+1,K)$ for $n\geq 1$ an integer, and $K$ a division ring. A regulus[4] $R$ in $\Sigma $ is a collection of pairwise disjoint $n$-dimensional subspaces with the following properties:
1. $R$ contains at least 3 elements
2. Every line meeting three elements of $R$, called a transversal, meets every element of $R$
3. Every point of a transversal to $R$ lies on some element of $R$
Any three pairwise disjoint $n$-dimensional subspaces in $\Sigma $ lie in a unique regulus.[5] A spread $S$ of $\Sigma $ is regular if for any three distinct $n$-dimensional subspaces of $S$, all the members of the unique regulus determined by them are contained in $S$. Regular spreads are significant in the theory of translation planes, in that they generate Moufang planes in general, and Desarguesian planes in the finite case when the order of the ambient field is greater than $2$. All spreads of $PG(2n+1,2)$ are trivially regular, since a regulus only contains three elements.
Constructing a regular spread
Construction of a regular spread is most easily seen using an algebraic model. Letting $V$ be a $(2n+2)$-dimensional vector space over a field $F$, one can model the $k$-dimensional subspaces of $PG(2n+1,F)$ using the $(k+1)$-dimensional subspaces of $V$; this model uses homogeneous coordinates to represent points and hyperplanes. Incidence is defined by intersection, with subspaces intersecting in only the zero vector considered disjoint; in this model, the zero vector of $V$ is effectively ignored.
Let $F$ be a field and $E$ an $n$-dimensional extension field of $F$. Consider $V=E\oplus E$ as a $2n$-dimensional vector space over $F$, which provides a model for the projective space $PG(2n-1,F)$ as above. Each element of $V$ can be written uniquely as $(x,y)$ where $x,y\in E$. A regular spread is given by the set of $n$-dimensional projective spaces defined by $J(k)=\{(x,kx):x\in E\}$, for each $k\in E$, together with $J(\infty )=\{(0,y):y\in E\}$.[6]
Constructing spreads
Spread sets
The construction of a regular spread above is an instance of a more general construction of spreads, which uses the fact that field multiplication is a linear transformation over $E$ when considered as a vector space. Since $E$ is a finite $n$-dimensional extension over $F$, a linear transformation from $E$ to itself can be represented by an $n\times n$ matrix with entries in $F$. A spread set is a set $S$ of $n\times n$ matrices over $F$ with the following properties:
• $S$ contains the zero matrix and the identity matrix
• For any two distinct matrices $X$ and $Y$ in $S$, $X-Y$ is nonsingular
• For each pair of elements $a,b\in E$, there is a unique $X\in S$ such that $aX=b$
In the finite case, where $E$ is the field of order $q^{n}$ for some prime power $q$, the last condition is equivalent to the spread set containing $q^{n}$ matrices. Given a spread set $S$, one can create a spread as the set of $n$-dimensional projective spaces defined by $J(k)=\{(x,xM):x\in E\}$, for each $M\in S$, together with $J(\infty )=\{(0,y):y\in E\}$,[2] As a specific example, the following nine matrices represent $GF(9)$ as 2 × 2 matrices over $GF(3)$ and so provide a spread set of $AG(2,9)$.[6]
$\left[{\begin{matrix}0&0\\0&0\end{matrix}}\right],\left[{\begin{matrix}1&0\\0&1\end{matrix}}\right],\left[{\begin{matrix}2&0\\0&2\end{matrix}}\right],\left[{\begin{matrix}0&1\\2&0\end{matrix}}\right],\left[{\begin{matrix}1&1\\2&1\end{matrix}}\right],\left[{\begin{matrix}2&1\\2&2\end{matrix}}\right],\left[{\begin{matrix}0&2\\1&0\end{matrix}}\right],\left[{\begin{matrix}1&2\\1&1\end{matrix}}\right],\left[{\begin{matrix}2&2\\2&1\end{matrix}}\right]$
Another example of a spread set yields the Hall plane of order 9[6]
$\left[{\begin{matrix}0&0\\0&0\end{matrix}}\right],\left[{\begin{matrix}1&0\\0&1\end{matrix}}\right],\left[{\begin{matrix}2&0\\0&2\end{matrix}}\right],\left[{\begin{matrix}1&1\\1&2\end{matrix}}\right],\left[{\begin{matrix}2&2\\2&1\end{matrix}}\right],\left[{\begin{matrix}0&1\\2&0\end{matrix}}\right],\left[{\begin{matrix}0&2\\1&0\end{matrix}}\right],\left[{\begin{matrix}1&2\\2&2\end{matrix}}\right],\left[{\begin{matrix}2&1\\1&1\end{matrix}}\right]$
Modifying spreads
One common approach to creating new spreads is to start with a regular spread and modify it in some way. The techniques presented here are some of the more elementary examples of this approach.
Spreads of 3-space
One can create new spreads by starting with a spread and looking for a switching set, a subset of its elements that can be replaced with an alternate set of pairwise disjoint subspaces of the correct dimension. In $PG(3,K)$, a regulus forms a switching set, as the set of transversals of a regulus $R$ also form a regulus, called the opposite regulus of $R$. Removing the lines of a regulus in a spread and replacing them with the opposite regulus produces a new spread which is often non-isomorphic to the original. This process is a special case of a more general process called derivation or net replacement.[7]
Starting with a regular spread of $PG(3,q)$ and reversing any regulus produces a spread that yields a Hall plane. In more generality, the process can be applied independently to any collection of reguli in a regular spread, yielding a subregular spread[8]; the resulting translation plane is called a subregular plane. The André planes form a special subclass of subregular planes, of which the Hall planes are the simplest examples, arising by replacing a single regulus in a regular spread.
More complex switching sets have been constructed. Bruen[9] has explored the concept of a chain of reguli in a regular spread of $PG(3,q)$, $q$ odd, namely a set of $(q+3)/2$ reguli which pairwise meet in exactly 2 lines, so that every line contained in a regulus of the chain is contained in exactly two distinct reguli of the chain. Bruen constructed an example of a chain in the regular spread of $PG(3,5)$, and showed that it could be replaced by taking the union of exactly half of the lines from the opposite regulus of each regulus in the chain. Numerous examples of Bruen chains have appeared in the literature since, and Heden[10] has shown that any Bruen chain is replaceable using opposite half-reguli. Chains are known to exist in a regular spread of $PG(3,q)$ for all odd prime powers $q$ up to 37, except 29, and are known not to exist for $q\in \{29,41,43,47,49\}$.[11] It is conjectured that no additional Bruen chains exist.
Baker and Ebert[12] generalized the concept of a chain to a nest, which is a set of reguli in a regular spread such that every line contained in a regulus of the nest is contained in exactly two distinct reguli of the nest. Unlike a chain, two reguli in a nest are not required to meet in a pair of lines. Unlike chains, a nest in a regular spread need not be replaceable,[13] however several infinite families of replaceable nests are known.[14][15]
Higher-dimensional spreads
In higher dimensions a regulus cannot be reversed because the transversals do not have the correct dimension. There exist analogs to reguli, called norm surfaces, which can be reversed.[16] The higher-dimensional André planes can be obtained from spreads obtained by reversing these norm surfaces, and there also exist analogs of subregular spreads which do not give rise to André planes.[17][18]
Geometric techniques
There are several known ways to construct spreads of $PG(3,q)$ from other geometrical objects without reference to an initial regular spread. Some well-studied approaches to this are given below.
Flocks of quadratic cones
In $PG(3,q)$, a quadratic cone is the union of the set of lines containing a fixed point P (the vertex) and a point on a conic in a plane not passing through P. Since a conic has $q+1$ points, a quadratic cone has $q(q+1)+1$ points. As with traditional geometric conic sections, a plane of $PG(3,q)$ can meet a quadratic cone in either a point, a conic, a line or a line pair. A flock of a quadratic cone is a set of $q$ planes whose intersections with the quadratic cone are pairwise disjoint conics. The classic construction of a flock is to pick a line $m$ that does not meet the quadratic cone, and take the $q$ planes through $m$ that do not contain the vertex of the cone; such a flock is called linear.
Fisher and Thas[19] show how to construct a spread of $PG(3,q)$ from a flock of a quadratic cone using the Klein correspondence, and show that the resulting spread is regular if and only if the initial flock is linear. Many infinite families of flocks of quadratic cones are known, as are numerous sporadic examples.[20]
Every spread arising from a flock of a quadratic cone is the union of $q$ reguli which all meet in a fixed line $m$. Much like with a regular spread, any of these reguli can be replaced with its opposite to create several potentially new spreads.[21]
Hyperbolic fibrations
In $PG(3,q)$ a hyperbolic fibration is a partition of the space into $q-1$ pairwise disjoint hyperbolic quadrics and two lines disjoint from all of the quadrics and each other. Since a hyperbolic quadric consists of the points covered by a regulus and its opposite, a hyperbolic fibration yields $2^{q-1}$ different spreads.
All spreads yielding André planes, including the regular spread, are obtainable from a hyperbolic fibration (specifically an algebraic pencil generated by any two of the quadrics), as articulated by André.[1] Using nest replacement, Ebert[22] found a family of spreads in which a hyperbolic fibration was identified. Baker, et al.[23] provide an explicit example of a construction of a hyperbolic fibration. A much more robust source of hyperbolic fibrations was identified by Baker, et al.,[24] where the authors developed a correspondence between flocks of quadratic cones and hyperbolic fibrations; interestingly, the spreads generated by a flock of a quadratic cone are not generally isomorphic to the spreads generated from the corresponding hyperbolic fibration.
Subgeometry partitions
Hirschfeld and Thas[25] note that for any odd integer $n\geq 3$, a partition of $PG(n-1,q^{2})$ into subgeometries isomorphic to $PG(n-1,q)$ gives rise to a spread of $PG(2n-1,q)$, where each subgeometry of the partition corresponds to a regulus of the new spread.
The "classical" subgeometry partitions of $PG(n-1,q^{2})$ can be generated using suborbits of a Singer cycle, but this simply generates a regular spread.[26] Yff[27] published the non-classical subgeometry partition, namely a partition of $PG(2,9)$ into 7 copies of $PG(2,3)$, that admit a cyclic group permuting the subplanes. Baker, et al.[28] provide several infinite families of partitions of $PG(2,q^{2})$ into subplanes, with the same cyclic group action.
Partial spreads
A partial spread of a projective space $PG(d,K)$ is a set of pairwise disjoint $r$-dimensional subspaces in the space; hence a spread is just a partial spread where every point of the space is covered. A partial spread is called complete or maximal if there is no larger partial spread that contains it; equivalently, there is no $r$-dimensional subspace disjoint from all members of the partial spread. As with spreads, the most well-studied case is partial spreads of lines of the finite projective space $PG(3,q)$, where a full spread has size $q^{2}+1$. Mesner[29] showed that any partial spread of lines in $PG(3,q)$ with size greater than $q^{2}-{\sqrt {q}}$ cannot be complete; indeed, it must be a subset of a unique spread. For a lower bound, Bruen[30] showed that a complete partial spread of lines in $PG(3,q)$ with size at most $q+{\sqrt {q}}$ lines cannot be complete; there will necessarily be a line that can be added to a partial spread of this size. Bruen also provides examples of complete partial spreads of lines in $PG(3,q)$ with sizes $q^{2}-q+1$ and $q^{2}-q+2$ for all $q>2$.
Spreads of classical polar spaces
The classical polar spaces are all embedded in some projective space $PG(d,K)$ as the set of totally isotropic subspaces of a sesquilinear or quadratic form on the vector space underlying the projective space. A particularly interesting class of partial spreads of $PG(d,K)$ are those that consist strictly of maximal subspaces of a classical polar space embedded in the projective space. Such partial spreads that cover all of the points of the polar space are called spreads of the polar space.
From the perspective of the theory of translation planes, the symplectic polar space is of particular interest, as its set of points are all of the points in $PG(2n+1,K)$, and its maximal subspaces are of dimension $n$. Hence a spread of the symplectic polar space is also a spread of the entire projective space, and can be used as noted above to create a translation plane. Several examples of symplectic spreads are known; see Ball, et al.[31]
References
1. André, Johannes (1954), "Über nicht-Desarguessche Ebenen mit transitiver Translationsgruppe", Mathematische Zeitschrift, 60: 156–186, doi:10.1007/BF01187370, ISSN 0025-5874, MR 0063056, S2CID 123661471
2. Bruck, R. H.; Bose, R. C. (1964), "The Construction of Translation Planes from Projective Spaces" (PDF), Journal of Algebra, 1: 85–102, doi:10.1016/0021-8693(64)90010-9
3. This is ultimately a consequence of the fact that a finite field of order $p^{d+1}$ has a subfield of order $p^{r+1}$ if and only if $r+1\mid d+1$.
4. This notion generalizes that of a classical regulus, which is one of the two families of ruling lines on a hyperboloid of one sheet in 3-dimensional space
5. Bruck, R. H.; Bose, R. C. (1966), "Linear Representations of Projective Planes in Projective Spaces" (PDF), Journal of Algebra, 4: 117–172, doi:10.1016/0021-8693(66)90054-8, page 163
6. Moorhouse, Eric (2007), Incidence Geometry (PDF), archived from the original (PDF) on 2013-10-29
7. Johnson, Norman L.; Jha, Vikram; Biliotti, Mauro (2007), Handbook of Finite Translation Planes, Chapman&Hall/CRC, ISBN 978-1-58488-605-1, page 49
8. Bruck, R.H. (1969), R.C.Bose and T.A. Dowling (ed.), "Construction Problems of finite projective planes", Combinatorial Mathematics and Its Applications, Univ. of North Carolina Press, pp. 426–514
9. Bruen, A.A. (1978). "Inversive geometry and some translation planes, I". Geometriae Dedicata. 7: 81–98. doi:10.1007/BF00181353. S2CID 122632353.
10. Heden, O. (1995). "On Bruen chains" (PDF). Discrete Mathematics. 146 (1–3): 69–96. doi:10.1016/0012-365X(94)00058-0.
11. Johnson, Norman L.; Jha, Vikram; Biliotti, Mauro (2007). "Nests". Handbook of Finite Translation Planes. Boca Raton, FL: Chapman & Hall/CRC. ISBN 978-1-58488-605-1.
12. Baker, R. D.; Ebert, G. L. (1988). "Nests of size $q-1$ and another family of translation planes". Journal of the London Mathematical Society. 38 (2): 341–355. doi:10.1112/jlms/s2-38.2.341.
13. Ebert, G. L. (1988). "Some nonreplaceable nests". Combinatorics '88. Research Lecture Notes in Mathematics, Mediterranean, Rende. 1: 353–372.
14. Baker, R. D.; Ebert, G. L. (1988). "A new class of translation planes". Annals of Discrete Mathematics. Amsterdam: North-Holland. 37: 7–20. doi:10.1016/S0167-5060(08)70220-6. ISBN 9780444703699.
15. Baker, R. D.; Ebert, G. L. (1996). "Filling the nest gaps". Finite Fields and Applications. 2 (1): 45–61.
16. Bruck, R.H. (1973). "Circle geometry in higher dimensions. II". Geometriae Dedicata. 2 (2). doi:10.1007/BF00147854. ISSN 0046-5755. S2CID 189889878.
17. Dover, Jeremy (1998). "Subregular Spreads of PG(2n+1,q)". Finite Fields and Their Applications. 4 (4): 362–380. doi:10.1006/ffta.1998.0222.
18. Culbert, Craig; Ebert, Gary (2005). "Circle geometry and three-dimensional subregular translation planes". Innovations in Incidence Geometry: Algebraic, Topological and Combinatorial. 1 (1): 3–18. doi:10.2140/iig.2005.1.3. ISSN 1781-6475.
19. Fisher, J. Chris; Thas, Joseph A. (1979). "Flocks in PG(3,q)". Mathematische Zeitschrift. 169 (1): 1–11. doi:10.1007/BF01214908. ISSN 0025-5874. S2CID 121796426.
20. Johnson, Norman L.; Jha, Vikram; Biliotti, Mauro (2007). "Infinite Classes of Flocks". Handbook of Finite Translation Planes. Boca Raton, FL: Chapman & Hall/CRC. ISBN 978-1-58488-605-1.
21. A specific example of this phenomenon is illustrated in Dover, Jeremy M. (2019-02-27). "A genealogy of the translation planes of order 25". arXiv:1902.07838 [math.CO]. to relate two translation planes of order 25 found by computer search in Czerwinski, Terry; Oakden, David (1992). "The translation planes of order twenty-five". Journal of Combinatorial Theory, Series A. 59 (2): 193–217. doi:10.1016/0097-3165(92)90065-3.
22. Ebert, G. L. (1989). "Spreads Admitting Regular Elliptic Covers". European Journal of Combinatorics. 10 (4): 319–330. doi:10.1016/S0195-6698(89)80004-6.
23. Baker, R. D.; Dover, J. M.; Ebert, G. L.; Wantz, K. L. (1999). "Hyperbolic Fibrations of PG(3,q)". European Journal of Combinatorics. 20 (1): 1–16. doi:10.1006/eujc.1998.0249.
24. Baker, R. D.; Ebert, G. L.; Penttila, Tim (2005). "Hyperbolic Fibrations and q-Clans". Designs, Codes and Cryptography. 34 (2–3): 295–305. doi:10.1007/s10623-004-4861-8. ISSN 0925-1022. S2CID 21853272.
25. Hirschfeld, J. W. P.; Thas, J. A. (1991). General Galois geometries. London: Oxford University Press. p. 206. ISBN 978-1-4471-6790-7. OCLC 936691484.
26. Bruck, R. H. (1960). "Quadratic extensions of cyclic planes". In Bellman, Richard; Hall, Marshall (eds.). Combinatorial Analysis. Proceedings of Symposia in Applied Mathematics. Vol. 10. Providence, Rhode Island: American Mathematical Society. pp. 15–44. doi:10.1090/psapm/010. ISBN 978-0-8218-1310-2.
27. Yff, Peter (1977). "On subplane partitions of a finite projective plane". Journal of Combinatorial Theory, Series A. 22 (1): 118–122. doi:10.1016/0097-3165(77)90072-3.
28. Baker, Ronald D.; Dover, Jeremy M.; Ebert, Gary L.; Wantz, Kenneth L. (2000). "Baer subgeometry partitions". Journal of Geometry. 67 (1–2): 23–34. doi:10.1007/BF01220294. ISSN 0047-2468. S2CID 121116940.
29. Mesner, Dale M. (1967). "Sets of Disjoint Lines in PG(3, q)". Canadian Journal of Mathematics. 19: 273–280. doi:10.4153/CJM-1967-019-5. ISSN 0008-414X. S2CID 123550829.
30. Bruen, A. (1971). "Partial Spreads and Replaceable Nets". Canadian Journal of Mathematics. 23 (3): 381–391. doi:10.4153/CJM-1971-039-x. ISSN 0008-414X. S2CID 124356288.
31. Ball, Simeon; Bamberg, John; Lavrauw, Michel; Penttila, Tim (May 2004). "Symplectic Spreads". Designs, Codes and Cryptography. 32 (1–3): 9–14. doi:10.1023/B:DESI.0000029209.24742.89. ISSN 0925-1022. S2CID 8228870.
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Wikipedia
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Spread of a matrix
In mathematics, and more specifically matrix theory, the spread of a matrix is the largest distance in the complex plane between any two eigenvalues of the matrix.
Definition
Let $A$ be a square matrix with eigenvalues $\lambda _{1},\ldots ,\lambda _{n}$. That is, these values $\lambda _{i}$ are the complex numbers such that there exists a vector $v_{i}$ on which $A$ acts by scalar multiplication:
$Av_{i}=\lambda _{i}v_{i}.$
Then the spread of $A$ is the non-negative number
$s(A)=\max\{|\lambda _{i}-\lambda _{j}|:i,j=1,\ldots n\}.$
Examples
• For the zero matrix and the identity matrix, the spread is zero. The zero matrix has only zero as its eigenvalues, and the identity matrix has only one as its eigenvalues. In both cases, all eigenvalues are equal, so no two eigenvalues can be at nonzero distance from each other.
• For a projection, the only eigenvalues are zero and one. A projection matrix therefore has a spread that is either $0$ (if all eigenvalues are equal) or $1$ (if there are two different eigenvalues).
• All eigenvalues of a unitary matrix $A$ lie on the unit circle. Therefore, in this case, the spread is at most equal to the diameter of the circle, the number 2.
• The spread of a matrix depends only on the spectrum of the matrix (its multiset of eigenvalues). If a second matrix $B$ of the same size is invertible, then $BAB^{-1}$ has the same spectrum as $A$. Therefore, it also has the same spread as $A$.
See also
• Field of values
References
• Marvin Marcus and Henryk Minc, A survey of matrix theory and matrix inequalities, Dover Publications, 1992, ISBN 0-486-67102-X. Chap.III.4.
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Wikipedia
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Elastic pendulum
In physics and mathematics, in the area of dynamical systems, an elastic pendulum[1][2] (also called spring pendulum[3][4] or swinging spring) is a physical system where a piece of mass is connected to a spring so that the resulting motion contains elements of both a simple pendulum and a one-dimensional spring-mass system.[2] The system exhibits chaotic behaviour and is sensitive to initial conditions.[2] The motion of an elastic pendulum is governed by a set of coupled ordinary differential equations.
Analysis and interpretation
The system is much more complex than a simple pendulum, as the properties of the spring add an extra dimension of freedom to the system. For example, when the spring compresses, the shorter radius causes the spring to move faster due to the conservation of angular momentum. It is also possible that the spring has a range that is overtaken by the motion of the pendulum, making it practically neutral to the motion of the pendulum.
Lagrangian
The spring has the rest length $l_{0}$ and can be stretched by a length $x$. The angle of oscillation of the pendulum is $\theta $.
The Lagrangian $L$ is:
$L=T-V$
where $T$ is the kinetic energy and $V$ is the potential energy.
Hooke's law is the potential energy of the spring itself:
$V_{k}={\frac {1}{2}}kx^{2}$
where $k$ is the spring constant.
The potential energy from gravity, on the other hand, is determined by the height of the mass. For a given angle and displacement, the potential energy is:
$V_{g}=-gm(l_{0}+x)\cos \theta $
where $g$ is the gravitational acceleration.
The kinetic energy is given by:
$T={\frac {1}{2}}mv^{2}$
where $v$ is the velocity of the mass. To relate $v$ to the other variables, the velocity is written as a combination of a movement along and perpendicular to the spring:
$T={\frac {1}{2}}m({\dot {x}}^{2}+(l_{0}+x)^{2}{\dot {\theta }}^{2})$
So the Lagrangian becomes:[1]
$L=T-V_{k}-V_{g}$
$L[x,{\dot {x}},\theta ,{\dot {\theta }}]={\frac {1}{2}}m({\dot {x}}^{2}+(l_{0}+x)^{2}{\dot {\theta }}^{2})-{\frac {1}{2}}kx^{2}+gm(l_{0}+x)\cos \theta $
Equations of motion
With two degrees of freedom, for $x$ and $\theta $, the equations of motion can be found using two Euler-Lagrange equations:
${\partial L \over \partial x}-{\operatorname {d} \over \operatorname {d} t}{\partial L \over \partial {\dot {x}}}=0$
${\partial L \over \partial \theta }-{\operatorname {d} \over \operatorname {d} t}{\partial L \over \partial {\dot {\theta }}}=0$
For $x$:[1]
$m(l_{0}+x){\dot {\theta }}^{2}-kx+gm\cos \theta -m{\ddot {x}}=0$
${\ddot {x}}$ isolated:
${\ddot {x}}=(l_{0}+x){\dot {\theta }}^{2}-{\frac {k}{m}}x+g\cos \theta $
And for $\theta $:[1]
$-gm(l_{0}+x)\sin \theta -m(l_{0}+x)^{2}{\ddot {\theta }}-2m(l_{0}+x){\dot {x}}{\dot {\theta }}=0$
${\ddot {\theta }}$ isolated:
${\ddot {\theta }}=-{\frac {g}{l_{0}+x}}\sin \theta -{\frac {2{\dot {x}}}{l_{0}+x}}{\dot {\theta }}$
The elastic pendulum is now described with two coupled ordinary differential equations. These can be solved numerically. Furthermore, one can use analytical methods to study the intriguing phenomenon of order-chaos-order[6] in this system.
See also
• Double pendulum
• Duffing oscillator
• Pendulum (mathematics)
• Spring-mass system
References
1. Xiao, Qisong; et al. "Dynamics of the Elastic Pendulum" (PDF).
2. Pokorny, Pavel (2008). "Stability Condition for Vertical Oscillation of 3-dim Heavy Spring Elastic Pendulum" (PDF). Regular and Chaotic Dynamics. 13 (3): 155–165. Bibcode:2008RCD....13..155P. doi:10.1134/S1560354708030027. S2CID 56090968.
3. Sivasrinivas, Kolukula. "Spring Pendulum".
4. Hill, Christian (19 July 2017). "The spring pendulum".
5. Simionescu, P.A. (2014). Computer Aided Graphing and Simulation Tools for AutoCAD Users (1st ed.). Boca Raton, Florida: CRC Press. ISBN 978-1-4822-5290-3.
6. Anurag, Anurag; Basudeb, Mondal; Bhattacharjee, Jayanta Kumar; Chakraborty, Sagar (2020). "Understanding the order-chaos-order transition in the planar elastic pendulum". Physica D. 402: 132256. Bibcode:2020PhyD..40232256A. doi:10.1016/j.physd.2019.132256. S2CID 209905775.
Further reading
• Pokorny, Pavel (2008). "Stability Condition for Vertical Oscillation of 3-dim Heavy Spring Elastic Pendulum" (PDF). Regular and Chaotic Dynamics. 13 (3): 155–165. Bibcode:2008RCD....13..155P. doi:10.1134/S1560354708030027. S2CID 56090968.
• Pokorny, Pavel (2009). "Continuation of Periodic Solutions of Dissipative and Conservative Systems: Application to Elastic Pendulum" (PDF). Mathematical Problems in Engineering. 2009: 1–15. doi:10.1155/2009/104547.
External links
• Holovatsky V., Holovatska Y. (2019) "Oscillations of an elastic pendulum" (interactive animation), Wolfram Demonstrations Project, published February 19, 2019.
Chaos theory
Concepts
Core
• Attractor
• Bifurcation
• Fractal
• Limit set
• Lyapunov exponent
• Orbit
• Periodic point
• Phase space
• Anosov diffeomorphism
• Arnold tongue
• axiom A dynamical system
• Bifurcation diagram
• Box-counting dimension
• Correlation dimension
• Conservative system
• Ergodicity
• False nearest neighbors
• Hausdorff dimension
• Invariant measure
• Lyapunov stability
• Measure-preserving dynamical system
• Mixing
• Poincaré section
• Recurrence plot
• SRB measure
• Stable manifold
• Topological conjugacy
Theorems
• Ergodic theorem
• Liouville's theorem
• Krylov–Bogolyubov theorem
• Poincaré–Bendixson theorem
• Poincaré recurrence theorem
• Stable manifold theorem
• Takens's theorem
Theoretical
branches
• Bifurcation theory
• Control of chaos
• Dynamical system
• Ergodic theory
• Quantum chaos
• Stability theory
• Synchronization of chaos
Chaotic
maps (list)
Discrete
• Arnold's cat map
• Baker's map
• Complex quadratic map
• Coupled map lattice
• Duffing map
• Dyadic transformation
• Dynamical billiards
• outer
• Exponential map
• Gauss map
• Gingerbreadman map
• Hénon map
• Horseshoe map
• Ikeda map
• Interval exchange map
• Irrational rotation
• Kaplan–Yorke map
• Langton's ant
• Logistic map
• Standard map
• Tent map
• Tinkerbell map
• Zaslavskii map
Continuous
• Double scroll attractor
• Duffing equation
• Lorenz system
• Lotka–Volterra equations
• Mackey–Glass equations
• Rabinovich–Fabrikant equations
• Rössler attractor
• Three-body problem
• Van der Pol oscillator
Physical
systems
• Chua's circuit
• Convection
• Double pendulum
• Elastic pendulum
• FPUT problem
• Hénon–Heiles system
• Kicked rotator
• Multiscroll attractor
• Population dynamics
• Swinging Atwood's machine
• Tilt-A-Whirl
• Weather
Chaos
theorists
• Michael Berry
• Rufus Bowen
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• Chen Guanrong
• Leon O. Chua
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• Peter Grassberger
• Celso Grebogi
• Martin Gutzwiller
• Brosl Hasslacher
• Michel Hénon
• Svetlana Jitomirskaya
• Bryna Kra
• Edward Norton Lorenz
• Aleksandr Lyapunov
• Benoît Mandelbrot
• Hee Oh
• Edward Ott
• Henri Poincaré
• Mary Rees
• Otto Rössler
• David Ruelle
• Caroline Series
• Yakov Sinai
• Oleksandr Mykolayovych Sharkovsky
• Nina Snaith
• Floris Takens
• Audrey Terras
• Mary Tsingou
• Marcelo Viana
• Amie Wilkinson
• James A. Yorke
• Lai-Sang Young
Related
articles
• Butterfly effect
• Complexity
• Edge of chaos
• Predictability
• Santa Fe Institute
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Wikipedia
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Graduate Texts in Mathematics
Graduate Texts in Mathematics (GTM) (ISSN 0072-5285) is a series of graduate-level textbooks in mathematics published by Springer-Verlag. The books in this series, like the other Springer-Verlag mathematics series, are yellow books of a standard size (with variable numbers of pages). The GTM series is easily identified by a white band at the top of the book.
The books in this series tend to be written at a more advanced level than the similar Undergraduate Texts in Mathematics series, although there is a fair amount of overlap between the two series in terms of material covered and difficulty level.
List of books
1. Introduction to Axiomatic Set Theory, Gaisi Takeuti, Wilson M. Zaring (1982, 2nd ed., ISBN 978-1-4613-8170-9)
2. Measure and Category – A Survey of the Analogies between Topological and Measure Spaces, John C. Oxtoby (1980, 2nd ed., ISBN 978-0-387-90508-2)
3. Topological Vector Spaces, H. H. Schaefer, M. P. Wolff (1999, 2nd ed., ISBN 978-0-387-98726-2)
4. A Course in Homological Algebra, Peter Hilton, Urs Stammbach (1997, 2nd ed., ISBN 978-0-387-94823-2)
5. Categories for the Working Mathematician, Saunders Mac Lane (1998, 2nd ed., ISBN 978-0-387-98403-2)
6. Projective Planes, Daniel R. Hughes, Fred C. Piper, (1982, ISBN 978-3-540-90043-6)
7. A Course in Arithmetic, Jean-Pierre Serre (1996, ISBN 978-0-387-90040-7)
8. Axiomatic Set Theory, Gaisi Takeuti, Wilson M. Zaring, (1973, ISBN 978-3-540-90050-4)
9. Introduction to Lie Algebras and Representation Theory, James E. Humphreys (1997, ISBN 978-0-387-90053-7)
10. A Course in Simple-Homotopy Theory, Marshall. M. Cohen, (1973, ISBN 0-387-90056-X)
11. Functions of One Complex Variable I, John B. Conway (1978, 2nd ed., ISBN 978-0-387-90328-6)
12. Advanced Mathematical Analysis, Richard Beals (1973, ISBN 978-0-387-90065-0)
13. Rings and Categories of Modules, Frank W. Anderson, Kent R. Fuller (1992, 2nd ed., ISBN 978-0-387-97845-1)
14. Stable Mappings and Their Singularities, Martin Golubitsky, Victor Guillemin, (1974, ISBN 978-0-387-90072-8)
15. Lectures in Functional Analysis and Operator Theory, Sterling K. Berberian, (1974, ISBN 978-0-387-90080-3)
16. The Structure of Fields, David J. Winter, (1974, ISBN 978-3-540-90074-0)
17. Random Processes, Murray Rosenblatt, (1974, ISBN 978-0-387-90085-8)
18. Measure Theory, Paul R. Halmos (1974, ISBN 978-0-387-90088-9)
19. A Hilbert Space Problem Book, Paul R. Halmos (1982, 2nd ed., ISBN 978-0-387-90685-0)
20. Fibre Bundles, Dale Husemoller (1994, 3rd ed., ISBN 978-0-387-94087-8)
21. Linear Algebraic Groups, James E. Humphreys (1975, ISBN 978-0-387-90108-4)
22. An Algebraic Introduction to Mathematical Logic, Donald W. Barnes, John M. Mack (1975, ISBN 978-0-387-90109-1)
23. Linear Algebra, Werner H. Greub (1975, ISBN 978-0-387-90110-7)
24. Geometric Functional Analysis and Its Applications, Richard B. Holmes, (1975, ISBN 978-0-387-90136-7)
25. Real and Abstract Analysis, Edwin Hewitt, Karl Stromberg (1975, ISBN 978-0-387-90138-1)
26. Algebraic Theories, Ernest G. Manes, (1976, ISBN 978-3-540-90140-2)
27. General Topology, John L. Kelley (1975, ISBN 978-0-387-90125-1)
28. Commutative Algebra I, Oscar Zariski, Pierre Samuel (1975, ISBN 978-0-387-90089-6)
29. Commutative Algebra II, Oscar Zariski, Pierre Samuel (1975, ISBN 978-0-387-90171-8)
30. Lectures in Abstract Algebra I: Basic Concepts, Nathan Jacobson (1976, ISBN 978-0-387-90181-7)
31. Lectures in Abstract Algebra II: Linear Algebra, Nathan Jacobson (1984, ISBN 978-0-387-90123-7)
32. Lectures in Abstract Algebra III: Theory of Fields and Galois Theory, Nathan Jacobson (1976, ISBN 978-0-387-90168-8)
33. Differential Topology, Morris W. Hirsch (1976, ISBN 978-0-387-90148-0)
34. Principles of Random Walk, Frank Spitzer (1964, 2nd ed., ISBN 978-1-4757-4229-9)
35. Several Complex Variables and Banach Algebras, Herbert Alexander, John Wermer (1998, 3rd ed., ISBN 978-0-387-98253-3)
36. Linear Topological Spaces, John L. Kelley, Isaac Namioka (1982, ISBN 978-0-387-90169-5)
37. Mathematical Logic, J. Donald Monk (1976, ISBN 978-0-387-90170-1)
38. Several Complex Variables, H. Grauert, K. Fritzsche (1976, ISBN 978-0-387-90172-5)
39. An Invitation to $C^{*}$-Algebras, William Arveson (1976, ISBN 978-0-387-90176-3)
40. Denumerable Markov Chains, John G. Kemeny, J. Laurie Snell, Anthony W. Knapp, D.S. Griffeath (1976, ISBN 978-0-387-90177-0)
41. Modular Functions and Dirichlet Series in Number Theory, Tom M. Apostol (1989, 2nd ed., ISBN 978-0-387-97127-8)
42. Linear Representations of Finite Groups, Jean-Pierre Serre, Leonhard L. Scott (1977, ISBN 978-0-387-90190-9)
43. Rings of Continuous Functions, Leonard Gillman, Meyer Jerison (1976, ISBN 978-0-387-90198-5)
44. Elementary Algebraic Geometry, Keith Kendig (1977, ISBN 978-0-387-90199-2)[1]
45. Probability Theory I, M. Loève (1977, 4th ed, ISBN 978-0-387-90210-4)
46. Probability Theory II, M. Loève (1978, 4th ed, ISBN 978-0-387-90262-3)
47. Geometric Topology in Dimensions 2 and 3, Edwin E. Moise (1977, ISBN 978-0-387-90220-3)
48. General Relativity for Mathematicians, R. K. Sachs, H. Wu (1983, ISBN 978-0-387-90218-0)
49. Linear Geometry, K. W. Gruenberg, A. J. Weir (1977, 2nd ed., ISBN 978-0-387-90227-2)
50. Fermat's Last Theorem: A Genetic Introduction to Algebraic Number Theory, Harold M. Edwards (2000, ISBN 978-0-387-90230-2)
51. A Course in Differential Geometry, William Klingenberg, D. Hoffman (1983, ISBN 978-0-387-90255-5)
52. Algebraic Geometry, Robin Hartshorne (2010, ISBN 978-1-4419-2807-8)
53. A Course in Mathematical Logic for Mathematicians, Yu. I. Manin, Boris Zilber (2009, 2nd ed., ISBN 978-1-4419-0614-4)
54. Combinatorics with Emphasis on the Theory of Graphs, Mark E. Watkins, Jack E. Graver (1977, ISBN 978-0-387-90245-6)
55. Introduction to Operator Theory I: Elements of Functional Analysis, Arlen Brown, Carl Pearcy (1977, ISBN 978-0-387-90257-9)
56. Algebraic Topology: An Introduction, William S. Massey (1977, ISBN 978-0-387-90271-5)
57. Introduction to Knot Theory, Richard H. Crowell, Ralph H. Fox (1977, ISBN 978-0-387-90272-2)
58. p-adic Numbers, p-adic Analysis, and Zeta-Functions, Neal Koblitz (1984, 2nd ed., ISBN 978-0-387-96017-3)
59. Cyclotomic Fields, Serge Lang (1978, ISBN 978-0-387-90307-1)[2]
60. Mathematical Methods of Classical Mechanics, V. I. Arnold, A. Weinstein, K. Vogtmann (1989, 2nd ed., ISBN 978-0-387-96890-2)
61. Elements of Homotopy Theory, George W. Whitehead (1978, ISBN 978-0-387-90336-1)
62. Fundamentals of the Theory of Groups, M. I. Kargapolov, J. I. Merzljakov (1979, ISBN 978-1-4612-9966-0)
63. Graph Theory – An Introductory Course, Béla Bollobás (1979, ISBN 978-1-4612-9969-1)
64. Fourier Series – A Modern Introduction Volume 1, R. E. Edwards (1979, 2nd ed., ISBN 978-1-4612-6210-7)
65. Differential Analysis on Complex Manifolds, Raymond O. Wells, Jr. (2008, 3rd ed., ISBN 978-0-387-73891-8)
66. Introduction to Affine Group Schemes, W. C. Waterhouse (1979, ISBN 978-1-4612-6219-0)
67. Local Fields, Jean-Pierre Serre (1979, ISBN 978-0-387-90424-5)
68. Linear Operators in Hilbert Spaces, Joachim Weidmann (1980, ISBN 978-1-4612-6029-5)
69. Cyclotomic Fields II, Serge Lang (1980, ISBN 978-1-4684-0088-5)
70. Singular Homology Theory, William S. Massey (1980, ISBN 978-1-4684-9233-0)
71. Riemann Surfaces, Herschel Farkas, Irwin Kra (1992, 2nd ed., ISBN 978-0-387-97703-4)
72. Classical Topology and Combinatorial Group Theory, John Stillwell (1980, 2ed 1993, ISBN 978-0-3879-7970-0)
73. Algebra, Thomas W. Hungerford (1974, ISBN 978-0-387-90518-1)
74. Multiplicative Number Theory, Harold Davenport, Hugh Montgomery (2000, 3rd ed., ISBN 978-0-387-95097-6)
75. Basic Theory of Algebraic Groups and Lie Algebras, G. P. Hochschild (1981, ISBN 978-1-4613-8116-7)
76. Algebraic Geometry – An Introduction to Birational Geometry of Algebraic Varieties, Shigeru Iitaka (1982, ISBN 978-1-4613-8121-1)
77. Lectures on the Theory of Algebraic Numbers, E. T. Hecke (1981, ISBN 978-0-387-90595-2)
78. A Course in Universal Algebra, Burris, Stanley and Sankappanavar, H. P. (Online) (1981 ISBN 978-0-3879-0578-5)
79. An Introduction to Ergodic Theory, Peter Walters (1982, ISBN 978-0-387-95152-2)
80. A Course in the Theory of Groups, Derek J.S. Robinson (1996, 2nd ed., ISBN 978-0-387-94461-6)
81. Lectures on Riemann Surfaces, Otto Forster (1981, ISBN 978-0-387-90617-1)
82. Differential Forms in Algebraic Topology, Raoul Bott, Loring W. Tu (1982, ISBN 978-0-387-90613-3)
83. Introduction to Cyclotomic Fields, Lawrence C. Washington (1997, 2nd ed., ISBN 978-0-387-94762-4)
84. A Classical Introduction to Modern Number Theory, Kenneth Ireland, Michael Rosen (1990, 2nd ed., ISBN 978-0-387-97329-6)
85. Fourier Series – A Modern Introduction Volume 2, R. E. Edwards (1982, 2nd ed., ISBN 978-1-4613-8158-7)
86. Introduction to Coding Theory, J. H. van Lint (3rd ed 1998, ISBN 3-540-64133-5)
87. Cohomology of Groups, Kenneth S. Brown (1982, ISBN 978-1-4684-9329-0)
88. Associative Algebras, R. S. Pierce (1982, ISBN 978-1-4757-0165-4)
89. Introduction to Algebraic and Abelian Functions, Serge Lang (1982, 2nd ed., ISBN 978-0-387-90710-9)
90. An Introduction to Convex Polytopes, Arne Brondsted (1983, ISBN 978-1-4612-1148-8)
91. The Geometry of Discrete Groups, Alan F. Beardon (1983, 2nd print 1995, ISBN 978-1-4612-7022-5)
92. Sequences and Series in Banach Spaces, J. Diestel (1984, ISBN 978-1-4612-9734-5)
93. Modern Geometry — Methods and Applications Part I: The Geometry of Surfaces, Transformation Groups, and Fields, B. A. Dubrovin, Anatoly Timofeevich Fomenko, Sergei Novikov (1992, 2nd ed., ISBN 978-0-387-97663-1)
94. Foundations of Differentiable Manifolds and Lie Groups, Frank W. Warner (1983, ISBN 978-0-387-90894-6)
95. Probability-1, Probability-2, Albert N. Shiryaev (2016, 2019, 3rd ed., ISBN 978-0-387-72205-4, ISBN 978-0-387-72207-8)
96. A Course in Functional Analysis, John B. Conway (2007, 2nd ed., ISBN 978-0-387-97245-9)
97. Introduction to Elliptic Curves and Modular Forms, Neal I. Koblitz (1993, 2nd ed., ISBN 978-0-387-97966-3)
98. Representations of Compact Lie Groups, Theodor Bröcker, Tammo tom Dieck (1985, ISBN 978-3-540-13678-1)
99. Finite Reflection Groups, L.C. Grove, C.T. Benson (1985, 2nd ed., ISBN 978-0-387-96082-1)
100. Harmonic Analysis on Semigroups – Theory of Positive Definite and Related Functions, Christian Berg, Jens Peter Reus Christensen, Paul Ressel (1984, ISBN 978-0-387-90925-7)
101. Galois Theory, Harold M. Edwards (1984, ISBN 978-0-387-90980-6)
102. Lie Groups, Lie Algebras, and Their Representations, V. S. Varadarajan (1984, ISBN 978-0-387-90969-1)
103. Complex Analysis, Serge Lang (1999, 4th ed., ISBN 978-0-387-98592-3)
104. Modern Geometry — Methods and Applications Part II: The Geometry and Topology of Manifolds, B. A. Dubrovin, Anatoly Timofeevich Fomenko, Sergei Novikov (1985, ISBN 978-0-387-96162-0)
105. SL2(R), Serge Lang (1985, ISBN 978-0-387-96198-9)
106. The Arithmetic of Elliptic Curves, Joseph H. Silverman (2009, 2nd ed., ISBN 978-0-387-09493-9)
107. Applications of Lie Groups to Differential Equations, Peter J. Olver (2ed 1993, ISBN 978-1-4684-0276-6)
108. Holomorphic Functions and Integral Representations in Several Complex Variables, R. Michael Range (1986, ISBN 978-0-387-96259-7)
109. Univalent Functions and Teichmüller Spaces, O. Lehto (1987, ISBN 978-1-4613-8654-4)
110. Algebraic Number Theory, Serge Lang (1994, 2nd ed., ISBN 978-0-387-94225-4)
111. Elliptic Curves, Dale Husemöller (2004, 2nd ed., ISBN 978-0-387-95490-5)
112. Elliptic Functions, Serge Lang (1987, 2nd ed., ISBN 978-0-387-96508-6)
113. Brownian Motion and Stochastic Calculus, Ioannis Karatzas, Steven Shreve (2ed 2000, ISBN 978-0-387-97655-6)
114. A Course in Number Theory and Cryptography, Neal Koblitz (2ed 1994, ISBN 978-1-4684-0312-1)
115. Differential Geometry: Manifolds, Curves and Surfaces, Marcel Berger, Bernard Gostiaux (1988, ISBN 978-0-387-96626-7)
116. Measure and Integral — Volume 1, John L. Kelley, T.P. Srinivasan (1988, ISBN 978-0-387-96633-5)
117. Algebraic Groups and Class Fields, Jean-Pierre Serre (1988, ISBN 978-1-4612-6993-9)
118. Analysis Now, Gert K. Pedersen (1989, ISBN 978-0-387-96788-2)
119. An Introduction to Algebraic Topology, Joseph J. Rotman, (1988, ISBN 978-0-3879-6678-6)
120. Weakly Differentiable Functions — Sobolev Spaces and Functions of Bounded Variation, William P. Ziemer (1989, ISBN 978-0-387-97017-2)
121. Cyclotomic Fields I and II, Serge Lang (1990, Combined 2nd ed. ISBN 978-1-4612-6972-4)[3]
122. Theory of Complex Functions, Reinhold Remmert (1991, ISBN 978-0-387-97195-7)
123. Numbers, Heinz-Dieter Ebbinghaus et al. (1990, ISBN 978-0-387-97497-2)
124. Modern Geometry — Methods and Applications Part III: Introduction to Homology Theory, B. A. Dubrovin, Anatoly Timofeevich Fomenko, Sergei Novikov (1990, ISBN 978-0-387-97271-8)
125. Complex Variables — An Introduction, Carlos A. Berenstein, Roger Gay (1991, ISBN 978-0-387-97349-4)[4]
126. Linear Algebraic Groups, Armand Borel (1991, ISBN 978-1-4612-6954-0)
127. A Basic Course in Algebraic Topology, William S. Massey (1991, ISBN 978-0-3879-7430-9)
128. Partial Differential Equations, Jeffrey Rauch (1991, ISBN 978-1-4612-6959-5)
129. Representation Theory, William Fulton, Joe Harris (1991, ISBN 978-3-5400-0539-1)
130. Tensor Geometry — The Geometric Viewpoint and its Uses, Christopher T. J. Dodson, Timothy Poston (1991, 2nd ed., ISBN 978-3-540-52018-4)
131. A First Course in Noncommutative Rings, T. Y. Lam (2001, 2nd ed., ISBN 978-0-387-95183-6)
132. Iteration of Rational Functions — Complex Analytic Dynamical Systems, Alan F. Beardon (1991, ISBN 978-0-387-95151-5)
133. Algebraic Geometry, Joe Harris (1992, ISBN 978-0-387-97716-4)
134. Coding and Information Theory, Steven Roman (1992, ISBN 978-0-387-97812-3)
135. Advanced Linear Algebra, Steven Roman (2008, 3rd ed., ISBN 978-0-387-72828-5)
136. Algebra — An Approach via Module Theory, William Adkins, Steven Weintraub (1992, ISBN 978-0-387-97839-0)
137. Harmonic Function Theory, Sheldon Axler, Paul Bourdon, Wade Ramey (2001, 2nd ed., ISBN 978-0-387-95218-5)
138. A Course in Computational Algebraic Number Theory, Henri Cohen (1996, ISBN 0-387-55640-0)
139. Topology and Geometry, Glen E. Bredon (1993, ISBN 978-0-387-97926-7)
140. Optima and Equilibria, Jean-Pierre Aubin (1998, ISBN 978-3-642-08446-1)
141. Gröbner Bases — A Computational Approach to Commutative Algebra, Thomas Becker, Volker Weispfenning (1993, ISBN 978-0-387-97971-7)
142. Real and Functional Analysis, Serge Lang (1993, 3rd ed., ISBN 978-0-387-94001-4)
143. Measure Theory, J. L. Doob (1994, ISBN 978-0-387-94055-7)
144. Noncommutative Algebra, Benson Farb, R. Keith Dennis (1993, ISBN 978-0-387-94057-1)
145. Homology Theory — An Introduction to Algebraic Topology, James W. Vick (1994, 2nd ed., ISBN 978-0-387-94126-4)
146. Computability — A Mathematical Sketchbook, Douglas S. Bridges (1994, ISBN 978-0-387-94174-5)
147. Algebraic K-Theory and Its Applications, Jonathan Rosenberg (1994, ISBN 978-0-387-94248-3)
148. An Introduction to the Theory of Groups, Joseph J. Rotman (1995, 4th ed., ISBN 978-0-387-94285-8)
149. Foundations of Hyperbolic Manifolds, John G. Ratcliffe (2019, 3rd ed., ISBN 978-3-030-31597-9)
150. Commutative Algebra — with a View Toward Algebraic Geometry, David Eisenbud (1995, ISBN 978-0-387-94269-8)
151. Advanced Topics in the Arithmetic of Elliptic Curves, Joseph H. Silverman (1994, ISBN 978-0-387-94328-2)[5]
152. Lectures on Polytopes, Günter M. Ziegler (1995, ISBN 978-0-387-94365-7)
153. Algebraic Topology — A First Course, William Fulton (1995, ISBN 978-0-387-94327-5)
154. An Introduction to Analysis, Arlen Brown, Carl Pearcy (1995, ISBN 978-0-387-94369-5)
155. Quantum Groups, Christian Kassel (1995, ISBN 978-0-387-94370-1)
156. Classical Descriptive Set Theory, Alexander S. Kechris (1995, ISBN 978-0-387-94374-9)
157. Integration and Probability, Paul Malliavin (1995, ISBN 978-0-387-94409-8)[6]
158. Field Theory, Steven Roman (2006, 2nd ed., ISBN 978-0-387-27677-9)
159. Functions of One Complex Variable II, John B. Conway (1995, ISBN 978-0-387-94460-9)
160. Differential and Riemannian Manifolds, Serge Lang (1995, ISBN 978-0-387-94338-1)
161. Polynomials and Polynomial Inequalities, Peter Borwein, Tamas Erdelyi (1995, ISBN 978-0-387-94509-5)
162. Groups and Representations, J. L. Alperin, Rowen B. Bell (1995, ISBN 978-0-387-94526-2)
163. Permutation Groups, John D. Dixon, Brian Mortimer (1996, ISBN 978-0-387-94599-6)
164. Additive Number Theory The Classical Bases, Melvyn B. Nathanson (1996, ISBN 978-0-387-94656-6)
165. Additive Number Theory: Inverse Problems and the Geometry of Sumsets, Melvyn B. Nathanson (1996, ISBN 978-0-387-94655-9)
166. Differential Geometry — Cartan's Generalization of Klein's Erlangen Program, R. W. Sharpe (1997, ISBN 978-0-387-94732-7)
167. Field and Galois Theory, Patrick Morandi (1996, ISBN 978-0-387-94753-2)
168. Combinatorial Convexity and Algebraic Geometry, Guenter Ewald (1996, ISBN 978-1-4612-8476-5)
169. Matrix Analysis, Rajendra Bhatia (1997, ISBN 978-0-387-94846-1)
170. Sheaf Theory, Glen E. Bredon (1997, 2nd ed., ISBN 978-0-387-94905-5)
171. Riemannian Geometry, Peter Petersen (2016, 3rd ed., ISBN 978-3-319-26652-7)
172. Classical Topics in Complex Function Theory, Reinhold Remmert (1998, ISBN 978-0-387-98221-2)
173. Graph Theory, Reinhard Diestel (2017, 5th ed., ISBN 978-3-662-53621-6)
174. Foundations of Real and Abstract Analysis, Douglas S. Bridges (1998, ISBN 978-0-387-98239-7)
175. An Introduction to Knot Theory, W. B. Raymond Lickorish (1997, ISBN 978-1-4612-6869-7)
176. Introduction to Riemannian Manifolds, John M. Lee (2018, 2nd ed., ISBN 978-3-319-91754-2)[7]
177. Analytic Number Theory , Donald J. Newman (1998, ISBN 978-0-387-98308-0)
178. Nonsmooth Analysis and Control Theory, Francis H. Clarke, Yuri S. Ledyaev, Ronald J. Stern, Peter R. Wolenski (1998, ISBN 978-0-387-98336-3)
179. Banach Algebra Techniques in Operator Theory, Ronald G. Douglas (1998, 2nd ed., ISBN 978-0-387-98377-6)
180. A Course on Borel Sets, S. M. Srivastava (1998, ISBN 978-0-387-98412-4)
181. Numerical Analysis, Rainer Kress (1998, ISBN 978-0-387-98408-7)
182. Ordinary Differential Equations, Wolfgang Walter (1998, ISBN 978-0-387-98459-9)
183. An Introduction to Banach Space Theory, Robert E. Megginson (1998, ISBN 978-0-387-98431-5)
184. Modern Graph Theory, Béla Bollobás (1998, ISBN 978-0-387-98488-9)
185. Using Algebraic Geometry, David A. Cox, John Little, Donal O'Shea (2005, 2nd ed., ISBN 978-0-387-20706-3)
186. Fourier Analysis on Number Fields, Dinakar Ramakrishnan, Robert J. Valenza (1999, ISBN 978-0-387-98436-0)
187. Moduli of Curves, Joe Harris, Ian Morrison (1998, ISBN 978-0-387-98438-4)
188. Lectures on the Hyperreals – An Introduction to Nonstandard Analysis, Robert Goldblatt (1998, ISBN 978-0-387-98464-3)
189. Lectures on Modules and Rings, Tsit-Yuen Lam (1999, ISBN 978-0-387-98428-5)
190. Problems in Algebraic Number Theory, M. Ram Murty, Jody Indigo Esmonde (2005, 2nd ed., ISBN 978-0-387-22182-3)
191. Fundamentals of Differential Geometry, Serge Lang (1999, ISBN 978-0-387-98593-0)
192. Elements of Functional Analysis, Francis Hirsch, Gilles Lacombe (1999, ISBN 978-0-387-98524-4)
193. Advanced Topics in Computational Number Theory, Henri Cohen (2000, ISBN 0-387-98727-4)
194. One-Parameter Semigroups for Linear Evolution Equations, Klaus-Jochen Engel, Rainer Nagel (2000, ISBN 978-0-387-98463-6)
195. Elementary Methods in Number Theory, Melvyn B. Nathanson (2000, ISBN 978-0-387-98912-9)
196. Basic Homological Algebra, M. Scott Osborne (2000, ISBN 978-0-387-98934-1)
197. The Geometry of Schemes, Eisenbud, Joe Harris (2000, ISBN 978-0-387-98638-8)
198. A Course in p-adic Analysis, Alain M. Robert (2000, ISBN 978-0-387-98669-2)
199. Theory of Bergman Spaces, Hakan Hedenmalm, Boris Korenblum, Kehe Zhu (2000, ISBN 978-0-387-98791-0)
200. An Introduction to Riemann–Finsler Geometry, David Bao, Shiing-Shen Chern, Zhongmin Shen (2000, ISBN 978-1-4612-7070-6)
201. Diophantine Geometry, Marc Hindry, Joseph H. Silverman (2000, ISBN 978-0-387-98975-4)
202. Introduction to Topological Manifolds, John M. Lee (2011, 2nd ed., ISBN 978-1-4419-7939-1)
203. The Symmetric Group — Representations, Combinatorial Algorithms, and Symmetric Functions, Bruce E. Sagan (2001, 2nd ed., ISBN 978-0-387-95067-9)
204. Galois Theory, Jean-Pierre Escofier (2001, ISBN 978-0-387-98765-1)
205. Rational Homotopy Theory, Yves Félix, Stephen Halperin, Jean-Claude Thomas (2000, ISBN 978-0-387-95068-6)
206. Problems in Analytic Number Theory, M. Ram Murty (2007, 2nd ed., ISBN 978-0-387-95143-0)
207. Algebraic Graph Theory, Chris Godsil, Gordon Royle (2001, ISBN 978-0-387-95241-3)
208. Analysis for Applied Mathematics, Ward Cheney (2001, ISBN 978-0-387-95279-6)
209. A Short Course on Spectral Theory, William Arveson (2002, ISBN 978-0-387-95300-7)
210. Number Theory in Function Fields, Michael Rosen (2002, ISBN 978-0-387-95335-9)
211. Algebra, Serge Lang (2002, Revised 3rd ed, ISBN 978-0-387-95385-4)
212. Lectures on Discrete Geometry, Jiří Matoušek (2002, ISBN 978-0-387-95374-8)
213. From Holomorphic Functions to Complex Manifolds, Klaus Fritzsche, Hans Grauert (2002, ISBN 978-0-387-95395-3)
214. Partial Differential Equations, Jürgen Jost, (2013, 3rd ed., ISBN 978-1-4614-4808-2)
215. Algebraic Functions and Projective Curves, David M. Goldschmidt, (2003, ISBN 978-0-387-95432-5)
216. Matrices — Theory and Applications, Denis Serre, (2010, 2nd ed., ISBN 978-1-4419-7682-6)
217. Model Theory: An Introduction, David Marker, (2002, ISBN 978-0-387-98760-6)
218. Introduction to Smooth Manifolds, John M. Lee (2012, 2nd ed., ISBN 978-1-4419-9981-8)
219. The Arithmetic of Hyperbolic 3-Manifolds, Colin Maclachlan, Alan W. Reid, (2003, ISBN 978-0-387-98386-8)
220. Smooth Manifolds and Observables, Jet Nestruev, (2020, 2nd ed., ISBN 978-0-387-95543-8 )
221. Convex Polytopes, Branko Grünbaum (2003, 2nd ed., ISBN 978-0-387-40409-7)
222. Lie Groups, Lie Algebras, and Representations – An Elementary Introduction, Brian C. Hall, (2015, 2nd ed., ISBN 978-3-319-13466-6)
223. Fourier Analysis and its Applications, Anders Vretblad, (2003, ISBN 978-0-387-00836-3)
224. Metric Structures in Differential Geometry, Walschap, G., (2004, ISBN 978-0-387-20430-7)
225. Lie Groups, Daniel Bump, (2013, 2nd ed., ISBN 978-1-4614-8023-5)
226. Spaces of Holomorphic Functions in the Unit Ball, Kehe Zhu, (2005, ISBN 978-0-387-22036-9)
227. Combinatorial Commutative Algebra, Ezra Miller, Bernd Sturmfels, (2005, ISBN 978-0-387-22356-8)
228. A First Course in Modular Forms, Fred Diamond, J. Shurman, (2006, ISBN 978-0-387-23229-4)
229. The Geometry of Syzygies – A Second Course in Algebraic Geometry and Commutative Algebra, David Eisenbud (2005, ISBN 978-0-387-22215-8)
230. An Introduction to Markov Processes, Daniel W. Stroock, (2014, 2nd ed., ISBN 978-3-540-23499-9)
231. Combinatorics of Coxeter Groups, Anders Björner, Francisco Brenti, (2005, ISBN 978-3-540-44238-7)
232. An Introduction to Number Theory, Graham Everest, Thomas Ward., (2005, ISBN 978-1-85233-917-3)
233. Topics in Banach Space Theory, Albiac, F., Kalton, N. J., (2016, 2nd ed., ISBN 978-3-319-31555-3)
234. Analysis and Probability — Wavelets, Signals, Fractals, Jorgensen, P. E. T., (2006, ISBN 978-0-387-29519-0)
235. Compact Lie Groups, M. R. Sepanski, (2007, ISBN 978-0-387-30263-8)
236. Bounded Analytic Functions, Garnett, J., (2007, ISBN 978-0-387-33621-3)
237. An Introduction to Operators on the Hardy–Hilbert Space, Ruben A. Martinez-Avendano, Peter Rosenthal, (2007, ISBN 978-0-387-35418-7)
238. A Course in Enumeration, Martin Aigner, (2007, ISBN 978-3-540-39032-9)
239. Number Theory — Volume I: Tools and Diophantine Equations, Henri Cohen, (2007, ISBN 978-0-387-49922-2)
240. Number Theory — Volume II: Analytic and Modern Tools, Henri Cohen, (2007, ISBN 978-0-387-49893-5)
241. The Arithmetic of Dynamical Systems, Joseph H. Silverman, (2007, ISBN 978-0-387-69903-5)
242. Abstract Algebra, Grillet, Pierre Antoine, (2007, ISBN 978-0-387-71567-4)
243. Topological Methods in Group Theory, Geoghegan, Ross, (2007, ISBN 978-0-387-74611-1)
244. Graph Theory, Adrian Bondy, U.S.R. Murty, (2008, ISBN 978-1-84628-969-9)
245. Complex Analysis – In the Spirit of Lipman Bers, Rubí E. Rodríguez, Irwin Kra, Jane P. Gilman (2013, 2nd ed., ISBN 978-1-4899-9908-5)
246. A Course in Commutative Banach Algebras, Kaniuth, Eberhard, (2008, ISBN 978-0-387-72475-1)
247. Braid Groups, Kassel, Christian, Turaev, Vladimir, (2008, ISBN 978-0-387-33841-5)
248. Buildings Theory and Applications, Abramenko, Peter, Brown, Ken (2008, ISBN 978-0-387-78834-0)
249. Classical Fourier Analysis, Loukas Grafakos (2014, 3rd ed., ISBN 978-1-4939-1193-6)
250. Modern Fourier Analysis, Loukas Grafakos (2014, 3rd ed., ISBN 978-1-4939-1229-2)
251. The Finite Simple Groups, Robert A. Wilson (2009, ISBN 978-1-84800-987-5)
252. Distributions and Operators, Gerd Grubb, (2009, ISBN 978-0-387-84894-5)
253. Elementary Functional Analysis, MacCluer, Barbara D., (2009, ISBN 978-0-387-85528-8)
254. Algebraic Function Fields and Codes, Henning Stichtenoth, (2009, 2nd ed., ISBN 978-3-540-76877-7)
255. Symmetry, Representations, and Invariants, Goodman, Roe, Wallach, Nolan R., (2009, ISBN 978-0-387-79851-6)
256. A Course in Commutative Algebra, Kemper, Gregor, (2010, ISBN 978-3-642-03544-9)
257. Deformation Theory, Robin Hartshorne, (2010, ISBN 978-1-4419-1595-5)
258. Foundations of Optimization in Finite Dimensions, Osman Guler, (2010, ISBN 978-0-387-34431-7)
259. Ergodic Theory – with a view towards Number Theory, Manfred Einsiedler, Thomas Ward, (2011, ISBN 978-0-85729-020-5)
260. Monomial Ideals, Jürgen Herzog, Hibi Takayuki(2010, ISBN 978-0-85729-105-9)
261. Probability and Stochastics, Erhan Cinlar, (2011, ISBN 978-0-387-87858-4)
262. Essentials of Integration Theory for Analysis, Daniel W. Stroock, (2012, ISBN 978-1-4614-1134-5)
263. Analysis on Fock Spaces, Kehe Zhu, (2012, ISBN 978-1-4419-8800-3)
264. Functional Analysis, Calculus of Variations and Optimal Control, Francis H. Clarke, (2013, ISBN 978-1-4471-4819-7)
265. Unbounded Self-adjoint Operators on Hilbert Space, Konrad Schmüdgen, (2012, ISBN 978-94-007-4752-4)
266. Calculus Without Derivatives, Jean-Paul Penot, (2012, ISBN 978-1-4614-4537-1)
267. Quantum Theory for Mathematicians, Brian C. Hall, (2013, ISBN 978-1-4614-7115-8)
268. Geometric Analysis of the Bergman Kernel and Metric, Krantz, Steven G., (2013, ISBN 978-1-4614-7923-9)
269. Locally Convex Spaces, M Scott Osborne, (2014, ISBN 978-3-319-02044-0)
270. Fundamentals of Algebraic Topology, Steven Weintraub, (2014, ISBN 978-1-4939-1843-0)
271. Integer Programming, Michelangelo Conforti, Gérard P. Cornuéjols, Giacomo Zambelli, (2014, ISBN 978-3-319-11007-3)
272. Operator Theoretic Aspects of Ergodic Theory, Tanja Eisner, Bálint Farkas, Markus Haase, Rainer Nagel, (2015, ISBN 978-3-319-16897-5)
273. Homotopical Topology, Anatoly Fomenko, Dmitry Fuchs, (2016, 2nd ed., ISBN 978-3-319-23487-8)
274. Brownian Motion, Martingales, and Stochastic Calculus, Jean-François Le Gall, (2016, ISBN 978-3-319-31088-6)
275. Differential Geometry – Connections, Curvature, and Characteristic Classes, Loring W. Tu (2017, ISBN 978-3-319-55082-4)
276. Functional Analysis, Spectral Theory, and Applications, Manfred Einsiedler, Thomas Ward (2017, ISBN 978-3-319-58539-0)
277. The Moment Problem, Konrad Schmüdgen (2017, ISBN 978-3-319-64545-2)
278. Modern Real Analysis, William P. Ziemer (2017, 2nd ed., ISBN 978-3-319-64628-2)
279. Binomial Ideals, Jürgen Herzog, Takayuki Hibi, Hidefumi Ohsugi (2018, ISBN 978-3-319-95347-2)
280. Introduction to Real Analysis, Christopher Heil (2019, ISBN 978-3-030-26901-2)
281. Intersection Homology & Perverse Sheaves with Applications to Singularities, Laurenţiu G. Maxim (2019, ISBN 978-3-030-27644-7)
282. Measure, Integration & Real Analysis, Sheldon Axler (2020, ISBN 978-3-030-33143-6)
283. Basic Representation Theory of Algebras, Ibrahim Assem, Flávio U Coelho (2020, ISBN 978-3-030-35117-5)
284. Spectral Theory – Basic Concepts and Applications, David Borthwick (2020, ISBN 978-3-030-38001-4)
285. An Invitation to Unbounded Representations of ∗-Algebras on Hilbert Space, Konrad Schmüdgen (2020, ISBN 978-3-030-46365-6)
286. Lectures on Convex Geometry, Daniel Hug, Wolfgang Weil (2020, ISBN 978-3-030-50179-2)
287. Explorations in Complex Functions, Richard Beals, Roderick S. C. Wong (2020, ISBN 978-3-030-54532-1)
288. Quaternion Algebras, John Voight (2020, ISBN 978-3-030-56692-0)
289. Ergodic Dynamics – From Basic Theory to Applications, Jane M. Hawkins (2020, ISBN 978-3-030-59242-4)
290. Lessons in Enumerative Combinatorics, Omer Egecioglu , Adriano Garsia (2021, ISBN 978-3-030-71249-5)
291. Mathematical Logic, Heinz-Dieter Ebbinghaus, Jörg Flum, Wolfgang Thomas (2021, 3rd ed. ISBN 978-3-030-73839-6)
292. Random Walk, Brownian Motion and Martingales, Rabi Bhattacharya, Edward C. Waymire (2021, ISBN 978-3-030-78939-8)
293. Stationary Processes and Discrete Parameter Markov Processes, Rabi Bhattacharya, Edward C. Waymire (2022, ISBN 978-3-031-00941-9)
294. Partial Differential Equations, Wolfgang Arendt, Karsten Urban (2023, ISBN 978-3-031-13378-7)
295. Measure Theory, Probability, and Stochastic Processes, Jean-François Le Gall (2022, ISBN 978-3-031-14205-5)
296. Drinfeld Modules, Mihran Papikian (2023, ISBN 978-3-031-19706-2)
See also
• Graduate Studies in Mathematics
Notes
1. Also published from Dover Publications as the second edition. (2015, ISBN 978-0-486-78608-7)
2. This volume of the series with volume 69 were combined as volume 121.
3. Originally published as volumes 59 and 69 in this series.
4. A companion volume by the same authors: Complex Analysis and Special Topics in Harmonic Analysis (1995, ISBN 978-1-4613-8445-8).
5. This volume is subsequent to volume 106 in this series.
6. The problems and worked-out solutions book for all the exercises: Exercises and Solutions Manual for Integration and Probability by Paul Malliavin, Gerard Letac (1995, ISBN 978-0-387-94421-0)
7. The first edition is Riemannian Manifolds: An Introduction to Curvature.
External links
• Springer-Verlag's Summary of Graduate Texts in Mathematics
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Wikipedia
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Springer resolution
In mathematics, the Springer resolution is a resolution of the variety of nilpotent elements in a semisimple Lie algebra,[1][2] or the unipotent elements of a reductive algebraic group, introduced by Tonny Albert Springer in 1969.[3] The fibers of this resolution are called Springer fibers.[4]
If U is the variety of unipotent elements in a reductive group G, and X the variety of Borel subgroups B, then the Springer resolution of U is the variety of pairs (u,B) of U×X such that u is in the Borel subgroup B. The map to U is the projection to the first factor. The Springer resolution for Lie algebras is similar, except that U is replaced by the nilpotent elements of the Lie algebra of G and X replaced by the variety of Borel subalgebras.[5]
The Grothendieck–Springer resolution is defined similarly, except that U is replaced by the whole group G (or the whole Lie algebra of G). When restricted to the unipotent elements of G it becomes the Springer resolution.[6][7]
Examples
When G=SL(2), the Lie algebra Springer resolution is T*P1 → n, where n are the nilpotent elements of sl(2). In this example, n are the matrices x with tr(x2)=0, which is a two dimensional conical subvariety of sl(2). n has a unique singular point 0, the fibre above which in the Springer resolution is the zero section P1 .
References
1. Chriss, Neil; Ginzburg, Victor (1997), Representation theory and complex geometry, Boston, MA: Birkhäuser Boston, Inc., ISBN 0-8176-3792-3, MR 1433132
2. Dolgachev, Igor; Goldstein, Norman (1984), "On the Springer resolution of the minimal unipotent conjugacy class", Journal of Pure and Applied Algebra, 32 (1): 33–47, doi:10.1016/0022-4049(84)90012-4, hdl:2027.42/24847, MR 0739636
3. Springer, Tonny A. (1969), "The unipotent variety of a semi-simple group", Algebraic Geometry (Internat. Colloq., Tata Inst. Fund. Res., Bombay, 1968), Oxford Univ. Press, London, pp. 373–391, ISBN 978-0-19-635281-7, MR 0263830
4. Ginzburg, Victor (1998), "Geometric methods in the representation theory of Hecke algebras and quantum groups", Representation theories and algebraic geometry (Montreal, PQ, 1997), NATO Advanced Science Institutes Series C: Mathematical and Physical Sciences, vol. 514, Kluwer Acad. Publ., Dordrecht, pp. 127–183, arXiv:math/9802004, Bibcode:1998math......2004G, ISBN 0-7923-5193-2, MR 1649626
5. Springer, Tonny A. (1976), "Trigonometric sums, Green functions of finite groups and representations of Weyl groups", Inventiones Mathematicae, 36: 173–207, Bibcode:1976InMat..36..173S, doi:10.1007/BF01390009, MR 0442103, S2CID 121820241
6. Steinberg, Robert (1974), Conjugacy classes in algebraic groups, Lecture Notes in Mathematics, vol. 366, Berlin-New York: Springer-Verlag, doi:10.1007/BFb0067854, ISBN 978-3-540-06657-6, MR 0352279
7. Steinberg, Robert (1976), "On the desingularization of the unipotent variety", Inventiones Mathematicae, 36: 209–224, Bibcode:1976InMat..36..209S, doi:10.1007/BF01390010, MR 0430094, S2CID 120400717
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Wikipedia
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Joel Spruck
Joel Spruck (born 1946[1]) is a mathematician, J. J. Sylvester Professor of Mathematics at Johns Hopkins University, whose research concerns geometric analysis and elliptic partial differential equations.[2] He obtained his PhD from Stanford University with the supervision of Robert S. Finn in 1971.[3]
Mathematical contributions
Spruck is well known in the field of elliptic partial differential equations for his series of papers "The Dirichlet problem for nonlinear second-order elliptic equations," written in collaboration with Luis Caffarelli, Joseph J. Kohn, and Louis Nirenberg. These papers were among the first to develop a general theory of second-order elliptic differential equations which are fully nonlinear, with a regularity theory that extends to the boundary. Caffarelli, Nirenberg & Spruck (1985) has been particularly influential in the field of geometric analysis since many geometric partial differential equations are amenable to its methods.
With Basilis Gidas, Spruck studied positive solutions of subcritical second-order elliptic partial differential equations of Yamabe type. With Caffarelli, they studied the Yamabe equation on Euclidean space, proving a positive mass-style theorem on the asymptotic behavior of isolated singularities.
In 1974, Spruck and David Hoffman extended a mean curvature-based Sobolev inequality of James H. Michael and Leon Simon to the setting of submanifolds of Riemannian manifolds.[4] This has been useful for the study of many analytic problems in geometric settings, such as for Gerhard Huisken's study of mean curvature flow in Riemannian manifolds and for Richard Schoen and Shing-Tung Yau's study of the Jang equation in their resolution of the positive energy theorem in general relativity.[5][6]
In the late 80s, Stanley Osher and James Sethian developed the level-set method as a computational tool in numerical analysis.[7] In collaboration with Lawrence Evans, Spruck pioneered the rigorous study of the level-set flow, as adapted to the mean curvature flow. The level-set approach to mean curvature flow is important in the technical ease with topological change can occur along the flow. The same approach was independently developed by Yun Gang Chen, Yoshikazu Giga, and Shun'ichi Goto.[8] The works of Evans-Spruck and Chen-Giga-Goto found significant application in Gerhard Huisken and Tom Ilmanen's solution of the Riemannian Penrose inequality of general relativity and differential geometry, where they adapted the level-set approach to the inverse mean curvature flow.[9][10]
In 1994 Spruck was an invited speaker at the International Congress of Mathematicians in Zurich.[11]
Major publications
• Hoffman, David; Spruck, Joel. Sobolev and isoperimetric inequalities for Riemannian submanifolds. Comm. Pure Appl. Math. 27 (1974), 715–727.
• Gidas, B.; Spruck, J. A priori bounds for positive solutions of nonlinear elliptic equations. Comm. Partial Differential Equations 6 (1981), no. 8, 883–901.
• Gidas, B.; Spruck, J. Global and local behavior of positive solutions of nonlinear elliptic equations. Comm. Pure Appl. Math. 34 (1981), no. 4, 525–598.
• Caffarelli, L.; Nirenberg, L.; Spruck, J. The Dirichlet problem for nonlinear second-order elliptic equations. I. Monge-Ampère equation. Comm. Pure Appl. Math. 37 (1984), no. 3, 369–402.
• Caffarelli, L.; Kohn, J.J.; Nirenberg, L.; Spruck, J. The Dirichlet problem for nonlinear second-order elliptic equations. II. Complex Monge-Ampère, and uniformly elliptic, equations. Comm. Pure Appl. Math. 38 (1985), no. 2, 209–252.
• Caffarelli, L.; Nirenberg, L.; Spruck, J. The Dirichlet problem for nonlinear second-order elliptic equations. III. Functions of the eigenvalues of the Hessian. Acta Math. 155 (1985), no. 3–4, 261–301.
• Caffarelli, Luis A.; Gidas, Basilis; Spruck, Joel. Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth. Comm. Pure Appl. Math. 42 (1989), no. 3, 271–297.
• Evans, L.C.; Spruck, J. Motion of level sets by mean curvature. I. J. Differential Geom. 33 (1991), no. 3, 635–681.
• Spruck, Joel; Yang, Yi Song. Topological solutions in the self-dual Chern-Simons theory: existence and approximation. Ann. Inst. H. Poincaré Anal. Non Linéaire 12 (1995), no. 1, 75–97.
Prizes
• Simons Fellowship (2012–2013)[12]
• Fellow of the American Mathematical Society (2013 inauguration)[13]
• Guggenheim Fellowship (1999–2000)[14]
References
1. Tartar, Luc (December 3, 2009). The General Theory of Homogenization: A Personalized Introduction. Springer Science & Business Media. ISBN 9783642051951 – via Google Books.
2. "Joel Spruck". Mathematics.
3. Joel Spruck at the Mathematics Genealogy Project
4. Michael, J.H.; Simon, L.M. Sobolev and mean-value inequalities on generalized submanifolds of Rn. Comm. Pure Appl. Math. 26 (1973), 361–379.
5. Huisken, Gerhard. Contracting convex hypersurfaces in Riemannian manifolds by their mean curvature. Invent. Math. 84 (1986), no. 3, 463–480.
6. Schoen, Richard; Yau, Shing Tung. Proof of the positive mass theorem. II. Comm. Math. Phys. 79 (1981), no. 2, 231–260.
7. Osher, Stanley; Sethian, James A. Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations. J. Comput. Phys. 79 (1988), no. 1, 12–49.
8. Chen, Yun Gang; Giga, Yoshikazu; Goto, Shun'ichi. Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations. J. Differential Geom. 33 (1991), no. 3, 749–786.
9. Huisken, Gerhard; Ilmanen, Tom. The inverse mean curvature flow and the Riemannian Penrose inequality. J. Differential Geom. 59 (2001), no. 3, 353–437.
10. A more general version of the Riemannian Penrose inequality was found at the same time by Hubert Bray, who did not make use of level-set methods.
11. Spruck, Joel. Fully nonlinear elliptic equations and applications to geometry. In: Srishti D. Chatterji (ed.): Proceedings of the International Congress of Mathematicians. August 3–11, 1994, Zürich, Switzerland. vol. 2. Basel, Birkhäuser 1995, ISBN 3-7643-5153-5, pp. 1145–1152.
12. "Joel Spruck". Simons Foundation. July 13, 2017.
13. "Fellows of the American Mathematical Society". American Mathematical Society.
14. "John Simon Guggenheim Memorial Foundation Home Page". October 24, 2008. Archived from the original on 2008-10-24.
External links
• Joel Spruck publications indexed by Google Scholar
Authority control
International
• ISNI
• VIAF
National
• Norway
• France
• BnF data
• Israel
• United States
• Japan
Academics
• CiNii
• MathSciNet
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• zbMATH
Other
• IdRef
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Wikipedia
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Spt function
The spt function (smallest parts function) is a function in number theory that counts the sum of the number of smallest parts in each partition of a positive integer. It is related to the partition function.[1]
The first few values of spt(n) are:
1, 3, 5, 10, 14, 26, 35, 57, 80, 119, 161, 238, 315, 440, 589 ... (sequence A092269 in the OEIS)
Example
For example, there are five partitions of 4 (with smallest parts underlined):
4
3 + 1
2 + 2
2 + 1 + 1
1 + 1 + 1 + 1
These partitions have 1, 1, 2, 2, and 4 smallest parts, respectively. So spt(4) = 1 + 1 + 2 + 2 + 4 = 10.
Properties
Like the partition function, spt(n) has a generating function. It is given by
$S(q)=\sum _{n=1}^{\infty }\mathrm {spt} (n)q^{n}={\frac {1}{(q)_{\infty }}}\sum _{n=1}^{\infty }{\frac {q^{n}\prod _{m=1}^{n-1}(1-q^{m})}{1-q^{n}}}$
where $(q)_{\infty }=\prod _{n=1}^{\infty }(1-q^{n})$.
The function $S(q)$ is related to a mock modular form. Let $E_{2}(z)$ denote the weight 2 quasi-modular Eisenstein series and let $\eta (z)$ denote the Dedekind eta function. Then for $q=e^{2\pi iz}$, the function
${\tilde {S}}(z):=q^{-1/24}S(q)-{\frac {1}{12}}{\frac {E_{2}(z)}{\eta (z)}}$
is a mock modular form of weight 3/2 on the full modular group $SL_{2}(\mathbb {Z} )$ with multiplier system $\chi _{\eta }^{-1}$, where $\chi _{\eta }$ is the multiplier system for $\eta (z)$.
While a closed formula is not known for spt(n), there are Ramanujan-like congruences including
$\mathrm {spt} (5n+4)\equiv 0\mod (5)$
$\mathrm {spt} (7n+5)\equiv 0\mod (7)$
$\mathrm {spt} (13n+6)\equiv 0\mod (13).$
References
1. Andrews, George E. (2008-11-01). "The number of smallest parts in the partitions of n". 2008 (624): 133–142. doi:10.1515/CRELLE.2008.083. ISSN 1435-5345. {{cite journal}}: Cite journal requires |journal= (help)
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Quadrature (mathematics)
In mathematics, quadrature is a historical term for the process of determining area. This term is still used in the context of differential equations, where "solving an equation by quadrature" or "reduction to quadrature" means expressing its solution in terms of integrals.
Quadrature problems served as one of the main sources of problems in the development of calculus. They introduce important topics in mathematical analysis.
History
Antiquity
Greek mathematicians understood the determination of an area of a figure as the process of geometrically constructing a square having the same area (squaring), thus the name quadrature for this process. The Greek geometers were not always successful (see squaring the circle), but they did carry out quadratures of some figures whose sides were not simply line segments, such as the lune of Hippocrates and the parabola. By a certain Greek tradition, these constructions had to be performed using only a compass and straightedge, though not all Greek mathematicians adhered to this dictum.
For a quadrature of a rectangle with the sides a and b it is necessary to construct a square with the side $x={\sqrt {ab}}$ (the geometric mean of a and b). For this purpose it is possible to use the following: if one draws the circle with diameter made from joining line segments of lengths a and b, then the height (BH in the diagram) of the line segment drawn perpendicular to the diameter, from the point of their connection to the point where it crosses the circle, equals the geometric mean of a and b. A similar geometrical construction solves the problems of quadrature of a parallelogram and of a triangle.
Problems of quadrature for curvilinear figures are much more difficult. The quadrature of the circle with compass and straightedge was proved in the 19th century to be impossible.[1][2] Nevertheless, for some figures a quadrature can be performed. The quadratures of the surface of a sphere and a parabola segment discovered by Archimedes became the highest achievement of analysis in antiquity.
• The area of the surface of a sphere is equal to four times the area of the circle formed by a great circle of this sphere.
• The area of a segment of a parabola determined by a straight line cutting it is 4/3 the area of a triangle inscribed in this segment.
For the proofs of these results, Archimedes used the method of exhaustion attributed to Eudoxus.[3]
Medieval mathematics
In medieval Europe, quadrature meant the calculation of area by any method. Most often the method of indivisibles was used; it was less rigorous than the geometric constructions of the Greeks, but it was simpler and more powerful. With its help, Galileo Galilei and Gilles de Roberval found the area of a cycloid arch, Grégoire de Saint-Vincent investigated the area under a hyperbola (Opus Geometricum, 1647),[3]: 491 and Alphonse Antonio de Sarasa, de Saint-Vincent's pupil and commentator, noted the relation of this area to logarithms.[3]: 492 [4]
Integral calculus
John Wallis algebrised this method; he wrote in his Arithmetica Infinitorum (1656) some series which are equivalent to what is now called the definite integral, and he calculated their values. Isaac Barrow and James Gregory made further progress: quadratures for some algebraic curves and spirals. Christiaan Huygens successfully performed a quadrature of the surface area of some solids of revolution.
The quadrature of the hyperbola by Saint-Vincent and de Sarasa provided a new function, the natural logarithm, of critical importance. With the invention of integral calculus came a universal method for area calculation. In response, the term quadrature has become traditional, and instead the modern phrase finding the area is more commonly used for what is technically the computation of a univariate definite integral.
See also
• Gaussian quadrature
• Hyperbolic angle
• Numerical integration
• Quadratrix
• Tanh-sinh quadrature
Notes
1. Lindemann, F. (1882). "Über die Zahl π" [On the number π]. Mathematische Annalen (in German). 20: 213–225. doi:10.1007/bf01446522. S2CID 120469397.
2. Fritsch, Rudolf (1984). "The transcendence of π has been known for about a century—but who was the man who discovered it?". Results in Mathematics. 7 (2): 164–183. doi:10.1007/BF03322501. MR 0774394. S2CID 119986449.
3. Katz, Victor J. (1998). A History of Mathematics: An Introduction (2nd ed.). Addison Wesley Longman. ISBN 0-321-01618-1.
4. Enrique A. Gonzales-Velasco (2011) Journey through Mathematics, § 2.4 Hyperbolic Logarithms, page 117
References
• Boyer, C. B. (1989) A History of Mathematics, 2nd ed. rev. by Uta C. Merzbach. New York: Wiley, ISBN 0-471-09763-2 (1991 pbk ed. ISBN 0-471-54397-7).
• Eves, Howard (1990) An Introduction to the History of Mathematics, Saunders, ISBN 0-03-029558-0,
• Christiaan Huygens (1651) Theoremata de Quadratura Hyperboles, Ellipsis et Circuli
• Jean-Etienne Montucla (1873) History of the Quadrature of the Circle, J. Babin translator, William Alexander Myers editor, link from HathiTrust.
• Christoph Scriba (1983) "Gregory's Converging Double Sequence: a new look at the controversy between Huygens and Gregory over the 'analytical' quadrature of the circle", Historia Mathematica 10:274–85.
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Wikipedia
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Square-free element
In mathematics, a square-free element is an element r of a unique factorization domain R that is not divisible by a non-trivial square. This means that every s such that $s^{2}\mid r$ is a unit of R.
Alternate characterizations
Square-free elements may be also characterized using their prime decomposition. The unique factorization property means that a non-zero non-unit r can be represented as a product of prime elements
$r=p_{1}p_{2}\cdots p_{n}$
Then r is square-free if and only if the primes pi are pairwise non-associated (i.e. that it doesn't have two of the same prime as factors, which would make it divisible by a square number).
Examples
Common examples of square-free elements include square-free integers and square-free polynomials.
See also
• Prime number
References
• David Darling (2004) The Universal Book of Mathematics: From Abracadabra to Zeno's Paradoxes John Wiley & Sons
• Baker, R. C. "The square-free divisor problem." The Quarterly Journal of Mathematics 45.3 (1994): 269-277.
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Wikipedia
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Square-free integer
In mathematics, a square-free integer (or squarefree integer) is an integer which is divisible by no square number other than 1. That is, its prime factorization has exactly one factor for each prime that appears in it. For example, 10 = 2 ⋅ 5 is square-free, but 18 = 2 ⋅ 3 ⋅ 3 is not, because 18 is divisible by 9 = 32. The smallest positive square-free numbers are
1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 30, 31, 33, 34, 35, 37, 38, 39, ... (sequence A005117 in the OEIS)
Square-free factorization
Every positive integer $n$ can be factored in a unique way as
$n=\prod _{i=1}^{k}q_{i}^{i},$
where the $q_{i}$ different from one are square-free integers that are pairwise coprime. This is called the square-free factorization of n.
To construct the square-free factorization, let
$n=\prod _{j=1}^{h}p_{j}^{e_{j}}$
be the prime factorization of $n$, where the $p_{j}$ are distinct prime numbers. Then the factors of the square-free factorization are defined as
$q_{i}=\prod _{j:e_{j}=i}p_{j}.$
An integer is square-free if and only if $q_{i}=1$ for all $i>1$. An integer greater than one is the $k$th power of another integer if and only if $k$ is a divisor of all $i$ such that $q_{i}\neq 1.$
The use of the square-free factorization of integers is limited by the fact that its computation is as difficult as the computation of the prime factorization. More precisely every known algorithm for computing a square-free factorization computes also the prime factorization. This is a notable difference with the case of polynomials for which the same definitions can be given, but, in this case, the square-free factorization is not only easier to compute than the complete factorization, but it is the first step of all standard factorization algorithms.
Square-free factors of integers
The radical of an integer is its largest square-free factor, that is $\textstyle \prod _{i=1}^{k}q_{i}$ with notation of the preceding section. An integer is square-free if and only if it is equal to its radical.
Every positive integer $n$ can be represented in a unique way as the product of a powerful number (that is an integer such that is divisible by the square of every prime factor) and a square-free integer, which are coprime. In this factorization, the square-free factor is $q_{1},$ and the powerful number is $\textstyle \prod _{i=2}^{k}q_{i}^{i}.$
The square-free part of $n$ is $q_{1},$ which is the largest square-free divisor $k$ of $n$ that is coprime with $n/k$. The square-free part of an integer may be smaller than the largest square-free divisor, which is $\textstyle \prod _{i=1}^{k}q_{i}.$
Any arbitrary positive integer $n$ can be represented in a unique way as the product of a square and a square-free integer:
$n=m^{2}k$
In this factorization, $m$ is the largest divisor of $n$ such that $m^{2}$ is a divisor of $n$.
In summary, there are three square-free factors that are naturally associated to every integer: the square-free part, the above factor $k$, and the largest square-free factor. Each is a factor of the next one. All are easily deduced from the prime factorization or the square-free factorization: if
$n=\prod _{i=1}^{h}p_{i}^{e_{i}}=\prod _{i=1}^{k}q_{i}^{i}$
are the prime factorization and the square-free factorization of $n$, where $p_{1},\ldots ,p_{h}$ are distinct prime numbers, then the square-free part is
$\prod _{e_{i}=1}p_{i}=q_{1},$
The square-free factor such the quotient is a square is
$\prod _{e_{i}{\text{ odd}}}p_{i}=\prod _{i{\text{ odd}}}q_{i},$
and the largest square-free factor is
$\prod _{i=1}^{h}p_{i}=\prod _{i=1}^{k}q_{i}.$
For example, if $n=75600=2^{4}\cdot 3^{3}\cdot 5^{2}\cdot 7,$ one has $q_{1}=7,\;q_{2}=5,\;q_{3}=3,\;q_{4}=2.$ The square-free part is 7, the square-free factor such that the quotient is a square is 3 ⋅ 7 = 21, and the largest square-free factor is 2 ⋅ 3 ⋅ 5 ⋅ 7 = 210.
No algorithm is known for computing any of these square-free factors which is faster than computing the complete prime factorization. In particular, there is no known polynomial-time algorithm for computing the square-free part of an integer, or even for determining whether an integer is square-free.[1] In contrast, polynomial-time algorithms are known for primality testing.[2] This is a major difference between the arithmetic of the integers, and the arithmetic of the univariate polynomials, as polynomial-time algorithms are known for square-free factorization of polynomials (in short, the largest square-free factor of a polynomial is its quotient by the greatest common divisor of the polynomial and its formal derivative).[3]
Equivalent characterizations
A positive integer $n$ is square-free if and only if in the prime factorization of $n$, no prime factor occurs with an exponent larger than one. Another way of stating the same is that for every prime factor $p$ of $n$, the prime $p$ does not evenly divide $n/p$. Also $n$ is square-free if and only if in every factorization $n=ab$, the factors $a$ and $b$ are coprime. An immediate result of this definition is that all prime numbers are square-free.
A positive integer $n$ is square-free if and only if all abelian groups of order $n$ are isomorphic, which is the case if and only if any such group is cyclic. This follows from the classification of finitely generated abelian groups.
A integer $n$ is square-free if and only if the factor ring $\mathbb {Z} /n\mathbb {Z} $ (see modular arithmetic) is a product of fields. This follows from the Chinese remainder theorem and the fact that a ring of the form $\mathbb {Z} /k\mathbb {Z} $ is a field if and only if $k$ is prime.
For every positive integer $n$, the set of all positive divisors of $n$ becomes a partially ordered set if we use divisibility as the order relation. This partially ordered set is always a distributive lattice. It is a Boolean algebra if and only if $n$ is square-free.
A positive integer $n$ is square-free if and only if $\mu (n)\neq 0$, where $\mu $ denotes the Möbius function.
Dirichlet series
The absolute value of the Möbius function is the indicator function for the square-free integers – that is, |μ(n)| is equal to 1 if n is square-free, and 0 if it is not. The Dirichlet series of this indicator function is
$\sum _{n=1}^{\infty }{\frac {|\mu (n)|}{n^{s}}}={\frac {\zeta (s)}{\zeta (2s)}},$
where ζ(s) is the Riemann zeta function. This follows from the Euler product
${\frac {\zeta (s)}{\zeta (2s)}}=\prod _{p}{\frac {(1-p^{-2s})}{(1-p^{-s})}}=\prod _{p}(1+p^{-s}),$
where the products are taken over the prime numbers.
Distribution
Let Q(x) denote the number of square-free integers between 1 and x (OEIS: A013928 shifting index by 1). For large n, 3/4 of the positive integers less than n are not divisible by 4, 8/9 of these numbers are not divisible by 9, and so on. Because these ratios satisfy the multiplicative property (this follows from Chinese remainder theorem), we obtain the approximation:
${\begin{aligned}Q(x)&\approx x\prod _{p\ {\text{prime}}}\left(1-{\frac {1}{p^{2}}}\right)=x\prod _{p\ {\text{prime}}}{\frac {1}{(1-{\frac {1}{p^{2}}})^{-1}}}\\&=x\prod _{p\ {\text{prime}}}{\frac {1}{1+{\frac {1}{p^{2}}}+{\frac {1}{p^{4}}}+\cdots }}={\frac {x}{\sum _{k=1}^{\infty }{\frac {1}{k^{2}}}}}={\frac {x}{\zeta (2)}}={\frac {6x}{\pi ^{2}}}.\end{aligned}}$
This argument can be made rigorous for getting the estimate (using big O notation)
$Q(x)={\frac {6x}{\pi ^{2}}}+O\left({\sqrt {x}}\right).$
Sketch of a proof: the above characterization gives
$Q(x)=\sum _{n\leq x}\sum _{d^{2}\mid n}\mu (d)=\sum _{d\leq x}\mu (d)\sum _{n\leq x,d^{2}\mid n}1=\sum _{d\leq x}\mu (d)\left\lfloor {\frac {x}{d^{2}}}\right\rfloor ;$ ;}
observing that the last summand is zero for $d>{\sqrt {x}}$, it results that
${\begin{aligned}Q(x)&=\sum _{d\leq {\sqrt {x}}}\mu (d)\left\lfloor {\frac {x}{d^{2}}}\right\rfloor =\sum _{d\leq {\sqrt {x}}}{\frac {x\mu (d)}{d^{2}}}+O\left(\sum _{d\leq {\sqrt {x}}}1\right)=x\sum _{d\leq {\sqrt {x}}}{\frac {\mu (d)}{d^{2}}}+O({\sqrt {x}})\\&=x\sum _{d}{\frac {\mu (d)}{d^{2}}}+O\left(x\sum _{d>{\sqrt {x}}}{\frac {1}{d^{2}}}+{\sqrt {x}}\right)={\frac {x}{\zeta (2)}}+O({\sqrt {x}}).\end{aligned}}$
By exploiting the largest known zero-free region of the Riemann zeta function Arnold Walfisz improved the approximation to[4]
$Q(x)={\frac {6x}{\pi ^{2}}}+O\left(x^{1/2}\exp \left(-c{\frac {(\log x)^{3/5}}{(\log \log x)^{1/5}}}\right)\right),$
for some positive constant c.
Under the Riemann hypothesis, the error term can be reduced to[5]
$Q(x)={\frac {x}{\zeta (2)}}+O\left(x^{17/54+\varepsilon }\right)={\frac {6}{\pi ^{2}}}x+O\left(x^{17/54+\varepsilon }\right).$
In 2015 the error term was further reduced to[6]
$Q(x)={\frac {6}{\pi ^{2}}}x+O\left(x^{11/35+\varepsilon }\right).$
The asymptotic/natural density of square-free numbers is therefore
$\lim _{x\to \infty }{\frac {Q(x)}{x}}={\frac {6}{\pi ^{2}}}\approx 0.6079$
Therefore over 3/5 of the integers are square-free.
Likewise, if Q(x,n) denotes the number of n-free integers (e.g. 3-free integers being cube-free integers) between 1 and x, one can show[7]
$Q(x,n)={\frac {x}{\sum _{k=1}^{\infty }{\frac {1}{k^{n}}}}}+O\left({\sqrt[{n}]{x}}\right)={\frac {x}{\zeta (n)}}+O\left({\sqrt[{n}]{x}}\right).$
Since a multiple of 4 must have a square factor 4=22, it cannot occur that four consecutive integers are all square-free. On the other hand, there exist infinitely many integers n for which 4n +1, 4n +2, 4n +3 are all square-free. Otherwise, observing that 4n and at least one of 4n +1, 4n +2, 4n +3 among four could be non-square-free for sufficiently large n, half of all positive integers minus finitely many must be non-square-free and therefore
$Q(x)\leq {\frac {x}{2}}+C$ for some constant C,
contrary to the above asymptotic estimate for $Q(x)$.
There exist sequences of consecutive non-square-free integers of arbitrary length. Indeed, if n satisfies a simultaneous congruence
$n\equiv -i{\pmod {p_{i}^{2}}}\qquad (i=1,2,\ldots ,l)$
for distinct primes $p_{1},p_{2},\ldots ,p_{l}$, then each $n+i$ is divisible by pi 2.[8] On the other hand, the above-mentioned estimate $Q(x)=6x/\pi ^{2}+O\left({\sqrt {x}}\right)$ implies that, for some constant c, there always exists a square-free integer between x and $x+c{\sqrt {x}}$ for positive x. Moreover, an elementary argument allows us to replace $x+c{\sqrt {x}}$ by $x+cx^{1/5}\log x.$[9] The ABC conjecture would allow $x+x^{o(1)}$.[10]
Table of Q(x), 6/π2x, and R(x)
The table shows how $Q(x)$ and ${\frac {6}{\pi ^{2}}}x$ (with the latter rounded to one decimal place) compare at powers of 10.
$R(x)=Q(x)-{\frac {6}{\pi ^{2}}}x$ , also denoted as $\Delta (x)$.
$x$ $Q(x)$ ${\frac {6}{\pi ^{2}}}x$ $R(x)$
10 7 6.1 0.9
102 61 60.8 0.2
103 608 607.9 0.1
104 6,083 6,079.3 3.7
105 60,794 60,792.7 1.3
106 607,926 607,927.1 - 1.3
107 6,079,291 6,079,271.0 20.0
108 60,792,694 60,792,710.2 - 16.2
109 607,927,124 607,927,101.9 22.1
1010 6,079,270,942 6,079,271,018.5 - 76.5
1011 60,792,710,280 60,792,710,185.4 94.6
1012 607,927,102,274 607,927,101,854.0 420.0
1013 6,079,271,018,294 6,079,271,018,540.3 - 246.3
1014 60,792,710,185,947 60,792,710,185,402.7 544.3
1015 607,927,101,854,103 607,927,101,854,027.0 76.0
$R(x)$ changes its sign infinitely often as $x$ tends to infinity.[11]
The absolute value of $R(x)$ is astonishingly small compared with $x$.
Encoding as binary numbers
If we represent a square-free number as the infinite product
$\prod _{n=0}^{\infty }(p_{n+1})^{a_{n}},a_{n}\in \lbrace 0,1\rbrace ,{\text{ and }}p_{n}{\text{ is the }}n{\text{th prime}},$
then we may take those $a_{n}$ and use them as bits in a binary number with the encoding
$\sum _{n=0}^{\infty }{a_{n}}\cdot 2^{n}.$
The square-free number 42 has factorization 2 × 3 × 7, or as an infinite product 21 · 31 · 50 · 71 · 110 · 130 ··· Thus the number 42 may be encoded as the binary sequence ...001011 or 11 decimal. (The binary digits are reversed from the ordering in the infinite product.)
Since the prime factorization of every number is unique, so also is every binary encoding of the square-free integers.
The converse is also true. Since every positive integer has a unique binary representation it is possible to reverse this encoding so that they may be decoded into a unique square-free integer.
Again, for example, if we begin with the number 42, this time as simply a positive integer, we have its binary representation 101010. This decodes to 20 · 31 · 50 · 71 · 110 · 131 = 3 × 7 × 13 = 273.
Thus binary encoding of squarefree numbers describes a bijection between the nonnegative integers and the set of positive squarefree integers.
(See sequences A019565, A048672 and A064273 in the OEIS.)
Erdős squarefree conjecture
The central binomial coefficient
${2n \choose n}$
is never squarefree for n > 4. This was proven in 1985 for all sufficiently large integers by András Sárközy,[12] and for all integers > 4 in 1996 by Olivier Ramaré and Andrew Granville.[13]
Squarefree core
Let us call "t-free" a positive integer that has no t-th power in its divisors. In particular, the 2-free integers are the square-free integers.
The multiplicative function $\mathrm {core} _{t}(n)$ maps every positive integer n to the quotient of n by its largest divisor that is a t-th power. That is,
$\mathrm {core} _{t}(p^{e})=p^{e{\bmod {t}}}.$
The integer $\mathrm {core} _{t}(n)$ is t-free, and every t-free integer is mapped to itself by the function $\mathrm {core} _{t}.$
The Dirichlet generating function of the sequence $\left(\mathrm {core} _{t}(n)\right)_{n\in \mathbb {N} }$ is
$\sum _{n\geq 1}{\frac {\mathrm {core} _{t}(n)}{n^{s}}}={\frac {\zeta (ts)\zeta (s-1)}{\zeta (ts-t)}}$.
See also OEIS: A007913 (t=2), OEIS: A050985 (t=3) and OEIS: A053165 (t=4).
Notes
1. Adleman, Leonard M.; McCurley, Kevin S. (1994). "Open problems in number theoretic complexity, II". In Adleman, Leonard M.; Huang, Ming-Deh A. (eds.). Algorithmic Number Theory, First International Symposium, ANTS-I, Ithaca, NY, USA, May 6–9, 1994, Proceedings. Lecture Notes in Computer Science. Vol. 877. Springer. pp. 291–322. doi:10.1007/3-540-58691-1_70.
2. Agrawal, Manindra; Kayal, Neeraj; Saxena, Nitin (1 September 2004). "PRIMES is in P" (PDF). Annals of Mathematics. 160 (2): 781–793. doi:10.4007/annals.2004.160.781. ISSN 0003-486X. MR 2123939. Zbl 1071.11070.
3. Richards, Chelsea (2009). Algorithms for factoring square-free polynomials over finite fields (PDF) (Master's thesis). Canada: Simon Fraser University.
4. Walfisz, A. (1963). Weylsche Exponentialsummen in der neueren Zahlentheorie. Berlin: VEB Deutscher Verlag der Wissenschaften.
5. Jia, Chao Hua. "The distribution of square-free numbers", Science in China Series A: Mathematics 36:2 (1993), pp. 154–169. Cited in Pappalardi 2003, A Survey on k-freeness; also see Kaneenika Sinha, "Average orders of certain arithmetical functions", Journal of the Ramanujan Mathematical Society 21:3 (2006), pp. 267–277.
6. Liu, H.-Q. (2016). "On the distribution of squarefree numbers". Journal of Number Theory. 159: 202–222. doi:10.1016/j.jnt.2015.07.013.
7. Linfoot, E. H.; Evelyn, C. J. A. (1929). "On a Problem in the Additive Theory of Numbers". Mathematische Zeitschrift. 30: 443–448. doi:10.1007/BF01187781. S2CID 120604049.
8. Parent, D. P. (1984). Exercises in Number Theory. Problem Books in Mathematics. Springer-Verlag New York. doi:10.1007/978-1-4757-5194-9. ISBN 978-1-4757-5194-9.
9. Filaseta, Michael; Trifonov, Ognian (1992). "On gaps between squarefree numbers. II". Journal of the London Mathematical Society. Second Series. 45 (2): 215–221. doi:10.1112/jlms/s2-45.2.215. MR 1171549.
10. Andrew, Granville (1998). "ABC allows us to count squarefrees". Int. Math. Res. Not. 1998 (19): 991–1009. doi:10.1155/S1073792898000592.
11. Minoru, Tanaka (1979). "Experiments concerning the distribution of squarefree numbers". Proceedings of the Japan Academy, Series A, Mathematical Sciences. 55 (3). doi:10.3792/pjaa.55.101. S2CID 121862978.
12. Sárközy, A. (1985). "On divisors of binomial coefficients. I". Journal of Number Theory. 20 (1): 70–80. doi:10.1016/0022-314X(85)90017-4. MR 0777971.
13. Ramaré, Olivier; Granville, Andrew (1996). "Explicit bounds on exponential sums and the scarcity of squarefree binomial coefficients". Mathematika. 43 (1): 73–107. doi:10.1112/S0025579300011608.
References
• Shapiro, Harold N. (1983). Introduction to the theory of numbers. Oxford University Press Dover Publications. ISBN 978-0-486-46669-9.
• Granville, Andrew; Ramaré, Olivier (1996). "Explicit bounds on exponential sums and the scarcity of squarefree binomial coefficients". Mathematika. 43: 73–107. CiteSeerX 10.1.1.55.8. doi:10.1112/S0025579300011608. MR 1401709. Zbl 0868.11009.
• Guy, Richard K. (2004). Unsolved problems in number theory (3rd ed.). Springer-Verlag. ISBN 978-0-387-20860-2. Zbl 1058.11001.
• "OEIS Wiki". Retrieved 24 September 2021.
Divisibility-based sets of integers
Overview
• Integer factorization
• Divisor
• Unitary divisor
• Divisor function
• Prime factor
• Fundamental theorem of arithmetic
Factorization forms
• Prime
• Composite
• Semiprime
• Pronic
• Sphenic
• Square-free
• Powerful
• Perfect power
• Achilles
• Smooth
• Regular
• Rough
• Unusual
Constrained divisor sums
• Perfect
• Almost perfect
• Quasiperfect
• Multiply perfect
• Hemiperfect
• Hyperperfect
• Superperfect
• Unitary perfect
• Semiperfect
• Practical
• Erdős–Nicolas
With many divisors
• Abundant
• Primitive abundant
• Highly abundant
• Superabundant
• Colossally abundant
• Highly composite
• Superior highly composite
• Weird
Aliquot sequence-related
• Untouchable
• Amicable (Triple)
• Sociable
• Betrothed
Base-dependent
• Equidigital
• Extravagant
• Frugal
• Harshad
• Polydivisible
• Smith
Other sets
• Arithmetic
• Deficient
• Friendly
• Solitary
• Sublime
• Harmonic divisor
• Descartes
• Refactorable
• Superperfect
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Wikipedia
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Square-free polynomial
In mathematics, a square-free polynomial is a polynomial defined over a field (or more generally, an integral domain) that does not have as a divisor any square of a non-constant polynomial.[1] A univariate polynomial is square free if and only if it has no multiple root in an algebraically closed field containing its coefficients. This motivates that, in applications in physics and engineering, a square-free polynomial is commonly called a polynomial with no repeated roots.
In the case of univariate polynomials, the product rule implies that, if p2 divides f, then p divides the formal derivative f ' of f. The converse is also true and hence, $f$ is square-free if and only if $1$ is a greatest common divisor of the polynomial and its derivative.[2]
A square-free decomposition or square-free factorization of a polynomial is a factorization into powers of square-free polynomials
$f=a_{1}a_{2}^{2}a_{3}^{3}\cdots a_{n}^{n}=\prod _{k=1}^{n}a_{k}^{k}\,$
where those of the ak that are non-constant are pairwise coprime square-free polynomials (here, two polynomials are said coprime is their greatest common divisor is a constant; in other words that is the coprimality over the field of fractions of the coefficients that is considered).[1] Every non-zero polynomial admits a square-free factorization, which is unique up to the multiplication and division of the factors by non-zero constants. The square-free factorization is much easier to compute than the complete factorization into irreducible factors, and is thus often preferred when the complete factorization is not really needed, as for the partial fraction decomposition and the symbolic integration of rational fractions. Square-free factorization is the first step of the polynomial factorization algorithms that are implemented in computer algebra systems. Therefore, the algorithm of square-free factorization is basic in computer algebra.
Over a field of characteristic 0, the quotient of $f$ by its GCD with its derivative is the product of the $a_{i}$ in the above square-free decomposition. Over a perfect field of non-zero characteristic p, this quotient is the product of the $a_{i}$ such that i is not a multiple of p. Further GCD computations and exact divisions allow computing the square-free factorization (see square-free factorization over a finite field). In characteristic zero, a better algorithm is known, Yun's algorithm, which is described below.[1] Its computational complexity is, at most, twice that of the GCD computation of the input polynomial and its derivative. More precisely, if $T_{n}$ is the time needed to compute the GCD of two polynomials of degree $n$ and the quotient of these polynomial by the GCD, then $2T_{n}$ is an upper bound for the time needed to compute the square free decomposition.
There are also known algorithms for the computation of the square-free decomposition of multivariate polynomials, that proceed generally by considering a multivariate polynomial as a univariate polynomial with polynomial coefficients, and applying recursively a univariate algorithm.[3]
Yun's algorithm
This section describes Yun's algorithm for the square-free decomposition of univariate polynomials over a field of characteristic 0.[1] It proceeds by a succession of GCD computations and exact divisions.
The input is thus a non-zero polynomial f, and the first step of the algorithm consists of computing the GCD a0 of f and its formal derivative f'.
If
$f=a_{1}a_{2}^{2}a_{3}^{3}\cdots a_{k}^{k}$
is the desired factorization, we have thus
$a_{0}=a_{2}^{1}a_{3}^{2}\cdots a_{k}^{k-1},$
$f/a_{0}=a_{1}a_{2}a_{3}\cdots a_{k}$
and
$f'/a_{0}=\sum _{i=1}^{k}ia_{i}'a_{1}\cdots a_{i-1}a_{i+1}\cdots a_{k}.$
If we set $b_{1}=f/a_{0}$, $c_{1}=f'/a_{0}$ and $d_{1}=c_{1}-b_{1}'$, we get that
$\gcd(b_{1},d_{1})=a_{1},$
$b_{2}=b_{1}/a_{1}=a_{2}a_{3}\cdots a_{n},$
and
$c_{2}=d_{1}/a_{1}=\sum _{i=2}^{k}(i-1)a_{i}'a_{2}\cdots a_{i-1}a_{i+1}\cdots a_{k}.$
Iterating this process until $b_{k+1}=1$ we find all the $a_{i}.$
This is formalized into an algorithm as follows:
$a_{0}:=\gcd(f,f');\quad b_{1}:=f/a_{0};\quad c_{1}:=f'/a_{0};\quad d_{1}:=c_{1}-b_{1}';\quad i:=1;$
repeat
$a_{i}:=\gcd(b_{i},d_{i});\quad b_{i+1}:=b_{i}/a_{i};\quad c_{i+1}:=d_{i}/a_{i};\quad i:=i+1;\quad d_{i}:=c_{i}-b_{i}';$
until $b=1;$
Output $a_{1},\ldots ,a_{i-1}.$
The degree of $c_{i}$ and $d_{i}$ is one less than the degree of $b_{i}.$ As $f$ is the product of the $b_{i},$ the sum of the degrees of the $b_{i}$ is the degree of $f.$ As the complexity of GCD computations and divisions increase more than linearly with the degree, it follows that the total running time of the "repeat" loop is less than the running time of the first line of the algorithm, and that the total running time of Yun's algorithm is upper bounded by twice the time needed to compute the GCD of $f$ and $f'$ and the quotient of $f$ and $f'$ by their GCD.
Square root
In general, a polynomial has no square root. More precisely, most polynomials cannot be written as the square of another polynomial.
A polynomial has a square root if and only if all exponents of the square-free decomposition are even. In this case, a square root is obtained by dividing these exponents by 2.
Thus the problem of deciding if a polynomial has a square root, and of computing it if it exists, is a special case of square-free factorization.
Notes
References
1. Yun, David Y.Y. (1976). "On square-free decomposition algorithms". SYMSAC '76 Proceedings of the third ACM symposium on Symbolic and algebraic computation. Association for Computing Machinery. pp. 26–35. doi:10.1145/800205.806320. ISBN 978-1-4503-7790-4. S2CID 12861227.
2. Dummit, David S.; Foote, Richard M. (2004). Abstract Algebra. p. 547. ISBN 978-81-265-3228-5.
3. Gianni, P.; Trager, B. (1996). "Square-Free Algorithms in Positive Characteristic". Applicable Algebra in Engineering, Communication and Computing. 7 (1): 1–14. doi:10.1007/BF01613611. S2CID 36886948.
Polynomials and polynomial functions
By degree
• Zero polynomial (degree undefined or −1 or −∞)
• Constant function (0)
• Linear function (1)
• Linear equation
• Quadratic function (2)
• Quadratic equation
• Cubic function (3)
• Cubic equation
• Quartic function (4)
• Quartic equation
• Quintic function (5)
• Sextic equation (6)
• Septic equation (7)
By properties
• Univariate
• Bivariate
• Multivariate
• Monomial
• Binomial
• Trinomial
• Irreducible
• Square-free
• Homogeneous
• Quasi-homogeneous
Tools and algorithms
• Factorization
• Greatest common divisor
• Division
• Horner's method of evaluation
• Resultant
• Discriminant
• Gröbner basis
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Pyramid (geometry)
In geometry, a pyramid (from Ancient Greek πυραμίς (puramís))[1][2] is a polyhedron formed by connecting a polygonal base and a point, called the apex. Each base edge and apex form a triangle, called a lateral face. It is a conic solid with polygonal base. A pyramid with an n-sided base has n + 1 vertices, n + 1 faces, and 2n edges. All pyramids are self-dual.
Regular-based right pyramids
Example: square pyramid
Facesn triangles
1 n-sided polygon
Edges2n
Verticesn + 1
Schläfli symbol( ) ∨ {n}
Conway notationYn
Symmetry groupCnv, [1,n], (*nn), order 2n
Rotation groupCn, [1,n]+, (nn), order n
Dual polyhedronself-dual
Propertiesconvex
A right pyramid has its apex directly above the centroid of its base. Nonright pyramids are called oblique pyramids. A regular pyramid has a regular polygon base and is usually implied to be a right pyramid.[3][4]
When unspecified, a pyramid is usually assumed to be a regular square pyramid, like the physical pyramid structures. A triangle-based pyramid is more often called a tetrahedron.
Among oblique pyramids, like acute and obtuse triangles, a pyramid can be called acute if its apex is above the interior of the base and obtuse if its apex is above the exterior of the base. A right-angled pyramid has its apex above an edge or vertex of the base. In a tetrahedron these qualifiers change based on which face is considered the base.
Pyramids are a class of the prismatoids. Pyramids can be doubled into bipyramids by adding a second offset point on the other side of the base plane.
Right pyramids with a regular base
A right pyramid with a regular base has isosceles triangle sides, with symmetry is Cnv or [1,n], with order 2n. It can be given an extended Schläfli symbol ( ) ∨ {n}, representing a point, ( ), joined (orthogonally offset) to a regular polygon, {n}. A join operation creates a new edge between all pairs of vertices of the two joined figures.[5]
The trigonal or triangular pyramid with all equilateral triangle faces becomes the regular tetrahedron, one of the Platonic solids. A lower symmetry case of the triangular pyramid is C3v, which has an equilateral triangle base, and 3 identical isosceles triangle sides. The square and pentagonal pyramids can also be composed of regular convex polygons, in which case they are Johnson solids.
If all edges of a square pyramid (or any convex polyhedron) are tangent to a sphere so that the average position of the tangential points are at the center of the sphere, then the pyramid is said to be canonical, and it forms half of a regular octahedron.
Pyramids with a hexagon or higher base must be composed of isosceles triangles. A hexagonal pyramid with equilateral triangles would be a completely flat figure, and a heptagonal or higher would have the triangles not meet at all.
Regular pyramids
Digonal Triangular Square Pentagonal Hexagonal Heptagonal Octagonal Enneagonal Decagonal...
Improper Regular Equilateral Isosceles
Right star pyramids
Right pyramids with regular star polygon bases are called star pyramids.[6] For example, the pentagrammic pyramid has a pentagram base and 5 intersecting triangle sides.
Right pyramids with an irregular base
A right pyramid can be named as ( )∨P, where ( ) is the apex point, ∨ is a join operator, and P is a base polygon.
An isosceles triangle right tetrahedron can be written as ( )∨[( )∨{ }] as the join of a point to an isosceles triangle base, as [( )∨( )]∨{ } or { }∨{ } as the join (orthogonal offsets) of two orthogonal segments, a digonal disphenoid, containing 4 isosceles triangle faces. It has C1v symmetry from two different base-apex orientations, and C2v in its full symmetry.
A rectangular right pyramid, written as ( )∨[{ }×{ }], and a rhombic pyramid, as ( )∨[{ }+{ }], both have symmetry C2v.
Right pyramids
Rectangular pyramid Rhombic pyramid
Volume
See also: Cone (geometry) – Volume
The volume of a pyramid (also any cone) is $ V={\tfrac {1}{3}}bh$, where b is the area of the base and h the height from the base to the apex. This works for any polygon, regular or non-regular, and any location of the apex, provided that h is measured as the perpendicular distance from the plane containing the base. In 499 AD Aryabhata, a mathematician-astronomer from the classical age of Indian mathematics and Indian astronomy, used this method in the Aryabhatiya (section 2.6).
The formula can be formally proved using calculus. By similarity, the linear dimensions of a cross-section parallel to the base increase linearly from the apex to the base. The scaling factor (proportionality factor) is $ 1-{\tfrac {y}{h}}$, or $ {\tfrac {h-y}{h}}$, where h is the height and y is the perpendicular distance from the plane of the base to the cross-section. Since the area of any cross-section is proportional to the square of the shape's scaling factor, the area of a cross-section at height y is $ b{\tfrac {(h-y)^{2}}{h^{2}}}$, or since both b and h are constants, $ {\tfrac {b}{h^{2}}}(h-y)^{2}$. The volume is given by the integral
${\frac {b}{h^{2}}}\int _{0}^{h}(h-y)^{2}\,dy={\frac {-b}{3h^{2}}}(h-y)^{3}{\bigg |}_{0}^{h}={\tfrac {1}{3}}bh.$
The same equation, $V={\tfrac {1}{3}}bh$, also holds for cones with any base. This can be proven by an argument similar to the one above; see volume of a cone.
For example, the volume of a pyramid whose base is an n-sided regular polygon with side length s and whose height is h is
$V={\frac {n}{12}}hs^{2}\cot {\frac {\pi }{n}}.$
The formula can also be derived exactly without calculus for pyramids with rectangular bases. Consider a unit cube. Draw lines from the center of the cube to each of the 8 vertices. This partitions the cube into 6 equal square pyramids of base area 1 and height 1/2. As 1 of 6 identical pyramids within the unit cube with volume 1, each pyramid clearly has volume of 1/6. If we assume that the volume formula will be proportional to both height and base, the proportionality constant must be 1/3. From this we deduce that pyramid volume = height × base area / 3.
Next, expand the cube uniformly in three directions by unequal amounts so that the resulting rectangular solid edges are a, b and c, with solid volume abc. Under our assumption of volume proportionality to height and base, each of the 6 pyramids within are likewise expanded. And each pyramid has the same volume abc/6. Since pairs of pyramids have heights a/2, b/2 and c/2, we see that pyramid volume = height × base area / 3 again.
When the side triangles are equilateral, the formula for the volume is
$V={\frac {1}{12}}ns^{3}\cot \left({\frac {\pi }{n}}\right){\sqrt {1-{\frac {1}{4\sin ^{2}{\tfrac {\pi }{n}}}}}}.$
This formula only applies for n = 2, 3, 4 and 5; and it also covers the case n = 6, for which the volume equals zero (i.e., the pyramid height is zero).
Surface area
The surface area of a pyramid is $ SA=B+{\tfrac {1}{2}}PL$, where B is the base area, P is the base perimeter, and the slant height $ L={\sqrt {h^{2}+r^{2}}}$, where h is the pyramid altitude and r is the inradius of the base.
Centroid
The centroid of a pyramid is located on the line segment that connects the apex to the centroid of the base. For a solid pyramid, the centroid is 1/4 the distance from the base to the apex.
n-dimensional pyramids
A 2-dimensional pyramid is a triangle, formed by a base edge connected to a noncolinear point called an apex.
A 4-dimensional pyramid is called a polyhedral pyramid, constructed by a polyhedron in a 3-space hyperplane of 4-space with another point off that hyperplane.
Higher-dimensional pyramids are constructed similarly.
The family of simplices represent pyramids in any dimension, increasing from triangle, tetrahedron, 5-cell, 5-simplex, etc. A n-dimensional simplex has the minimum n+1 vertices, with all pairs of vertices connected by edges, all triples of vertices defining faces, all quadruples of points defining tetrahedral cells, etc.
Polyhedral pyramid
In 4-dimensional geometry, a polyhedral pyramid is a 4-polytope constructed by a base polyhedron cell and an apex point. The lateral facets are pyramid cells, each constructed by one face of the base polyhedron and the apex. The vertices and edges of polyhedral pyramids form examples of apex graphs, graphs formed by adding one vertex (the apex) to a planar graph (the graph of the base). The dual of a polyhedral pyramid is another polyhedral pyramid, with a dual base.
The regular 5-cell (or 4-simplex) is an example of a tetrahedral pyramid. Uniform polyhedra with circumradii less than 1 can be make polyhedral pyramids with regular tetrahedral sides. A polyhedron with v vertices, e edges, and f faces can be the base on a polyhedral pyramid with v+1 vertices, e+v edges, f+e faces, and 1+f cells.
A 4D polyhedral pyramid with axial symmetry can be visualized in 3D with a Schlegel diagram—a 3D projection that places the apex at the center of the base polyhedron.
Equilateral uniform polyhedron-based pyramids (Schlegel diagram)
Symmetry [1,1,4] [1,2,3] [1,3,3] [1,4,3] [1,5,3]
Name Square-pyramidal pyramid Triangular prism pyramid Tetrahedral pyramid Cubic pyramid Octahedral pyramid Icosahedral pyramid
Segmentochora
index[7]
K4.4 K4.7 K4.1 K4.26.1 K4.3 K4.84
Height 0.707107 0.645497 0.790569 0.500000 0.707107 0.309017
Image
(Base)
Base Square
pyramid
Triangular
prism
Tetrahedron Cube Octahedron Icosahedron
Any convex 4-polytope can be divided into polyhedral pyramids by adding an interior point and creating one pyramid from each facet to the center point. This can be useful for computing volumes.
The 4-dimensional hypervolume of a polyhedral pyramid is 1/4 of the volume of the base polyhedron times its perpendicular height, compared to the area of a triangle being 1/2 the length of the base times the height and the volume of a pyramid being 1/3 the area of the base times the height.
The 3-dimensional surface volume of a polyhedral pyramid is $ SV=B+{\tfrac {1}{3}}AL$, where B is the base volume, A is the base surface area, and L is the slant height (height of the lateral pyramidal cells) $ L={\sqrt {h^{2}+r^{2}}}$, where h is the height and r is the inradius.
See also
• Bipyramid
• Cone (geometry)
• Trigonal pyramid (chemistry)
• Frustum
References
1. "Henry George Liddell, Robert Scott, A Greek-English Lexicon, πυραμίς", www.perseus.tufts.edu
2. The word meant "a kind of cake of roasted wheat-grains preserved in honey"; the Egyptian pyramids were named after its form. See Beekes, Robert S. (2009), Etymological Dictionary of Greek, Brill, p. 1261.
3. Kern, William F.; Bland, James R. (1938), Solid Mensuration with proofs, p. 46
4. Frye, Albert Irvin (1913), Civil Engineers' Pocket Book: A Reference-book for Engineers, Contractors, and Students, Containing Rules, Data, Methods, Formulas and Tables, D. Van Nostrand Company, p. 248
5. Johnson, Norman W. (2018), Geometries and Transformations, ISBN 978-1-107-10340-5. See Chapter 11: Finite symmetry groups, 11.3 Pyramids, Prisms, and Antiprisms
6. Wenninger, Magnus J. (1974), Polyhedron Models, Cambridge University Press, p. 50, ISBN 978-0-521-09859-5, archived from the original on 2013-12-11
7. Convex Segmentochora Archived 2014-04-19 at the Wayback Machine Dr. Richard Klitzing, Symmetry: Culture and Science, Vol. 11, Nos. 1–4, 139–181, 2000
External links
Wikimedia Commons has media related to Pyramids (geometry).
• Weisstein, Eric W. "Pyramid". MathWorld.
Convex polyhedra
Platonic solids (regular)
• tetrahedron
• cube
• octahedron
• dodecahedron
• icosahedron
Archimedean solids
(semiregular or uniform)
• truncated tetrahedron
• cuboctahedron
• truncated cube
• truncated octahedron
• rhombicuboctahedron
• truncated cuboctahedron
• snub cube
• icosidodecahedron
• truncated dodecahedron
• truncated icosahedron
• rhombicosidodecahedron
• truncated icosidodecahedron
• snub dodecahedron
Catalan solids
(duals of Archimedean)
• triakis tetrahedron
• rhombic dodecahedron
• triakis octahedron
• tetrakis hexahedron
• deltoidal icositetrahedron
• disdyakis dodecahedron
• pentagonal icositetrahedron
• rhombic triacontahedron
• triakis icosahedron
• pentakis dodecahedron
• deltoidal hexecontahedron
• disdyakis triacontahedron
• pentagonal hexecontahedron
Dihedral regular
• dihedron
• hosohedron
Dihedral uniform
• prisms
• antiprisms
duals:
• bipyramids
• trapezohedra
Dihedral others
• pyramids
• truncated trapezohedra
• gyroelongated bipyramid
• cupola
• bicupola
• frustum
• bifrustum
• rotunda
• birotunda
• prismatoid
• scutoid
Degenerate polyhedra are in italics.
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Algebra extension
In abstract algebra, an algebra extension is the ring-theoretic equivalent of a group extension.
For the ring-theoretic equivalent of a field extension, see Subring#Ring extensions.
Not to be confused with Algebraic extension.
Precisely, a ring extension of a ring R by an abelian group I is a pair (E, $\phi $) consisting of a ring E and a ring homomorphism $\phi $ that fits into the short exact sequence of abelian groups:
$0\to I\to E{\overset {\phi }{{}\to {}}}R\to 0.$[1]
Note I is then isomorphic to a two-sided ideal of E. Given a commutative ring A, an A-extension or an extension of an A-algebra is defined in the same way by replacing "ring" with "algebra over A" and "abelian groups" with "A-modules".
An extension is said to be trivial or to split if $\phi $ splits; i.e., $\phi $ admits a section that is a ring homomorphism.[2] (see § Example: trivial extension).
A morphism between extensions of R by I, over say A, is an algebra homomorphism E → E' that induces the identities on I and R. By the five lemma, such a morphism is necessarily an isomorphism, and so two extensions are equivalent if there is a morphism between them.
Example: trivial extension
Let R be a commutative ring and M an R-module. Let E = R ⊕ M be the direct sum of abelian groups. Define the multiplication on E by
$(a,x)\cdot (b,y)=(ab,ay+bx).$
Note that identifying (a, x) with a + εx where ε squares to zero and expanding out (a + εx)(b + εy) yields the above formula; in particular we see that E is a ring. It is sometimes called the algebra of dual numbers. Alternatively, E can be defined as $\operatorname {Sym} (M)/\bigoplus _{n\geq 2}\operatorname {Sym} ^{n}(M)$ where $\operatorname {Sym} (M)$ is the symmetric algebra of M.[3] We then have the short exact sequence
$0\to M\to E{\overset {p}{{}\to {}}}R\to 0$
where p is the projection. Hence, E is an extension of R by M. It is trivial since $r\mapsto (r,0)$ is a section (note this section is a ring homomorphism since $(1,0)$ is the multiplicative identity of E). Conversely, every trivial extension E of R by I is isomorphic to $R\oplus I$ if $I^{2}=0$. Indeed, identifying $R$ as a subring of E using a section, we have $(E,\phi )\simeq (R\oplus I,p)$ via $e\mapsto (\phi (e),e-\phi (e))$.[1]
One interesting feature of this construction is that the module M becomes an ideal of some new ring. In his book Local Rings, Nagata calls this process the principle of idealization.[4]
Square-zero extension
Especially in deformation theory, it is common to consider an extension R of a ring (commutative or not) by an ideal whose square is zero. Such an extension is called a square-zero extension, a square extension or just an extension. For a square-zero ideal I, since I is contained in the left and right annihilators of itself, I is a $R/I$-bimodule.
More generally, an extension by a nilpotent ideal is called a nilpotent extension. For example, the quotient $R\to R_{\mathrm {red} }$ of a Noetherian commutative ring by the nilradical is a nilpotent extension.
In general,
$0\to I^{n}/I^{n-1}\to R/I^{n-1}\to R/I^{n}\to 0$
is a square-zero extension. Thus, a nilpotent extension breaks up into successive square-zero extensions. Because of this, it is usually enough to study square-zero extensions in order to understand nilpotent extensions.
See also
• Formally smooth map
• The Wedderburn principal theorem, a statement about an extension by the Jacobson radical.
References
1. Sernesi 2007, 1.1.1.
2. Typical references require sections be homomorphisms without elaborating whether 1 is preserved. But since we need to be able to identify R as a subring of E (see the trivial extension example), it seems 1 needs to be preserved.
3. Anderson, D. D.; Winders, M. (March 2009). "Idealization of a Module". Journal of Commutative Algebra. 1 (1): 3–56. doi:10.1216/JCA-2009-1-1-3. ISSN 1939-2346. S2CID 120720674.
4. Nagata, Masayoshi (1962), Local Rings, Interscience Tracts in Pure and Applied Mathematics, vol. 13, New York-London: Interscience Publishers a division of John Wiley & Sons, ISBN 0-88275-228-6, MR 0155856
• Sernesi, Edoardo (20 April 2007). Deformations of Algebraic Schemes. Springer Science & Business Media. ISBN 978-3-540-30615-3.
Further reading
• algebra extension at nLab
• infinitesimal extension at nLab
• Extension of an associative algebra at Encyclopedia of Mathematics
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Square
In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length adjacent sides. It is the only regular polygon whose internal angle, central angle, and external angle are all equal (90°), and whose diagonals are all equal in length. A square with vertices ABCD would be denoted $\square $ ABCD.[1]
Square
A regular quadrilateral
TypeRegular polygon
Edges and vertices4
Schläfli symbol{4}
Coxeter–Dynkin diagrams
Symmetry groupDihedral (D4), order 2×4
Internal angle (degrees)90°
PropertiesConvex, cyclic, equilateral, isogonal, isotoxal
Dual polygonSelf
Characterizations
A quadrilateral is a square if and only if it is any one of the following:[2][3]
• A rectangle with two adjacent equal sides
• A rhombus with a right vertex angle
• A rhombus with all angles equal
• A parallelogram with one right vertex angle and two adjacent equal sides
• A quadrilateral with four equal sides and four right angles
• A quadrilateral where the diagonals are equal, and are the perpendicular bisectors of each other (i.e., a rhombus with equal diagonals)
• A convex quadrilateral with successive sides a, b, c, d whose area is $A={\tfrac {1}{2}}(a^{2}+c^{2})={\tfrac {1}{2}}(b^{2}+d^{2}).$[4]: Corollary 15
Properties
A square is a special case of a rhombus (equal sides, opposite equal angles), a kite (two pairs of adjacent equal sides), a trapezoid (one pair of opposite sides parallel), a parallelogram (all opposite sides parallel), a quadrilateral or tetragon (four-sided polygon), and a rectangle (opposite sides equal, right-angles), and therefore has all the properties of all these shapes, namely:[5]
• All four internal angles of a square are equal (each being 360°/4 = 90°, a right angle).
• The central angle of a square is equal to 90° (360°/4).
• The external angle of a square is equal to 90°.
• The diagonals of a square are equal and bisect each other, meeting at 90°.
• The diagonal of a square bisects its internal angle, forming adjacent angles of 45°.
• All four sides of a square are equal.
• Opposite sides of a square are parallel.
• A square has Schläfli symbol {4}. A truncated square, t{4}, is an octagon, {8}. An alternated square, h{4}, is a digon, {2}.
• The square is the n = 2 case of the families of n-hypercubes and n-orthoplexes.
Perimeter and area
The perimeter of a square whose four sides have length $\ell $ is
$P=4\ell $
and the area A is
$A=\ell ^{2}.$[1]
Since four squared equals sixteen, a four by four square has an area equal to its perimeter. The only other quadrilateral with such a property is that of a three by six rectangle.
In classical times, the second power was described in terms of the area of a square, as in the above formula. This led to the use of the term square to mean raising to the second power.
The area can also be calculated using the diagonal d according to
$A={\frac {d^{2}}{2}}.$
In terms of the circumradius R, the area of a square is
$A=2R^{2};$
since the area of the circle is $\pi R^{2},$ the square fills $2/\pi \approx 0.6366$ of its circumscribed circle.
In terms of the inradius r, the area of the square is
$A=4r^{2};$
hence the area of the inscribed circle is $\pi /4\approx 0.7854$ of that of the square.
Because it is a regular polygon, a square is the quadrilateral of least perimeter enclosing a given area. Dually, a square is the quadrilateral containing the largest area within a given perimeter.[6] Indeed, if A and P are the area and perimeter enclosed by a quadrilateral, then the following isoperimetric inequality holds:
$16A\leq P^{2}$
with equality if and only if the quadrilateral is a square.
Other facts
• The diagonals of a square are ${\sqrt {2}}$ (about 1.414) times the length of a side of the square. This value, known as the square root of 2 or Pythagoras' constant,[1] was the first number proven to be irrational.
• A square can also be defined as a parallelogram with equal diagonals that bisect the angles.
• If a figure is both a rectangle (right angles) and a rhombus (equal edge lengths), then it is a square.
• A square has a larger area than any other quadrilateral with the same perimeter.[7]
• A square tiling is one of three regular tilings of the plane (the others are the equilateral triangle and the regular hexagon).
• The square is in two families of polytopes in two dimensions: hypercube and the cross-polytope. The Schläfli symbol for the square is {4}.
• The square is a highly symmetric object. There are four lines of reflectional symmetry and it has rotational symmetry of order 4 (through 90°, 180° and 270°). Its symmetry group is the dihedral group D4.
• A square can be inscribed inside any regular polygon. The only other polygon with this property is the equilateral triangle.
• If the inscribed circle of a square ABCD has tangency points E on AB, F on BC, G on CD, and H on DA, then for any point P on the inscribed circle,[8]
$2(PH^{2}-PE^{2})=PD^{2}-PB^{2}.$
• If $d_{i}$ is the distance from an arbitrary point in the plane to the i-th vertex of a square and $R$ is the circumradius of the square, then[9]
${\frac {d_{1}^{4}+d_{2}^{4}+d_{3}^{4}+d_{4}^{4}}{4}}+3R^{4}=\left({\frac {d_{1}^{2}+d_{2}^{2}+d_{3}^{2}+d_{4}^{2}}{4}}+R^{2}\right)^{2}.$
• If $L$ and $d_{i}$ are the distances from an arbitrary point in the plane to the centroid of the square and its four vertices respectively, then [10]
$d_{1}^{2}+d_{3}^{2}=d_{2}^{2}+d_{4}^{2}=2(R^{2}+L^{2})$
and
$d_{1}^{2}d_{3}^{2}+d_{2}^{2}d_{4}^{2}=2(R^{4}+L^{4}),$
where $R$ is the circumradius of the square.
Coordinates and equations
The coordinates for the vertices of a square with vertical and horizontal sides, centered at the origin and with side length 2 are (±1, ±1), while the interior of this square consists of all points (xi, yi) with −1 < xi < 1 and −1 < yi < 1. The equation
$\max(x^{2},y^{2})=1$
specifies the boundary of this square. This equation means "x2 or y2, whichever is larger, equals 1." The circumradius of this square (the radius of a circle drawn through the square's vertices) is half the square's diagonal, and is equal to ${\sqrt {2}}.$ Then the circumcircle has the equation
$x^{2}+y^{2}=2.$
Alternatively the equation
$\left|x-a\right|+\left|y-b\right|=r.$
can also be used to describe the boundary of a square with center coordinates (a, b), and a horizontal or vertical radius of r. The square is therefore the shape of a topological ball according to the L1 distance metric.
Construction
The following animations show how to construct a square using a compass and straightedge. This is possible as 4 = 22, a power of two.
Square at a given side length,
right angle by using Thales' theorem
Square at a given diagonal
Symmetry
The square has Dih4 symmetry, order 8. There are 2 dihedral subgroups: Dih2, Dih1, and 3 cyclic subgroups: Z4, Z2, and Z1.
A square is a special case of many lower symmetry quadrilaterals:
• A rectangle with two adjacent equal sides
• A quadrilateral with four equal sides and four right angles
• A parallelogram with one right angle and two adjacent equal sides
• A rhombus with a right angle
• A rhombus with all angles equal
• A rhombus with equal diagonals
These 6 symmetries express 8 distinct symmetries on a square. John Conway labels these by a letter and group order.[11]
Each subgroup symmetry allows one or more degrees of freedom for irregular quadrilaterals. r8 is full symmetry of the square, and a1 is no symmetry. d4 is the symmetry of a rectangle, and p4 is the symmetry of a rhombus. These two forms are duals of each other, and have half the symmetry order of the square. d2 is the symmetry of an isosceles trapezoid, and p2 is the symmetry of a kite. g2 defines the geometry of a parallelogram.
Only the g4 subgroup has no degrees of freedom, but can seen as a square with directed edges.
Squares inscribed in triangles
Main article: Triangle § Squares
Every acute triangle has three inscribed squares (squares in its interior such that all four of a square's vertices lie on a side of the triangle, so two of them lie on the same side and hence one side of the square coincides with part of a side of the triangle). In a right triangle two of the squares coincide and have a vertex at the triangle's right angle, so a right triangle has only two distinct inscribed squares. An obtuse triangle has only one inscribed square, with a side coinciding with part of the triangle's longest side.
The fraction of the triangle's area that is filled by the square is no more than 1/2.
Squaring the circle
Squaring the circle, proposed by ancient geometers, is the problem of constructing a square with the same area as a given circle, by using only a finite number of steps with compass and straightedge.
In 1882, the task was proven to be impossible as a consequence of the Lindemann–Weierstrass theorem, which proves that pi (π) is a transcendental number rather than an algebraic irrational number; that is, it is not the root of any polynomial with rational coefficients.
Non-Euclidean geometry
In non-Euclidean geometry, squares are more generally polygons with 4 equal sides and equal angles.
In spherical geometry, a square is a polygon whose edges are great circle arcs of equal distance, which meet at equal angles. Unlike the square of plane geometry, the angles of such a square are larger than a right angle. Larger spherical squares have larger angles.
In hyperbolic geometry, squares with right angles do not exist. Rather, squares in hyperbolic geometry have angles of less than right angles. Larger hyperbolic squares have smaller angles.
Examples:
Two squares can tile the sphere with 2 squares around each vertex and 180-degree internal angles. Each square covers an entire hemisphere and their vertices lie along a great circle. This is called a spherical square dihedron. The Schläfli symbol is {4,2}.
Six squares can tile the sphere with 3 squares around each vertex and 120-degree internal angles. This is called a spherical cube. The Schläfli symbol is {4,3}.
Squares can tile the hyperbolic plane with 5 around each vertex, with each square having 72-degree internal angles. The Schläfli symbol is {4,5}. In fact, for any n ≥ 5 there is a hyperbolic tiling with n squares about each vertex.
Crossed square
A crossed square is a faceting of the square, a self-intersecting polygon created by removing two opposite edges of a square and reconnecting by its two diagonals. It has half the symmetry of the square, Dih2, order 4. It has the same vertex arrangement as the square, and is vertex-transitive. It appears as two 45-45-90 triangle with a common vertex, but the geometric intersection is not considered a vertex.
A crossed square is sometimes likened to a bow tie or butterfly. the crossed rectangle is related, as a faceting of the rectangle, both special cases of crossed quadrilaterals.[12]
The interior of a crossed square can have a polygon density of ±1 in each triangle, dependent upon the winding orientation as clockwise or counterclockwise.
A square and a crossed square have the following properties in common:
• Opposite sides are equal in length.
• The two diagonals are equal in length.
• It has two lines of reflectional symmetry and rotational symmetry of order 2 (through 180°).
It exists in the vertex figure of a uniform star polyhedra, the tetrahemihexahedron.
Graphs
The K4 complete graph is often drawn as a square with all 6 possible edges connected, hence appearing as a square with both diagonals drawn. This graph also represents an orthographic projection of the 4 vertices and 6 edges of the regular 3-simplex (tetrahedron).
See also
• Cube
• Pythagorean theorem
• Square lattice
• Square number
• Square root
• Squaring the square
• Squircle
• Unit square
References
1. Weisstein, Eric W. "Square". mathworld.wolfram.com. Retrieved 2020-09-02.
2. Zalman Usiskin and Jennifer Griffin, "The Classification of Quadrilaterals. A Study of Definition", Information Age Publishing, 2008, p. 59, ISBN 1-59311-695-0.
3. "Problem Set 1.3". jwilson.coe.uga.edu. Retrieved 2017-12-12.
4. Josefsson, Martin, "Properties of equidiagonal quadrilaterals" Forum Geometricorum, 14 (2014), 129–144.
5. "Quadrilaterals - Square, Rectangle, Rhombus, Trapezoid, Parallelogram". www.mathsisfun.com. Retrieved 2020-09-02.
6. Chakerian, G.D. "A Distorted View of Geometry." Ch. 7 in Mathematical Plums (R. Honsberger, editor). Washington, DC: Mathematical Association of America, 1979: 147.
7. Lundsgaard Hansen, Martin. "Vagn Lundsgaard Hansen". www2.mat.dtu.dk. Retrieved 2017-12-12.
8. "Geometry classes, Problem 331. Square, Point on the Inscribed Circle, Tangency Points. Math teacher Master Degree. College, SAT Prep. Elearning, Online math tutor, LMS". gogeometry.com. Retrieved 2017-12-12.
9. Park, Poo-Sung. "Regular polytope distances", Forum Geometricorum 16, 2016, 227–232. http://forumgeom.fau.edu/FG2016volume16/FG201627.pdf
10. Meskhishvili, Mamuka (2021). "Cyclic Averages of Regular Polygonal Distances" (PDF). International Journal of Geometry. 10: 58–65.
11. John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, (2008) The Symmetries of Things, ISBN 978-1-56881-220-5 (Chapter 20, Generalized Schaefli symbols, Types of symmetry of a polygon pp. 275–278)
12. Wells, Christopher J. "Quadrilaterals". www.technologyuk.net. Retrieved 2017-12-12.
External links
Wikimedia Commons has media related to Square (geometry).
• Animated course (Construction, Circumference, Area)
• Definition and properties of a square With interactive applet
• Animated applet illustrating the area of a square
Polygons (List)
Triangles
• Acute
• Equilateral
• Ideal
• Isosceles
• Kepler
• Obtuse
• Right
Quadrilaterals
• Antiparallelogram
• Bicentric
• Crossed
• Cyclic
• Equidiagonal
• Ex-tangential
• Harmonic
• Isosceles trapezoid
• Kite
• Orthodiagonal
• Parallelogram
• Rectangle
• Right kite
• Right trapezoid
• Rhombus
• Square
• Tangential
• Tangential trapezoid
• Trapezoid
By number
of sides
1–10 sides
• Monogon (1)
• Digon (2)
• Triangle (3)
• Quadrilateral (4)
• Pentagon (5)
• Hexagon (6)
• Heptagon (7)
• Octagon (8)
• Nonagon (Enneagon, 9)
• Decagon (10)
11–20 sides
• Hendecagon (11)
• Dodecagon (12)
• Tridecagon (13)
• Tetradecagon (14)
• Pentadecagon (15)
• Hexadecagon (16)
• Heptadecagon (17)
• Octadecagon (18)
• Icosagon (20)
>20 sides
• Icositrigon (23)
• Icositetragon (24)
• Triacontagon (30)
• 257-gon
• Chiliagon (1000)
• Myriagon (10,000)
• 65537-gon
• Megagon (1,000,000)
• Apeirogon (∞)
Star polygons
• Pentagram
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• Heptagram
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• Dodecagram
Classes
• Concave
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• Isogonal
• Isotoxal
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• Rectilinear
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• Reinhardt
• Simple
• Skew
• Star-shaped
• Tangential
• Weakly simple
Fundamental convex regular and uniform polytopes in dimensions 2–10
Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron Octahedron • Cube Demicube Dodecahedron • Icosahedron
Uniform polychoron Pentachoron 16-cell • Tesseract Demitesseract 24-cell 120-cell • 600-cell
Uniform 5-polytope 5-simplex 5-orthoplex • 5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex • 6-cube 6-demicube 122 • 221
Uniform 7-polytope 7-simplex 7-orthoplex • 7-cube 7-demicube 132 • 231 • 321
Uniform 8-polytope 8-simplex 8-orthoplex • 8-cube 8-demicube 142 • 241 • 421
Uniform 9-polytope 9-simplex 9-orthoplex • 9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex • 10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplex • n-cube n-demicube 1k2 • 2k1 • k21 n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds
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Polyomino
A polyomino is a plane geometric figure formed by joining one or more equal squares edge to edge. It is a polyform whose cells are squares. It may be regarded as a finite subset of the regular square tiling.
"Polyominoes" redirects here. For the book by Solomon Golomb, see Polyominoes: Puzzles, Patterns, Problems, and Packings.
Polyominoes have been used in popular puzzles since at least 1907, and the enumeration of pentominoes is dated to antiquity.[1] Many results with the pieces of 1 to 6 squares were first published in Fairy Chess Review between the years 1937 to 1957, under the name of "dissection problems." The name polyomino was invented by Solomon W. Golomb in 1953,[2] and it was popularized by Martin Gardner in a November 1960 "Mathematical Games" column in Scientific American.[3]
Related to polyominoes are polyiamonds, formed from equilateral triangles; polyhexes, formed from regular hexagons; and other plane polyforms. Polyominoes have been generalized to higher dimensions by joining cubes to form polycubes, or hypercubes to form polyhypercubes.
In statistical physics, the study of polyominoes and their higher-dimensional analogs (which are often referred to as lattice animals in this literature) is applied to problems in physics and chemistry. Polyominoes have been used as models of branched polymers and of percolation clusters.[4]
Like many puzzles in recreational mathematics, polyominoes raise many combinatorial problems. The most basic is enumerating polyominoes of a given size. No formula has been found except for special classes of polyominoes. A number of estimates are known, and there are algorithms for calculating them.
Polyominoes with holes are inconvenient for some purposes, such as tiling problems. In some contexts polyominoes with holes are excluded, allowing only simply connected polyominoes.[5]
Enumeration of polyominoes
Free, one-sided, and fixed polyominoes
There are three common ways of distinguishing polyominoes for enumeration:[6][7]
• free polyominoes are distinct when none is a rigid transformation (translation, rotation, reflection or glide reflection) of another (pieces that can be picked up and flipped over). Translating, rotating, reflecting, or glide reflecting a free polyomino does not change its shape.
• one-sided polyominoes are distinct when none is a translation or rotation of another (pieces that cannot be flipped over). Translating or rotating a one-sided polyomino does not change its shape.
• fixed polyominoes are distinct when none is a translation of another (pieces that can be neither flipped nor rotated). Translating a fixed polyomino will not change its shape.
The following table shows the numbers of polyominoes of various types with n cells.
n name free one-sided fixed
total with holes without holes
1monomino10111
2domino10112
3tromino20226
4tetromino505719
5pentomino120121863
6hexomino3503560216
7heptomino1081107196760
8octomino36963637042,725
9nonomino1,285371,2482,5009,910
10decomino4,6551954,4609,18936,446
11undecomino17,07397916,09433,896135,268
12dodecomino63,6004,66358,937126,759505,861
OEIS sequence A000105 A001419 A000104 A000988 A001168
As of 2004, Iwan Jensen has enumerated the fixed polyominoes up to n = 56 – approximately 6.915×1031.[8]
Free polyominoes were enumerated in 2007 up to n = 28 by Tomás Oliveira e Silva,[9] in 2012 up to n = 45 by Toshihiro Shirakawa,[10] and in 2023 up to n = 50 by John Mason.[11]
The above OEIS sequences, with the exception of A001419, include the count of 1 for the number of null-polyominoes; a null-polyomino is one that is formed of zero squares.
Symmetries of polyominoes
The dihedral group D4 is the group of symmetries (symmetry group) of a square. This group contains four rotations and four reflections. It is generated by alternating reflections about the x-axis and about a diagonal. One free polyomino corresponds to at most 8 fixed polyominoes, which are its images under the symmetries of D4. However, those images are not necessarily distinct: the more symmetry a free polyomino has, the fewer distinct fixed counterparts it has. Therefore, a free polyomino that is invariant under some or all non-trivial symmetries of D4 may correspond to only 4, 2 or 1 fixed polyominoes. Mathematically, free polyominoes are equivalence classes of fixed polyominoes under the group D4.
Polyominoes have the following possible symmetries;[12] the least number of squares needed in a polyomino with that symmetry is given in each case:
• 8 fixed polyominoes for each free polyomino:
• no symmetry (4)
• 4 fixed polyominoes for each free polyomino:
• mirror symmetry with respect to one of the grid line directions (4)
• mirror symmetry with respect to a diagonal line (3)
• 2-fold rotational symmetry: C2 (4)
• 2 fixed polyominoes for each free polyomino:
• symmetry with respect to both grid line directions, and hence also 2-fold rotational symmetry: D2 (2) (also known as the Klein four-group)
• symmetry with respect to both diagonal directions, and hence also 2-fold rotational symmetry: D2 (7)
• 4-fold rotational symmetry: C4 (8)
• 1 fixed polyomino for each free polyomino:
• all symmetry of the square: D4 (1).
In the same way, the number of one-sided polyominoes depends on polyomino symmetry as follows:
• 2 one-sided polyominoes for each free polyomino:
• no symmetry
• 2-fold rotational symmetry: C2
• 4-fold rotational symmetry: C4
• 1 one-sided polyomino for each free polyomino:
• all symmetry of the square: D4
• mirror symmetry with respect to one of the grid line directions
• mirror symmetry with respect to a diagonal line
• symmetry with respect to both grid line directions, and hence also 2-fold rotational symmetry: D2
• symmetry with respect to both diagonal directions, and hence also 2-fold rotational symmetry: D2.
The following table shows the numbers of polyominoes with n squares, sorted by symmetry groups.
n none mirror
90°
mirror
45°
C2 D2
90°
D2
45°
C4 D4
100000001
200001000
300101000
411011001
552211001
6206252000
7849743100
8316235184111
91,1963826194002
104,4619022738100
1116,750147917310200
1262,8783417927815333
OEIS sequence A006749 A006746 A006748 A006747 A056877 A056878 A144553 A142886
[13]
Inductive algorithms
Each polyomino of size n+1 can be obtained by adding a square to a polyomino of size n. This leads to algorithms for generating polyominoes inductively.
Most simply, given a list of polyominoes of size n, squares may be added next to each polyomino in each possible position, and the resulting polyomino of size n+1 added to the list if not a duplicate of one already found; refinements in ordering the enumeration and marking adjacent squares that should not be considered reduce the number of cases that need to be checked for duplicates.[14] This method may be used to enumerate either free or fixed polyominoes.
A more sophisticated method, described by Redelmeier, has been used by many authors as a way of not only counting polyominoes (without requiring that all polyominoes of size n be stored in size to enumerate those of size n+1), but also proving upper bounds on their number. The basic idea is that we begin with a single square, and from there, recursively add squares. Depending on the details, it may count each n-omino n times, once from starting from each of its n squares, or may be arranged to count each once only.
The simplest implementation involves adding one square at a time. Beginning with an initial square, number the adjacent squares, clockwise from the top, 1, 2, 3, and 4. Now pick a number between 1 and 4, and add a square at that location. Number the unnumbered adjacent squares, starting with 5. Then, pick a number larger than the previously picked number, and add that square. Continue picking a number larger than the number of the current square, adding that square, and then numbering the new adjacent squares. When n squares have been created, an n-omino has been created.
This method ensures that each fixed polyomino is counted exactly n times, once for each starting square. It can be optimized so that it counts each polyomino only once, rather than n times. Starting with the initial square, declare it to be the lower-left square of the polyomino. Simply do not number any square that is on a lower row, or left of the square on the same row. This is the version described by Redelmeier.
If one wishes to count free polyominoes instead, then one may check for symmetries after creating each n-omino. However, it is faster[15] to generate symmetric polyominoes separately (by a variation of this method)[16] and so determine the number of free polyominoes by Burnside's lemma.
Transfer-matrix method
The most modern algorithm for enumerating the fixed polyominoes was discovered by Iwan Jensen.[17] An improvement on Andrew Conway's method,[18] it is exponentially faster than the previous methods (however, its running time is still exponential in n).
Both Conway's and Jensen's versions of the transfer-matrix method involve counting the number of polyominoes that have a certain width. Computing the number for all widths gives the total number of polyominoes. The basic idea behind the method is that possible beginning rows are considered, and then to determine the minimum number of squares needed to complete the polyomino of the given width. Combined with the use of generating functions, this technique is able to count many polyominoes at once, thus enabling it to run many times faster than methods that have to generate every polyomino.
Although it has excellent running time, the tradeoff is that this algorithm uses exponential amounts of memory (many gigabytes of memory are needed for n above 50), is much harder to program than the other methods, and can't currently be used to count free polyominoes.
Fixed polyominoes
Theoretical arguments and numerical calculations support the estimate for the number of fixed polyominoes of size n
$A_{n}\sim {\frac {c\lambda ^{n}}{n}}$
where λ = 4.0626 and c = 0.3169.[19] However, this result is not proven and the values of λ and c are only estimates.
The known theoretical results are not nearly as specific as this estimate. It has been proven that
$\lim _{n\rightarrow \infty }(A_{n})^{\frac {1}{n}}=\lambda $
exists. In other words, An grows exponentially. The best known lower bound for λ, found in 2016, is 4.00253.[20] The best known upper bound is λ < 4.5252.[21]
To establish a lower bound, a simple but highly effective method is concatenation of polyominoes. Define the upper-right square to be the rightmost square in the uppermost row of the polyomino. Define the bottom-left square similarly. Then, the upper-right square of any polyomino of size n can be attached to the bottom-left square of any polyomino of size m to produce a unique (n+m)-omino. This proves AnAm ≤ An+m. Using this equation, one can show λ ≥ (An)1/n for all n. Refinements of this procedure combined with data for An produce the lower bound given above.
The upper bound is attained by generalizing the inductive method of enumerating polyominoes. Instead of adding one square at a time, one adds a cluster of squares at a time. This is often described as adding twigs. By proving that every n-omino is a sequence of twigs, and by proving limits on the combinations of possible twigs, one obtains an upper bound on the number of n-ominoes. For example, in the algorithm outlined above, at each step we must choose a larger number, and at most three new numbers are added (since at most three unnumbered squares are adjacent to any numbered square). This can be used to obtain an upper bound of 6.75. Using 2.8 million twigs, Klarner and Rivest obtained an upper bound of 4.65,[22] which was subsequently improved by Barequet and Shalah to 4.5252.[21]
Free polyominoes
Approximations for the number of fixed polyominoes and free polyominoes are related in a simple way. A free polyomino with no symmetries (rotation or reflection) corresponds to 8 distinct fixed polyominoes, and for large n, most n-ominoes have no symmetries. Therefore, the number of fixed n-ominoes is approximately 8 times the number of free n-ominoes. Moreover, this approximation is exponentially more accurate as n increases.[12]
Special classes of polyominoes
Exact formulas are known for enumerating polyominoes of special classes, such as the class of convex polyominoes and the class of directed polyominoes.
The definition of a convex polyomino is different from the usual definition of convexity, but is similar to the definition used for the orthogonal convex hull. A polyomino is said to be vertically or column convex if its intersection with any vertical line is convex (in other words, each column has no holes). Similarly, a polyomino is said to be horizontally or row convex if its intersection with any horizontal line is convex. A polyomino is said to be convex if it is row and column convex.[23]
A polyomino is said to be directed if it contains a square, known as the root, such that every other square can be reached by movements of up or right one square, without leaving the polyomino.
Directed polyominoes,[24] column (or row) convex polyominoes,[25] and convex polyominoes[26] have been effectively enumerated by area n, as well as by some other parameters such as perimeter, using generating functions.
A polyomino is equable if its area equals its perimeter. An equable polyomino must be made from an even number of squares; every even number greater than 15 is possible. For instance, the 16-omino in the form of a 4 × 4 square and the 18-omino in the form of a 3 × 6 rectangle are both equable. For polyominoes with fewer than 15 squares, the perimeter always exceeds the area.[27]
Tiling with polyominoes
In recreational mathematics, challenges are often posed for tiling a prescribed region, or the entire plane, with polyominoes,[28] and related problems are investigated in mathematics and computer science.
Tiling regions with sets of polyominoes
Puzzles commonly ask for tiling a given region with a given set of polyominoes, such as the 12 pentominoes. Golomb's and Gardner's books have many examples. A typical puzzle is to tile a 6×10 rectangle with the twelve pentominoes; the 2339 solutions to this were found in 1960.[29] Where multiple copies of the polyominoes in the set are allowed, Golomb defines a hierarchy of different regions that a set may be able to tile, such as rectangles, strips, and the whole plane, and shows that whether polyominoes from a given set can tile the plane is undecidable, by mapping sets of Wang tiles to sets of polyominoes.[30]
Because the general problem of tiling regions of the plane with sets of polyominoes is NP-complete,[31] tiling with more than a few pieces rapidly becomes intractable and so the aid of a computer is required. The traditional approach to tiling finite regions of the plane uses a technique in computer science called backtracking.[32]
In Jigsaw Sudokus a square grid is tiled with polynomino-shaped regions (sequence A172477 in the OEIS).
Tiling regions with copies of a single polyomino
Another class of problems asks whether copies of a given polyomino can tile a rectangle, and if so, what rectangles they can tile.[33] These problems have been extensively studied for particular polyominoes,[34] and tables of results for individual polyominoes are available.[35] Klarner and Göbel showed that for any polyomino there is a finite set of prime rectangles it tiles, such that all other rectangles it tiles can be tiled by those prime rectangles.[36][37] Kamenetsky and Cooke showed how various disjoint (called "holey") polyominoes can tile rectangles.[38]
Beyond rectangles, Golomb gave his hierarchy for single polyominoes: a polyomino may tile a rectangle, a half strip, a bent strip, an enlarged copy of itself, a quadrant, a strip, a half plane, the whole plane, certain combinations, or none of these. There are certain implications among these, both obvious (for example, if a polyomino tiles the half plane then it tiles the whole plane) and less so (for example, if a polyomino tiles an enlarged copy of itself, then it tiles the quadrant). Polyominoes of size up to 6 are characterized in this hierarchy (with the status of one hexomino, later found to tile a rectangle, unresolved at that time).[39]
In 2001 Cristopher Moore and John Michael Robson showed that the problem of tiling one polyomino with copies of another is NP-complete.[40][41]
Tiling the plane with copies of a single polyomino
Tiling the plane with copies of a single polyomino has also been much discussed. It was noted in 1965 that all polyominoes up to hexominoes[42] and all but four heptominoes tile the plane.[43] It was then established by David Bird that all but 26 octominoes tile the plane.[44] Rawsthorne found that all but 235 polyominoes of size 9 tile,[45] and such results have been extended to higher area by Rhoads (to size 14)[46] and others. Polyominoes tiling the plane have been classified by the symmetries of their tilings and by the number of aspects (orientations) in which the tiles appear in them.[47][48]
The study of which polyominoes can tile the plane has been facilitated using the Conway criterion: except for two nonominoes, all tiling polyominoes up to size 9 form a patch of at least one tile satisfying it, with higher-size exceptions more frequent.[49]
Several polyominoes can tile larger copies of themselves, and repeating this process recursively gives a rep-tile tiling of the plane. For instance, for every positive integer n, it is possible to combine n2 copies of the L-tromino, L-tetromino, or P-pentomino into a single larger shape similar to the smaller polyomino from which it was formed.[50]
Tiling a common figure with various polyominoes
The compatibility problem is to take two or more polyominoes and find a figure that can be tiled with each. Polyomino compatibility has been widely studied since the 1990s. Jorge Luis Mireles and Giovanni Resta have published websites of systematic results,[51][52] and Livio Zucca shows results for some complicated cases like three different pentominoes.[53] The general problem can be hard. The first compatibility figure for the L and X pentominoes was published in 2005 and had 80 tiles of each kind.[54] Many pairs of polyominoes have been proved incompatible by systematic exhaustion. No algorithm is known for deciding whether two arbitrary polyominoes are compatible.
Polyominoes in puzzles and games
In addition to the tiling problems described above, there are recreational mathematics puzzles that require folding a polyomino to create other shapes. Gardner proposed several simple games with a set of free pentominoes and a chessboard. Some variants of the Sudoku puzzle use nonomino-shaped regions on the grid. The video game Tetris is based on the seven one-sided tetrominoes (spelled "Tetriminos" in the game), and the board game Blokus uses all of the free polyominoes up to pentominoes.
Etymology
The word polyomino and the names of the various sizes of polyomino are all back-formations from the word domino, a common game piece consisting of two squares, with the first letter d- fancifully interpreted as a version of the prefix di- meaning "two." The name domino for the game piece is believed to come from the spotted masquerade garment domino, from Latin dominus.[55]
Most of the numerical prefixes are Greek. Polyominoes of size 9 and 11 more often take the Latin prefixes nona- (nonomino) and undeca- (undecomino) than the Greek prefixes ennea- (enneomino) and hendeca- (hendecomino).
See also
• Percolation theory, the mathematical study of random subsets of integer grids. The finite connected components of these subsets form polyominoes.
• Young diagram, a special kind of polyomino used in number theory to describe integer partitions and in group theory and applications in mathematical physics to describe representations of the symmetric group.
• Blokus, a board game using polyominoes.
• Squaregraph, a kind of undirected graph including as a special case the graphs of vertices and edges of polyominoes.
• Polycube, its analogue in three dimensions.
Notes
1. Golomb (Polyominoes, Preface to the First Edition) writes "the observation that there are twelve distinctive patterns (the pentominoes) that can be formed by five connected stones on a Go board … is attributed to an ancient master of that game".
2. Golomb, Solomon W. (1994). Polyominoes (2nd ed.). Princeton, New Jersey: Princeton University Press. ISBN 978-0-691-02444-8.
3. Gardner, M. (November 1960). "More about the shapes that can be made with complex dominoes (Mathematical Games)". Scientific American. 203 (5): 186–201. doi:10.1038/scientificamerican1160-186. JSTOR 24940703.
4. Whittington, S. G.; Soteros, C. E. (1990). "Lattice Animals: Rigorous Results and Wild Guesses". In Grimmett, G.; Welsh, D. (eds.). Disorder in Physical Systems. Oxford University Press.
5. Grünbaum, Branko; Shephard, G.C. (1987). Tilings and Patterns. New York: W.H. Freeman and Company. ISBN 978-0-7167-1193-3.
6. Redelmeier, D. Hugh (1981). "Counting polyominoes: yet another attack". Discrete Mathematics. 36 (2): 191–203. doi:10.1016/0012-365X(81)90237-5.
7. Golomb, chapter 6
8. Iwan Jensen. "Series for lattice animals or polyominoes". Archived from the original on 2007-06-12. Retrieved 2007-05-06.
9. Tomás Oliveira e Silva. "Animal enumerations on the {4,4} Euclidean tiling". Archived from the original on 2007-04-23. Retrieved 2007-05-06.
10. "Harmonic Magic Square, Enumeration of Polyominoes considering the symmetry" (PDF).
11. "Counting size 50 polyominoes" (PDF).
12. Redelmeier, section 3
13. Redelmeier, D.Hugh (1981). "Counting polyominoes: Yet another attack". Discrete Mathematics. 36 (2): 191–203. doi:10.1016/0012-365X(81)90237-5.
14. Golomb, pp. 73–79
15. Redelmeier, section 4
16. Redelmeier, section 6
17. Jensen, Iwan (February 2001). "Enumerations of Lattice Animals and Trees". Journal of Statistical Physics. 102 (3–4): 865–881. arXiv:cond-mat/0007239. Bibcode:2001JSP...102..865J. doi:10.1023/A:1004855020556. S2CID 10549375.
18. Conway, Andrew (1995). "Enumerating 2D percolation series by the finite-lattice method: theory". Journal of Physics A: Mathematical and General. 28 (2): 335–349. Bibcode:1995JPhA...28..335C. doi:10.1088/0305-4470/28/2/011. Zbl 0849.05003.
19. Jensen, Iwan; Guttmann, Anthony J. (2000). "Statistics of lattice animals (polyominoes) and polygons". Journal of Physics A: Mathematical and General. 33 (29): L257–L263. arXiv:cond-mat/0007238v1. Bibcode:2000JPhA...33L.257J. doi:10.1088/0305-4470/33/29/102. S2CID 6461687.
20. Barequet, Gill; Rote, Gunter; Shalah, Mira. "λ > 4: An Improved Lower Bound on the Growth Constant of Polyominoes". Retrieved 2017-02-02. {{cite journal}}: Cite journal requires |journal= (help)
21. Barequet, Gill; Shalah, Mira (2022). "Improved upper bounds on the growth constants of polyominoes and polycubes". Algorithmica. 84 (12): 3559–3586. doi:10.1007/s00453-022-00948-6.
22. Klarner, D.A.; Rivest, R.L. (1973). "A procedure for improving the upper bound for the number of n-ominoes" (PDF). Canadian Journal of Mathematics. 25 (3): 585–602. CiteSeerX 10.1.1.309.9151. doi:10.4153/CJM-1973-060-4. S2CID 121448572. Archived from the original (PDF of technical report version) on 2006-11-26. Retrieved 2007-05-11.
23. Wilf, Herbert S. (1994). Generatingfunctionology (2nd ed.). Boston, MA: Academic Press. p. 151. ISBN 978-0-12-751956-2. Zbl 0831.05001.
24. Bousquet-Mélou, Mireille (1998). "New enumerative results on two-dimensional directed animals". Discrete Mathematics. 180 (1–3): 73–106. doi:10.1016/S0012-365X(97)00109-X.
25. Delest, M.-P. (1988). "Generating functions for column-convex polyominoes". Journal of Combinatorial Theory, Series A. 48 (1): 12–31. doi:10.1016/0097-3165(88)90071-4.
26. Bousquet-Mélou, Mireille; Fédou, Jean-Marc (1995). "The generating function of convex polyominoes: The resolution of a q-differential system". Discrete Mathematics. 137 (1–3): 53–75. doi:10.1016/0012-365X(93)E0161-V.
27. Picciotto, Henri (1999), Geometry Labs, MathEducationPage.org, p. 208.
28. Martin, George E. (1996). Polyominoes: A guide to puzzles and problems in tiling (2nd ed.). Mathematical Association of America. ISBN 978-0-88385-501-0.
29. C.B. Haselgrove; Jenifer Haselgrove (October 1960). "A Computer Program for Pentominoes" (PDF). Eureka. 23: 16–18.
30. Golomb, Solomon W. (1970). "Tiling with Sets of Polyominoes". Journal of Combinatorial Theory. 9: 60–71. doi:10.1016/S0021-9800(70)80055-2.
31. E.D. Demaine; M.L. Demaine (June 2007). "Jigsaw Puzzles, Edge Matching, and Polyomino Packing: Connections and Complexity". Graphs and Combinatorics. 23: 195–208. doi:10.1007/s00373-007-0713-4. S2CID 17190810.
32. S.W. Golomb; L.D. Baumert (1965). "Backtrack Programming". Journal of the ACM. 12 (4): 516–524. doi:10.1145/321296.321300.
33. Golomb, Polyominoes, chapter 8
34. Reid, Michael. "References for Rectifiable Polyominoes". Archived from the original on 2004-01-16. Retrieved 2007-05-11.
35. Reid, Michael. "List of known prime rectangles for various polyominoes". Archived from the original on 2007-04-16. Retrieved 2007-05-11.
36. Klarner, D.A.; Göbel, F. (1969). "Packing boxes with congruent figures". Indagationes Mathematicae. 31: 465–472.
37. Klarner, David A. (February 1973). "A Finite Basis Theorem Revisited" (PDF). Stanford University Technical Report STAN-CS-73–338. Archived from the original (PDF) on 2007-10-23. Retrieved 2007-05-12.
38. Kamenetsky, Dmitry; Cooke, Tristrom (2015). "Tiling rectangles with holey polyominoes". arXiv:1411.2699 [cs.CG].
39. Golomb, Solomon W. (1966). "Tiling with Polyominoes". Journal of Combinatorial Theory. 1 (2): 280–296. doi:10.1016/S0021-9800(66)80033-9.
40. Moore, Cristopher; Robson, John Michael (2001). "Hard Tiling Problems with Simple Tiles" (PDF). Archived from the original (PDF) on 2013-06-17.
41. Petersen, Ivars (September 25, 1999), "Math Trek: Tiling with Polyominoes", Science News, archived from the original on March 20, 2008, retrieved March 11, 2012.
42. Gardner, Martin (July 1965). "On the relation between mathematics and the ordered patterns of Op art". Scientific American. 213 (1): 100–104. doi:10.1038/scientificamerican1265-100.
43. Gardner, Martin (August 1965). "Thoughts on the task of communication with intelligent organisms on other worlds". Scientific American. 213 (2): 96–100. doi:10.1038/scientificamerican0865-96.
44. Gardner, Martin (August 1975). "More about tiling the plane: the possibilities of polyominoes, polyiamonds and polyhexes". Scientific American. 233 (2): 112–115. doi:10.1038/scientificamerican0875-112.
45. Rawsthorne, Daniel A. (1988). "Tiling complexity of small n-ominoes
(n<10)"
. Discrete Mathematics. 70: 71–75. doi:10.1016/0012-365X(88)90081-7.
46. Rhoads, Glenn C. (2003). Planar Tilings and the Search for an Aperiodic Prototile. PhD dissertation, Rutgers University.
47. Grünbaum and Shephard, section 9.4
48. Keating, K.; Vince, A. (1999). "Isohedral Polyomino Tiling of the Plane". Discrete & Computational Geometry. 21 (4): 615–630. doi:10.1007/PL00009442.
49. Rhoads, Glenn C. (2005). "Planar tilings by polyominoes, polyhexes, and polyiamonds". Journal of Computational and Applied Mathematics. 174 (2): 329–353. Bibcode:2005JCoAM.174..329R. doi:10.1016/j.cam.2004.05.002.
50. Niţică, Viorel (2003). "Rep-tiles revisited". MASS selecta. Providence, RI: American Mathematical Society. pp. 205–217. MR 2027179.
51. Mireles, J.L., "Poly2ominoes"
52. "Resta, G., "Polypolyominoes"". Archived from the original on 2011-02-22. Retrieved 2010-07-02.
53. "Zucca, L., "Triple Pentominoes"". Retrieved 2023-04-20.
54. Barbans, Uldis; Cibulis, Andris; Lee, Gilbert; Liu, Andy; Wainwright, Robert (2005). "Polyomino Number Theory (III)". In Cipra, Barry Arthur; Demaine, Erik D.; Demaine, Martin L.; Rodgers, Tom (eds.). Tribute to a Mathemagician. Wellesley, MA: A.K. Peters. pp. 131–136. ISBN 978-1-56881-204-5.
55. Oxford English Dictionary, 2nd edition, entry domino
External links
• Karl Dahlke's polyomino finite-rectangle tilings
• An implementation and description of Jensen's method
• A paper describing modern estimates (PS)
• Weisstein, Eric W. "Polyomino". MathWorld.
• MathPages – Notes on enumeration of polyominoes with various symmetries
• List of dissection problems in Fairy Chess Review
• Tetrads by Karl Scherer, Wolfram Demonstrations Project.
Polyforms
Polyominoes
• Domino
• Tromino
• Tetromino
• Pentomino
• Hexomino
• Heptomino
• Octomino
• Nonomino
• Decomino
Higher dimensions
• Polyominoid
• Polycube
Others
• Polyabolo
• Polydrafter
• Polyhex
• Polyiamond
• Pseudo-polyomino
• Polystick
Games and puzzles
• Blokus
• Soma cube
• Snake cube
• Tangram
• Hexastix
• Tantrix
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Solid angle
In geometry, a solid angle (symbol: Ω) is a measure of the amount of the field of view from some particular point that a given object covers. That is, it is a measure of how large the object appears to an observer looking from that point. The point from which the object is viewed is called the apex of the solid angle, and the object is said to subtend its solid angle at that point.
Not to be confused with spherical angle.
Solid angle
Common symbols
Ω
SI unitsteradian
Other units
Square degree
In SI base unitsm2/m2
Conserved?No
Derivations from
other quantities
$\Omega =A/r^{2}$
Dimension$1$
In the International System of Units (SI), a solid angle is expressed in a dimensionless unit called a steradian (symbol: sr). One steradian corresponds to one unit of area on the unit sphere surrounding the apex, so an object that blocks all rays from the apex would cover a number of steradians equal to the total surface area of the unit sphere, $4\pi $. Solid angles can also be measured in squares of angular measures such as degrees, minutes, and seconds.
A small object nearby may subtend the same solid angle as a larger object farther away. For example, although the Moon is much smaller than the Sun, it is also much closer to Earth. Indeed, as viewed from any point on Earth, both objects have approximately the same solid angle as well as apparent size. This is evident during a solar eclipse.
Definition and properties
See also: Spherical polygon area
An object's solid angle in steradians is equal to the area of the segment of a unit sphere, centered at the apex, that the object covers. Giving the area of a segment of a unit sphere in steradians is analogous to giving the length of an arc of a unit circle in radians. Just like a planar angle in radians is the ratio of the length of an arc to its radius, a solid angle in steradians is the ratio of the area covered on a sphere by an object to the area given by the square of the radius of said sphere. The formula is
$\Omega ={\frac {A}{r^{2}}},$
where $A$ is the spherical surface area and $r$ is the radius of the considered sphere.
Solid angles are often used in astronomy, physics, and in particular astrophysics. The solid angle of an object that is very far away is roughly proportional to the ratio of area to squared distance. Here "area" means the area of the object when projected along the viewing direction.
The solid angle of a sphere measured from any point in its interior is 4π sr, and the solid angle subtended at the center of a cube by one of its faces is one-sixth of that, or 2π/3 sr. Solid angles can also be measured in square degrees (1 sr = (180/π)2 square degrees), in square minutes and square seconds, or in fractions of the sphere (1 sr = 1/4π fractional area), also known as spat (1 sp = 4π sr).
In spherical coordinates there is a formula for the differential,
$d\Omega =\sin \theta \,d\theta \,d\varphi ,$
where θ is the colatitude (angle from the North Pole) and φ is the longitude.
The solid angle for an arbitrary oriented surface S subtended at a point P is equal to the solid angle of the projection of the surface S to the unit sphere with center P, which can be calculated as the surface integral:
$\Omega =\iint _{S}{\frac {{\hat {r}}\cdot {\hat {n}}}{r^{2}}}\,dS\ =\iint _{S}\sin \theta \,d\theta \,d\varphi ,$
where ${\hat {r}}={\vec {r}}/r$ is the unit vector corresponding to ${\vec {r}}$, the position vector of an infinitesimal area of surface dS with respect to point P, and where ${\hat {n}}$ represents the unit normal vector to dS. Even if the projection on the unit sphere to the surface S is not isomorphic, the multiple folds are correctly considered according to the surface orientation described by the sign of the scalar product ${\hat {r}}\cdot {\hat {n}}$.
Thus one can approximate the solid angle subtended by a small facet having flat surface area dS, orientation ${\hat {n}}$, and distance r from the viewer as:
$d\Omega =4\pi \left({\frac {dS}{A}}\right)\,({\hat {r}}\cdot {\hat {n}}),$
where the surface area of a sphere is A = 4πr2.
Practical applications
• Defining luminous intensity and luminance, and the correspondent radiometric quantities radiant intensity and radiance
• Calculating spherical excess E of a spherical triangle
• The calculation of potentials by using the boundary element method (BEM)
• Evaluating the size of ligands in metal complexes, see ligand cone angle
• Calculating the electric field and magnetic field strength around charge distributions
• Deriving Gauss's Law
• Calculating emissive power and irradiation in heat transfer
• Calculating cross sections in Rutherford scattering
• Calculating cross sections in Raman scattering
• The solid angle of the acceptance cone of the optical fiber
Solid angles for common objects
Cone, spherical cap, hemisphere
The solid angle of a cone with its apex at the apex of the solid angle, and with apex angle 2θ, is the area of a spherical cap on a unit sphere
$\Omega =2\pi \left(1-\cos \theta \right)\ =4\pi \sin ^{2}{\frac {\theta }{2}}.$
For small θ such that cos θ ≈ 1 − θ2/2 this reduces to πθ2, the area of a circle.
The above is found by computing the following double integral using the unit surface element in spherical coordinates:
${\begin{aligned}\int _{0}^{2\pi }\int _{0}^{\theta }\sin \theta '\,d\theta '\,d\phi &=\int _{0}^{2\pi }d\phi \int _{0}^{\theta }\sin \theta '\,d\theta '\\&=2\pi \int _{0}^{\theta }\sin \theta '\,d\theta '\\&=2\pi \left[-\cos \theta '\right]_{0}^{\theta }\\&=2\pi \left(1-\cos \theta \right).\end{aligned}}$
This formula can also be derived without the use of calculus. Over 2200 years ago Archimedes proved that the surface area of a spherical cap is always equal to the area of a circle whose radius equals the distance from the rim of the spherical cap to the point where the cap's axis of symmetry intersects the cap.[1] In the diagram this radius is given as
$2r\sin {\frac {\theta }{2}}.$
Hence for a unit sphere the solid angle of the spherical cap is given as
$\Omega =4\pi \sin ^{2}{\frac {\theta }{2}}=2\pi \left(1-\cos \theta \right).$
When θ = π/2, the spherical cap becomes a hemisphere having a solid angle 2π.
The solid angle of the complement of the cone is
$4\pi -\Omega =2\pi \left(1+\cos \theta \right)=4\pi \cos ^{2}{\frac {\theta }{2}}.$
This is also the solid angle of the part of the celestial sphere that an astronomical observer positioned at latitude θ can see as the Earth rotates. At the equator all of the celestial sphere is visible; at either pole, only one half.
The solid angle subtended by a segment of a spherical cap cut by a plane at angle γ from the cone's axis and passing through the cone's apex can be calculated by the formula[2]
$\Omega =2\left[\arccos \left({\frac {\sin \gamma }{\sin \theta }}\right)-\cos \theta \arccos \left({\frac {\tan \gamma }{\tan \theta }}\right)\right].$
For example, if γ = −θ, then the formula reduces to the spherical cap formula above: the first term becomes π, and the second π cos θ.
Tetrahedron
Let OABC be the vertices of a tetrahedron with an origin at O subtended by the triangular face ABC where ${\vec {a}}\ ,\,{\vec {b}}\ ,\,{\vec {c}}$ are the vector positions of the vertices A, B and C. Define the vertex angle θa to be the angle BOC and define θb, θc correspondingly. Let $\phi _{ab}$ be the dihedral angle between the planes that contain the tetrahedral faces OAC and OBC and define $\phi _{ac}$, $\phi _{bc}$ correspondingly. The solid angle Ω subtended by the triangular surface ABC is given by
$\Omega =\left(\phi _{ab}+\phi _{bc}+\phi _{ac}\right)\ -\pi .$
This follows from the theory of spherical excess and it leads to the fact that there is an analogous theorem to the theorem that "The sum of internal angles of a planar triangle is equal to π", for the sum of the four internal solid angles of a tetrahedron as follows:
$\sum _{i=1}^{4}\Omega _{i}=2\sum _{i=1}^{6}\phi _{i}\ -4\pi ,$
where $\phi _{i}$ ranges over all six of the dihedral angles between any two planes that contain the tetrahedral faces OAB, OAC, OBC and ABC.[3]
A useful formula for calculating the solid angle of the tetrahedron at the origin O that is purely a function of the vertex angles θa, θb, θc is given by L'Huilier's theorem[4][5] as
$\tan \left({\frac {1}{4}}\Omega \right)={\sqrt {\tan \left({\frac {\theta _{s}}{2}}\right)\tan \left({\frac {\theta _{s}-\theta _{a}}{2}}\right)\tan \left({\frac {\theta _{s}-\theta _{b}}{2}}\right)\tan \left({\frac {\theta _{s}-\theta _{c}}{2}}\right)}},$
where
$\theta _{s}={\frac {\theta _{a}+\theta _{b}+\theta _{c}}{2}}.$
Another interesting formula involves expressing the vertices as vectors in 3 dimensional space. Let ${\vec {a}}\ ,\,{\vec {b}}\ ,\,{\vec {c}}$ be the vector positions of the vertices A, B and C, and let a, b, and c be the magnitude of each vector (the origin-point distance). The solid angle Ω subtended by the triangular surface ABC is:[6][7]
$\tan \left({\frac {1}{2}}\Omega \right)={\frac {\left|{\vec {a}}\ {\vec {b}}\ {\vec {c}}\right|}{abc+\left({\vec {a}}\cdot {\vec {b}}\right)c+\left({\vec {a}}\cdot {\vec {c}}\right)b+\left({\vec {b}}\cdot {\vec {c}}\right)a}},$
where
$\left|{\vec {a}}\ {\vec {b}}\ {\vec {c}}\right|={\vec {a}}\cdot ({\vec {b}}\times {\vec {c}})$
denotes the scalar triple product of the three vectors and ${\vec {a}}\cdot {\vec {b}}$ denotes the scalar product.
Care must be taken here to avoid negative or incorrect solid angles. One source of potential errors is that the scalar triple product can be negative if a, b, c have the wrong winding. Computing the absolute value is a sufficient solution since no other portion of the equation depends on the winding. The other pitfall arises when the scalar triple product is positive but the divisor is negative. In this case returns a negative value that must be increased by π.
Pyramid
The solid angle of a four-sided right rectangular pyramid with apex angles a and b (dihedral angles measured to the opposite side faces of the pyramid) is
$\Omega =4\arcsin \left(\sin \left({a \over 2}\right)\sin \left({b \over 2}\right)\right).$
If both the side lengths (α and β) of the base of the pyramid and the distance (d) from the center of the base rectangle to the apex of the pyramid (the center of the sphere) are known, then the above equation can be manipulated to give
$\Omega =4\arctan {\frac {\alpha \beta }{2d{\sqrt {4d^{2}+\alpha ^{2}+\beta ^{2}}}}}.$
The solid angle of a right n-gonal pyramid, where the pyramid base is a regular n-sided polygon of circumradius r, with a pyramid height h is
$\Omega =2\pi -2n\arctan \left({\frac {\tan \left({\pi \over n}\right)}{\sqrt {1+{r^{2} \over h^{2}}}}}\right).$
The solid angle of an arbitrary pyramid with an n-sided base defined by the sequence of unit vectors representing edges {s1, s2}, ... sn can be efficiently computed by:[2]
$\Omega =2\pi -\arg \prod _{j=1}^{n}\left(\left(s_{j-1}s_{j}\right)\left(s_{j}s_{j+1}\right)-\left(s_{j-1}s_{j+1}\right)+i\left[s_{j-1}s_{j}s_{j+1}\right]\right).$
where parentheses (* *) is a scalar product and square brackets [* * *] is a scalar triple product, and i is an imaginary unit. Indices are cycled: s0 = sn and s1 = sn + 1. The complex products add the phase associated with each vertex angle of the polygon. However, a multiple of $2\pi $ is lost in the branch cut of $\arg $ and must be kept track of separately. Also, the running product of complex phases must scaled occasionally to avoid underflow in the limit of nearly parallel segments.
Latitude-longitude rectangle
The solid angle of a latitude-longitude rectangle on a globe is
$\left(\sin \phi _{\mathrm {N} }-\sin \phi _{\mathrm {S} }\right)\left(\theta _{\mathrm {E} }-\theta _{\mathrm {W} }\,\!\right)\;\mathrm {sr} ,$
where φN and φS are north and south lines of latitude (measured from the equator in radians with angle increasing northward), and θE and θW are east and west lines of longitude (where the angle in radians increases eastward).[8] Mathematically, this represents an arc of angle ϕN − ϕS swept around a sphere by θE − θW radians. When longitude spans 2π radians and latitude spans π radians, the solid angle is that of a sphere.
A latitude-longitude rectangle should not be confused with the solid angle of a rectangular pyramid. All four sides of a rectangular pyramid intersect the sphere's surface in great circle arcs. With a latitude-longitude rectangle, only lines of longitude are great circle arcs; lines of latitude are not.
Celestial objects
By using the definition of angular diameter, the formula for the solid angle of a celestial object can be defined in terms of the radius of the object, $ R$, and the distance from the observer to the object, $d$:
$\Omega =2\pi \left(1-{\frac {\sqrt {d^{2}-R^{2}}}{d}}\right):d\geq R.$
By inputting the appropriate average values for the Sun and the Moon (in relation to Earth), the average solid angle of the Sun is 6.794×10−5 steradians and the average solid angle of the Moon is 6.418×10−5 steradians. In terms of the total celestial sphere, the Sun and the Moon subtend average fractional areas of 0.0005406% (5.406 ppm) and 0.0005107% (5.107 ppm), respectively. As these solid angles are about the same size, the Moon can cause both total and annular solar eclipses depending on the distance between the Earth and the Moon during the eclipse.
Solid angles in arbitrary dimensions
The solid angle subtended by the complete (d − 1)-dimensional spherical surface of the unit sphere in d-dimensional Euclidean space can be defined in any number of dimensions d. One often needs this solid angle factor in calculations with spherical symmetry. It is given by the formula
$\Omega _{d}={\frac {2\pi ^{\frac {d}{2}}}{\Gamma \left({\frac {d}{2}}\right)}},$
where Γ is the gamma function. When d is an integer, the gamma function can be computed explicitly.[9] It follows that
$\Omega _{d}={\begin{cases}{\frac {1}{\left({\frac {d}{2}}-1\right)!}}2\pi ^{\frac {d}{2}}\ &d{\text{ even}}\\{\frac {\left({\frac {1}{2}}\left(d-1\right)\right)!}{(d-1)!}}2^{d}\pi ^{{\frac {1}{2}}(d-1)}\ &d{\text{ odd}}.\end{cases}}$
This gives the expected results of 4π steradians for the 3D sphere bounded by a surface of area 4πr2 and 2π radians for the 2D circle bounded by a circumference of length 2πr. It also gives the slightly less obvious 2 for the 1D case, in which the origin-centered 1D "sphere" is the interval [−r, r] and this is bounded by two limiting points.
The counterpart to the vector formula in arbitrary dimension was derived by Aomoto[10][11] and independently by Ribando.[12] It expresses them as an infinite multivariate Taylor series:
$\Omega =\Omega _{d}{\frac {\left|\det(V)\right|}{(4\pi )^{d/2}}}\sum _{{\vec {a}}\in \mathbb {N} _{0}^{\binom {d}{2}}}\left[{\frac {(-2)^{\sum _{i<j}a_{ij}}}{\prod _{i<j}a_{ij}!}}\prod _{i}\Gamma \left({\frac {1+\sum _{m\neq i}a_{im}}{2}}\right)\right]{\vec {\alpha }}^{\vec {a}}.$
Given d unit vectors ${\vec {v}}_{i}$ defining the angle, let V denote the matrix formed by combining them so the ith column is ${\vec {v}}_{i}$, and $\alpha _{ij}={\vec {v}}_{i}\cdot {\vec {v}}_{j}=\alpha _{ji},\alpha _{ii}=1$. The variables $\alpha _{ij},1\leq i<j\leq d$ form a multivariable ${\vec {\alpha }}=(\alpha _{12},\dotsc ,\alpha _{1d},\alpha _{23},\dotsc ,\alpha _{d-1,d})\in \mathbb {R} ^{\binom {d}{2}}$. For a "congruent" integer multiexponent ${\vec {a}}=(a_{12},\dotsc ,a_{1d},a_{23},\dotsc ,a_{d-1,d})\in \mathbb {N} _{0}^{\binom {d}{2}},$ define $ {\vec {\alpha }}^{\vec {a}}=\prod \alpha _{ij}^{a_{ij}}$. Note that here $\mathbb {N} _{0}$ = non-negative integers, or natural numbers beginning with 0. The notation $\alpha _{ji}$ for $j>i$ means the variable $\alpha _{ij}$, similarly for the exponents $a_{ji}$. Hence, the term $ \sum _{m\neq l}a_{lm}$ means the sum over all terms in ${\vec {a}}$ in which l appears as either the first or second index. Where this series converges, it converges to the solid angle defined by the vectors.
References
1. "Archimedes on Spheres and Cylinders". Math Pages. 2015.
2. Mazonka, Oleg (2012). "Solid Angle of Conical Surfaces, Polyhedral Cones, and Intersecting Spherical Caps". arXiv:1205.1396 [math.MG].
3. Hopf, Heinz (1940). "Selected Chapters of Geometry" (PDF). ETH Zurich: 1–2. Archived (PDF) from the original on 2018-09-21.
4. "L'Huilier's Theorem – from Wolfram MathWorld". Mathworld.wolfram.com. 2015-10-19. Retrieved 2015-10-19.
5. "Spherical Excess – from Wolfram MathWorld". Mathworld.wolfram.com. 2015-10-19. Retrieved 2015-10-19.
6. Eriksson, Folke (1990). "On the measure of solid angles". Math. Mag. 63 (3): 184–187. doi:10.2307/2691141. JSTOR 2691141.
7. Van Oosterom, A; Strackee, J (1983). "The Solid Angle of a Plane Triangle". IEEE Trans. Biomed. Eng. BME-30 (2): 125–126. doi:10.1109/TBME.1983.325207. PMID 6832789. S2CID 22669644.
8. "Area of a Latitude-Longitude Rectangle". The Math Forum @ Drexel. 2003.
9. Jackson, FM (1993). "Polytopes in Euclidean n-space". Bulletin of the Institute of Mathematics and Its Applications. 29 (11/12): 172–174.
10. Aomoto, Kazuhiko (1977). "Analytic structure of Schläfli function". Nagoya Math. J. 68: 1–16. doi:10.1017/s0027763000017839.
11. Beck, M.; Robins, S.; Sam, S. V. (2010). "Positivity theorems for solid-angle polynomials". Contributions to Algebra and Geometry. 51 (2): 493–507. arXiv:0906.4031. Bibcode:2009arXiv0906.4031B.
12. Ribando, Jason M. (2006). "Measuring Solid Angles Beyond Dimension Three". Discrete & Computational Geometry. 36 (3): 479–487. doi:10.1007/s00454-006-1253-4.
Further reading
• Jaffey, A. H. (1954). "Solid angle subtended by a circular aperture at point and spread sources: formulas and some tables". Rev. Sci. Instrum. 25 (4): 349–354. Bibcode:1954RScI...25..349J. doi:10.1063/1.1771061.
• Masket, A. Victor (1957). "Solid angle contour integrals, series, and tables". Rev. Sci. Instrum. 28 (3): 191. Bibcode:1957RScI...28..191M. doi:10.1063/1.1746479.
• Naito, Minoru (1957). "A method of calculating the solid angle subtended by a circular aperture". J. Phys. Soc. Jpn. 12 (10): 1122–1129. Bibcode:1957JPSJ...12.1122N. doi:10.1143/JPSJ.12.1122.
• Paxton, F. (1959). "Solid angle calculation for a circular disk". Rev. Sci. Instrum. 30 (4): 254. Bibcode:1959RScI...30..254P. doi:10.1063/1.1716590.
• Khadjavi, A. (1968). "Calculation of solid angle subtended by rectangular apertures". J. Opt. Soc. Am. 58 (10): 1417–1418. doi:10.1364/JOSA.58.001417.
• Gardner, R. P.; Carnesale, A. (1969). "The solid angle subtended at a point by a circular disk". Nucl. Instrum. Methods. 73 (2): 228–230. Bibcode:1969NucIM..73..228G. doi:10.1016/0029-554X(69)90214-6.
• Gardner, R. P.; Verghese, K. (1971). "On the solid angle subtended by a circular disk". Nucl. Instrum. Methods. 93 (1): 163–167. Bibcode:1971NucIM..93..163G. doi:10.1016/0029-554X(71)90155-8.
• Gotoh, H.; Yagi, H. (1971). "Solid angle subtended by a rectangular slit". Nucl. Instrum. Methods. 96 (3): 485–486. Bibcode:1971NucIM..96..485G. doi:10.1016/0029-554X(71)90624-0.
• Cook, J. (1980). "Solid angle subtended by a two rectangles". Nucl. Instrum. Methods. 178 (2–3): 561–564. Bibcode:1980NucIM.178..561C. doi:10.1016/0029-554X(80)90838-1.
• Asvestas, John S..; Englund, David C. (1994). "Computing the solid angle subtended by a planar figure". Opt. Eng. 33 (12): 4055–4059. Bibcode:1994OptEn..33.4055A. doi:10.1117/12.183402. Erratum ibid. vol 50 (2011) page 059801.
• Tryka, Stanislaw (1997). "Angular distribution of the solid angle at a point subtended by a circular disk". Opt. Commun. 137 (4–6): 317–333. Bibcode:1997OptCo.137..317T. doi:10.1016/S0030-4018(96)00789-4.
• Prata, M. J. (2004). "Analytical calculation of the solid angle subtended by a circular disc detector at a point cosine source". Nucl. Instrum. Methods Phys. Res. A. 521 (2–3): 576. arXiv:math-ph/0305034. Bibcode:2004NIMPA.521..576P. doi:10.1016/j.nima.2003.10.098. S2CID 15266291.
• Timus, D. M.; Prata, M. J.; Kalla, S. L.; Abbas, M. I.; Oner, F.; Galiano, E. (2007). "Some further analytical results on the solid angle subtended at a point by a circular disk using elliptic integrals". Nucl. Instrum. Methods Phys. Res. A. 580: 149–152. Bibcode:2007NIMPA.580..149T. doi:10.1016/j.nima.2007.05.055.
Wikimedia Commons has media related to Solid angle.
External links
• Arthur P. Norton, A Star Atlas, Gall and Inglis, Edinburgh, 1969.
• M. G. Kendall, A Course in the Geometry of N Dimensions, No. 8 of Griffin's Statistical Monographs & Courses, ed. M. G. Kendall, Charles Griffin & Co. Ltd, London, 1961
• Weisstein, Eric W. "Solid Angle". MathWorld.
Classical mechanics SI units
Linear/translational quantities Angular/rotational quantities
Dimensions 1 L L2 Dimensions 1 θ θ2
T time: t
s
absement: A
m s
T time: t
s
1 distance: d, position: r, s, x, displacement
m
area: A
m2
1 angle: θ, angular displacement: θ
rad
solid angle: Ω
rad2, sr
T−1 frequency: f
s−1, Hz
speed: v, velocity: v
m s−1
kinematic viscosity: ν,
specific angular momentum: h
m2 s−1
T−1 frequency: f
s−1, Hz
angular speed: ω, angular velocity: ω
rad s−1
T−2 acceleration: a
m s−2
T−2 angular acceleration: α
rad s−2
T−3 jerk: j
m s−3
T−3 angular jerk: ζ
rad s−3
M mass: m
kg
weighted position: M ⟨x⟩ = ∑ m x ML2 moment of inertia: I
kg m2
MT−1 Mass flow rate: ${\dot {m}}$
kg s−1
momentum: p, impulse: J
kg m s−1, N s
action: 𝒮, actergy: ℵ
kg m2 s−1, J s
ML2T−1 angular momentum: L, angular impulse: ΔL
kg m2 s−1
action: 𝒮, actergy: ℵ
kg m2 s−1, J s
MT−2 force: F, weight: Fg
kg m s−2, N
energy: E, work: W, Lagrangian: L
kg m2 s−2, J
ML2T−2 torque: τ, moment: M
kg m2 s−2, N m
energy: E, work: W, Lagrangian: L
kg m2 s−2, J
MT−3 yank: Y
kg m s−3, N s−1
power: P
kg m2 s−3, W
ML2T−3 rotatum: P
kg m2 s−3, N m s−1
power: P
kg m2 s−3, W
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Square bifrustum
The square bifrustum or square truncated bipyramid is the second in an infinite series of bifrustum polyhedra. It has 4 trapezoidal and 2 square faces.
Square bifrustum
TypeBifrustum
Faces8 trapezoids, 2 squares
Edges20
Vertices12
Symmetry groupD4h
Dual polyhedronelongated square bipyramid
Propertiesconvex
This polyhedron can be constructed by taking a square bipyramid (octahedron) and truncating the polar axis vertices, making it into two end-to-end frustums.
It is dual to the elongated square dipyramid.
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Square class
In mathematics, specifically abstract algebra, a square class of a field $F$ is an element of the square class group, the quotient group $F^{\times }/F^{\times 2}$ of the multiplicative group of nonzero elements in the field modulo the square elements of the field. Each square class is a subset of the nonzero elements (a coset of the multiplicative group) consisting of the elements of the form xy2 where x is some particular fixed element and y ranges over all nonzero field elements.[1]
For instance, if $F=\mathbb {R} $, the field of real numbers, then $F^{\times }$ is just the group of all nonzero real numbers (with the multiplication operation) and $F^{\times 2}$ is the subgroup of positive numbers (as every positive number has a real square root). The quotient of these two groups is a group with two elements, corresponding to two cosets: the set of positive numbers and the set of negative numbers. Thus, the real numbers have two square classes, the positive numbers and the negative numbers.[1]
Square classes are frequently studied in relation to the theory of quadratic forms.[2] The reason is that if $V$ is an $F$-vector space and $q:V\to F$ is a quadratic form and $v$ is an element of $V$ such that $q(v)=a\in F^{\times }$, then for all $u\in F^{\times }$, $q(uv)=au^{2}$ and thus it is sometimes more convenient to talk about the square classes which the quadratic form represents.
Every element of the square class group is an involution. It follows that, if the number of square classes of a field is finite, it must be a power of two.[2]
References
1. Salzmann, H. (2007), The Classical Fields: Structural Features of the Real and Rational Numbers, Encyclopedia of Mathematics and its Applications, vol. 112, Cambridge University Press, p. 295, ISBN 9780521865166.
2. Szymiczek, Kazimierz (1997), Bilinear Algebra: An Introduction to the Algebraic Theory of Quadratic Forms, Algebra, logic, and applications, vol. 7, CRC Press, pp. 29, 109, ISBN 9789056990763.
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Square cupola
In geometry, the square cupola, sometimes called lesser dome, is one of the Johnson solids (J4). It can be obtained as a slice of the rhombicuboctahedron. As in all cupolae, the base polygon has twice as many edges and vertices as the top; in this case the base polygon is an octagon.
Square cupola
TypeJohnson
J3 – J4 – J5
Faces4 triangles
1+4 squares
1 octagon
Edges20
Vertices12
Vertex configuration8(3.4.8)
4(3.43)
Symmetry groupC4v, [4], (*44)
Rotation groupC4, [4]+, (44)
Dual polyhedron-
Propertiesconvex
Net
A Johnson solid is one of 92 strictly convex polyhedra that is composed of regular polygon faces but are not uniform polyhedra (that is, they are not Platonic solids, Archimedean solids, prisms, or antiprisms). They were named by Norman Johnson, who first listed these polyhedra in 1966.[1]
Formulae
The following formulae for the circumradius, surface area, volume, and height can be used if all faces are regular, with edge length a:
$C=\left({\frac {1}{2}}{\sqrt {5+2{\sqrt {2}}}}\right)a\approx 1.39897a,$[2]
$A=\left(7+2{\sqrt {2}}+{\sqrt {3}}\right)a^{2}\approx 11.56048a^{2},$[3]
$V=\left(1+{\frac {2{\sqrt {2}}}{3}}\right)a^{3}\approx 1.94281a^{3}.$[4]
$h={\frac {\sqrt {2}}{2}}a\approx 0.70711a$[5]
Related polyhedra and honeycombs
Other convex cupolae
Family of convex cupolae
n2345678
Schläfli symbol{2} || t{2} {3} || t{3} {4} || t{4} {5} || t{5} {6} || t{6} {7} || t{7} {8} || t{8}
Cupola
Digonal cupola
Triangular cupola
Square cupola
Pentagonal cupola
Hexagonal cupola
(Flat)
Heptagonal cupola
(Non-regular face)
Octagonal cupola
(Non-regular face)
Related
uniform
polyhedra
Rhombohedron
Cuboctahedron
Rhombicuboctahedron
Rhombicosidodecahedron
Rhombitrihexagonal tiling
Rhombitriheptagonal tiling
Rhombitrioctagonal tiling
Dual polyhedron
The dual of the square cupola has 8 triangular and 4 kite faces:
Dual square cupola Net of dual 3D model
Crossed square cupola
The crossed square cupola is one of the nonconvex Johnson solid isomorphs, being topologically identical to the convex square cupola. It can be obtained as a slice of the nonconvex great rhombicuboctahedron or quasirhombicuboctahedron, analogously to how the square cupola may be obtained as a slice of the rhombicuboctahedron. As in all cupolae, the base polygon has twice as many edges and vertices as the top; in this case the base polygon is an octagram.
It may be seen as a cupola with a retrograde square base, so that the squares and triangles connect across the bases in the opposite way to the square cupola, hence intersecting each other.
Honeycombs
The square cupola is a component of several nonuniform space-filling lattices:
• with tetrahedra;
• with cubes and cuboctahedra; and
• with tetrahedra, square pyramids and various combinations of cubes, elongated square pyramids and elongated square bipyramids.[6]
References
1. Johnson, Norman W. (1966), "Convex polyhedra with regular faces", Canadian Journal of Mathematics, 18: 169–200, doi:10.4153/cjm-1966-021-8, MR 0185507, Zbl 0132.14603.
2. Wolfram Research, Inc. (2020). "Wolfram|Alpha Knowledgebase". Champaign, IL. PolyhedronData[{"Johnson", 4}, "Circumradius"] {{cite journal}}: Cite journal requires |journal= (help)
3. Wolfram Research, Inc. (2020). "Wolfram|Alpha Knowledgebase". Champaign, IL. PolyhedronData[{"Johnson", 4}, "SurfaceArea"] {{cite journal}}: Cite journal requires |journal= (help)
4. Wolfram Research, Inc. (2020). "Wolfram|Alpha Knowledgebase". Champaign, IL. PolyhedronData[{"Johnson", 4}, "Volume"] {{cite journal}}: Cite journal requires |journal= (help)
5. Sapiña, R. "Area and volume of the Johnson solid J4". Problemas y Ecuaciones (in Spanish). ISSN 2659-9899. Retrieved 2020-07-16.
6. "J4 honeycomb".
External links
• Eric W. Weisstein, Square cupola (Johnson solid) at MathWorld.
Johnson solids
Pyramids, cupolae and rotundae
• square pyramid
• pentagonal pyramid
• triangular cupola
• square cupola
• pentagonal cupola
• pentagonal rotunda
Modified pyramids
• elongated triangular pyramid
• elongated square pyramid
• elongated pentagonal pyramid
• gyroelongated square pyramid
• gyroelongated pentagonal pyramid
• triangular bipyramid
• pentagonal bipyramid
• elongated triangular bipyramid
• elongated square bipyramid
• elongated pentagonal bipyramid
• gyroelongated square bipyramid
Modified cupolae and rotundae
• elongated triangular cupola
• elongated square cupola
• elongated pentagonal cupola
• elongated pentagonal rotunda
• gyroelongated triangular cupola
• gyroelongated square cupola
• gyroelongated pentagonal cupola
• gyroelongated pentagonal rotunda
• gyrobifastigium
• triangular orthobicupola
• square orthobicupola
• square gyrobicupola
• pentagonal orthobicupola
• pentagonal gyrobicupola
• pentagonal orthocupolarotunda
• pentagonal gyrocupolarotunda
• pentagonal orthobirotunda
• elongated triangular orthobicupola
• elongated triangular gyrobicupola
• elongated square gyrobicupola
• elongated pentagonal orthobicupola
• elongated pentagonal gyrobicupola
• elongated pentagonal orthocupolarotunda
• elongated pentagonal gyrocupolarotunda
• elongated pentagonal orthobirotunda
• elongated pentagonal gyrobirotunda
• gyroelongated triangular bicupola
• gyroelongated square bicupola
• gyroelongated pentagonal bicupola
• gyroelongated pentagonal cupolarotunda
• gyroelongated pentagonal birotunda
Augmented prisms
• augmented triangular prism
• biaugmented triangular prism
• triaugmented triangular prism
• augmented pentagonal prism
• biaugmented pentagonal prism
• augmented hexagonal prism
• parabiaugmented hexagonal prism
• metabiaugmented hexagonal prism
• triaugmented hexagonal prism
Modified Platonic solids
• augmented dodecahedron
• parabiaugmented dodecahedron
• metabiaugmented dodecahedron
• triaugmented dodecahedron
• metabidiminished icosahedron
• tridiminished icosahedron
• augmented tridiminished icosahedron
Modified Archimedean solids
• augmented truncated tetrahedron
• augmented truncated cube
• biaugmented truncated cube
• augmented truncated dodecahedron
• parabiaugmented truncated dodecahedron
• metabiaugmented truncated dodecahedron
• triaugmented truncated dodecahedron
• gyrate rhombicosidodecahedron
• parabigyrate rhombicosidodecahedron
• metabigyrate rhombicosidodecahedron
• trigyrate rhombicosidodecahedron
• diminished rhombicosidodecahedron
• paragyrate diminished rhombicosidodecahedron
• metagyrate diminished rhombicosidodecahedron
• bigyrate diminished rhombicosidodecahedron
• parabidiminished rhombicosidodecahedron
• metabidiminished rhombicosidodecahedron
• gyrate bidiminished rhombicosidodecahedron
• tridiminished rhombicosidodecahedron
Elementary solids
• snub disphenoid
• snub square antiprism
• sphenocorona
• augmented sphenocorona
• sphenomegacorona
• hebesphenomegacorona
• disphenocingulum
• bilunabirotunda
• triangular hebesphenorotunda
(See also List of Johnson solids, a sortable table)
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Square gyrobicupola
In geometry, the square gyrobicupola is one of the Johnson solids (J29). Like the square orthobicupola (J28), it can be obtained by joining two square cupolae (J4) along their bases. The difference is that in this solid, the two halves are rotated 45 degrees with respect to one another.
Square gyrobicupola
TypeBicupola,
Johnson
J28 – J29 – J30
Faces8 triangles
2+8 squares
Edges32
Vertices16
Vertex configuration8(3.4.3.4)
8(3.43)
Symmetry groupD4d
Dual polyhedronElongated square trapezohedron
Propertiesconvex
Net
A Johnson solid is one of 92 strictly convex polyhedra that is composed of regular polygon faces but are not uniform polyhedra (that is, they are not Platonic solids, Archimedean solids, prisms, or antiprisms). They were named by Norman Johnson, who first listed these polyhedra in 1966.[1]
The square gyrobicupola is the second in an infinite set of gyrobicupolae.
Related to the square gyrobicupola is the elongated square gyrobicupola. This polyhedron is created when an octagonal prism is inserted between the two halves of the square gyrobicupola. It is argued whether or not the elongated square gyrobicupola is an Archimedean solid because, although it meets every other standard necessary to be an Archimedean solid, it is not highly symmetric.
Formulae
The following formulae for volume and surface area can be used if all faces are regular, with edge length a:[2]
$V=\left(2+{\frac {4{\sqrt {2}}}{3}}\right)a^{3}\approx 3.88562...a^{3}$
$A=2\left(5+{\sqrt {3}}\right)a^{2}\approx 13.4641...a^{2}$
Related polyhedra and honeycombs
The square gyrobicupola forms space-filling honeycombs with tetrahedra, cubes and cuboctahedra; and with tetrahedra, square pyramids, and elongated square bipyramids. (The latter unit can be decomposed into elongated square pyramids, cubes, and/or square pyramids).[3]
References
1. Johnson, Norman W. (1966), "Convex polyhedra with regular faces", Canadian Journal of Mathematics, 18: 169–200, doi:10.4153/cjm-1966-021-8, MR 0185507, Zbl 0132.14603.
2. Stephen Wolfram, "Triangular gyrobicupola" from Wolfram Alpha. Retrieved July 23, 2010.
3. "J29 honeycomb".
External links
• Eric W. Weisstein, Square gyrobicupola (Johnson solid) at MathWorld.
Johnson solids
Pyramids, cupolae and rotundae
• square pyramid
• pentagonal pyramid
• triangular cupola
• square cupola
• pentagonal cupola
• pentagonal rotunda
Modified pyramids
• elongated triangular pyramid
• elongated square pyramid
• elongated pentagonal pyramid
• gyroelongated square pyramid
• gyroelongated pentagonal pyramid
• triangular bipyramid
• pentagonal bipyramid
• elongated triangular bipyramid
• elongated square bipyramid
• elongated pentagonal bipyramid
• gyroelongated square bipyramid
Modified cupolae and rotundae
• elongated triangular cupola
• elongated square cupola
• elongated pentagonal cupola
• elongated pentagonal rotunda
• gyroelongated triangular cupola
• gyroelongated square cupola
• gyroelongated pentagonal cupola
• gyroelongated pentagonal rotunda
• gyrobifastigium
• triangular orthobicupola
• square orthobicupola
• square gyrobicupola
• pentagonal orthobicupola
• pentagonal gyrobicupola
• pentagonal orthocupolarotunda
• pentagonal gyrocupolarotunda
• pentagonal orthobirotunda
• elongated triangular orthobicupola
• elongated triangular gyrobicupola
• elongated square gyrobicupola
• elongated pentagonal orthobicupola
• elongated pentagonal gyrobicupola
• elongated pentagonal orthocupolarotunda
• elongated pentagonal gyrocupolarotunda
• elongated pentagonal orthobirotunda
• elongated pentagonal gyrobirotunda
• gyroelongated triangular bicupola
• gyroelongated square bicupola
• gyroelongated pentagonal bicupola
• gyroelongated pentagonal cupolarotunda
• gyroelongated pentagonal birotunda
Augmented prisms
• augmented triangular prism
• biaugmented triangular prism
• triaugmented triangular prism
• augmented pentagonal prism
• biaugmented pentagonal prism
• augmented hexagonal prism
• parabiaugmented hexagonal prism
• metabiaugmented hexagonal prism
• triaugmented hexagonal prism
Modified Platonic solids
• augmented dodecahedron
• parabiaugmented dodecahedron
• metabiaugmented dodecahedron
• triaugmented dodecahedron
• metabidiminished icosahedron
• tridiminished icosahedron
• augmented tridiminished icosahedron
Modified Archimedean solids
• augmented truncated tetrahedron
• augmented truncated cube
• biaugmented truncated cube
• augmented truncated dodecahedron
• parabiaugmented truncated dodecahedron
• metabiaugmented truncated dodecahedron
• triaugmented truncated dodecahedron
• gyrate rhombicosidodecahedron
• parabigyrate rhombicosidodecahedron
• metabigyrate rhombicosidodecahedron
• trigyrate rhombicosidodecahedron
• diminished rhombicosidodecahedron
• paragyrate diminished rhombicosidodecahedron
• metagyrate diminished rhombicosidodecahedron
• bigyrate diminished rhombicosidodecahedron
• parabidiminished rhombicosidodecahedron
• metabidiminished rhombicosidodecahedron
• gyrate bidiminished rhombicosidodecahedron
• tridiminished rhombicosidodecahedron
Elementary solids
• snub disphenoid
• snub square antiprism
• sphenocorona
• augmented sphenocorona
• sphenomegacorona
• hebesphenomegacorona
• disphenocingulum
• bilunabirotunda
• triangular hebesphenorotunda
(See also List of Johnson solids, a sortable table)
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Square knot (mathematics)
In knot theory, the square knot is a composite knot obtained by taking the connected sum of a trefoil knot with its reflection. It is closely related to the granny knot, which is also a connected sum of two trefoils. Because the trefoil knot is the simplest nontrivial knot, the square knot and the granny knot are the simplest of all composite knots.
Square knot
Three-dimensional view
Common nameReef knot
Crossing no.6
Stick no.8
A–B notation$3_{1}\#3_{1}^{*}$
Other
alternating, composite, pretzel, slice, amphichiral, tricolorable
The square knot is the mathematical version of the common reef knot.
Construction
The square knot can be constructed from two trefoil knots, one of which must be left-handed and the other right-handed. Each of the two knots is cut, and then the loose ends are joined together pairwise. The resulting connected sum is the square knot.
It is important that the original trefoil knots be mirror images of one another. If two identical trefoil knots are used instead, the result is a granny knot.
Properties
The square knot is amphichiral, meaning that it is indistinguishable from its own mirror image. The crossing number of a square knot is six, which is the smallest possible crossing number for a composite knot.
The Alexander polynomial of the square knot is
$\Delta (t)=(t-1+t^{-1})^{2},\,$
which is simply the square of the Alexander polynomial of a trefoil knot. Similarly, the Alexander–Conway polynomial of a square knot is
$\nabla (z)=(z^{2}+1)^{2}.$
These two polynomials are the same as those for the granny knot. However, the Jones polynomial for the square knot is
$V(q)=(q^{-1}+q^{-3}-q^{-4})(q+q^{3}-q^{4})=-q^{3}+q^{2}-q+3-q^{-1}+q^{-2}-q^{-3}.\,$
This is the product of the Jones polynomials for the right-handed and left-handed trefoil knots, and is different from the Jones polynomial for a granny knot.
The knot group of the square knot is given by the presentation
$\langle x,y,z\mid xyx=yxy,xzx=zxz\rangle .\,$[1]
This is isomorphic to the knot group of the granny knot, and is the simplest example of two different knots with isomorphic knot groups.
Unlike the granny knot, the square knot is a ribbon knot, and it is therefore also a slice knot.
References
1. Weisstein, Eric W. "Square Knot". MathWorld.
Knot theory (knots and links)
Hyperbolic
• Figure-eight (41)
• Three-twist (52)
• Stevedore (61)
• 62
• 63
• Endless (74)
• Carrick mat (818)
• Perko pair (10161)
• (−2,3,7) pretzel (12n242)
• Whitehead (52
1
)
• Borromean rings (63
2
)
• L10a140
• Conway knot (11n34)
Satellite
• Composite knots
• Granny
• Square
• Knot sum
Torus
• Unknot (01)
• Trefoil (31)
• Cinquefoil (51)
• Septafoil (71)
• Unlink (02
1
)
• Hopf (22
1
)
• Solomon's (42
1
)
Invariants
• Alternating
• Arf invariant
• Bridge no.
• 2-bridge
• Brunnian
• Chirality
• Invertible
• Crosscap no.
• Crossing no.
• Finite type invariant
• Hyperbolic volume
• Khovanov homology
• Genus
• Knot group
• Link group
• Linking no.
• Polynomial
• Alexander
• Bracket
• HOMFLY
• Jones
• Kauffman
• Pretzel
• Prime
• list
• Stick no.
• Tricolorability
• Unknotting no. and problem
Notation
and operations
• Alexander–Briggs notation
• Conway notation
• Dowker–Thistlethwaite notation
• Flype
• Mutation
• Reidemeister move
• Skein relation
• Tabulation
Other
• Alexander's theorem
• Berge
• Braid theory
• Conway sphere
• Complement
• Double torus
• Fibered
• Knot
• List of knots and links
• Ribbon
• Slice
• Sum
• Tait conjectures
• Twist
• Wild
• Writhe
• Surgery theory
• Category
• Commons
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Square lattice
In mathematics, the square lattice is a type of lattice in a two-dimensional Euclidean space. It is the two-dimensional version of the integer lattice, denoted as $\mathbb {Z} ^{2}$.[1] It is one of the five types of two-dimensional lattices as classified by their symmetry groups;[2] its symmetry group in IUC notation as p4m,[3] Coxeter notation as [4,4],[4] and orbifold notation as *442.[5]
Square lattices
Upright square
Simple
diagonal square
Centered
Two orientations of an image of the lattice are by far the most common. They can conveniently be referred to as the upright square lattice and diagonal square lattice; the latter is also called the centered square lattice.[6] They differ by an angle of 45°. This is related to the fact that a square lattice can be partitioned into two square sub-lattices, as is evident in the colouring of a checkerboard.
Symmetry
The square lattice's symmetry category is wallpaper group p4m. A pattern with this lattice of translational symmetry cannot have more, but may have less symmetry than the lattice itself. An upright square lattice can be viewed as a diagonal square lattice with a mesh size that is √2 times as large, with the centers of the squares added. Correspondingly, after adding the centers of the squares of an upright square lattice one obtains a diagonal square lattice with a mesh size that is √2 times as small as that of the original lattice. A pattern with 4-fold rotational symmetry has a square lattice of 4-fold rotocenters that is a factor √2 finer and diagonally oriented relative to the lattice of translational symmetry.
With respect to reflection axes there are three possibilities:
• None. This is wallpaper group p4.
• In four directions. This is wallpaper group p4m.
• In two perpendicular directions. This is wallpaper group p4g. The points of intersection of the reflexion axes form a square grid which is as fine as, and oriented the same as, the square lattice of 4-fold rotocenters, with these rotocenters at the centers of the squares formed by the reflection axes.
p4, [4,4]+, (442) p4g, [4,4+], (4*2) p4m, [4,4], (*442)
Wallpaper group p4, with the arrangement within a primitive cell of the 2- and 4-fold rotocenters (also applicable for p4g and p4m).
Fundamental domain
Wallpaper group p4g. There are reflection axes in two directions, not through the 4-fold rotocenters.
Fundamental domain
Wallpaper group p4m. There are reflection axes in four directions, through the 4-fold rotocenters. In two directions the reflection axes are oriented the same as, and as dense as, those for p4g, but shifted. In the other two directions they are linearly a factor √2 denser.
Fundamental domain
Crystal classes
The square lattice class names, Schönflies notation, Hermann-Mauguin notation, orbifold notation, Coxeter notation, and wallpaper groups are listed in the table below.
Geometric class, point group Arithmetic
class
Wallpaper groups
Schön.IntlOrb.Cox.
C44(44)[4]+ None p4
(442)
D44mm(*44)[4] Both p4m
(*442)
p4g
(4*2)
See also
• Centered square number
• Euclid's orchard
• Gaussian integer
• Hexagonal lattice
• Quincunx
• Square tiling
References
1. Conway, John; Sloane, Neil J. A. (1999), Sphere Packings, Lattices and Groups, Springer, p. 106, ISBN 9780387985855.
2. Golubitsky, Martin; Stewart, Ian (2003), The Symmetry Perspective: From Equilibrium to Chaos in Phase Space and Physical Space, Progress in Mathematics, vol. 200, Springer, p. 129, ISBN 9783764321710.
3. Field, Michael; Golubitsky, Martin (2009), Symmetry in Chaos: A Search for Pattern in Mathematics, Art, and Nature (2nd ed.), SIAM, p. 47, ISBN 9780898717709.
4. Johnson, Norman W.; Weiss, Asia Ivić (1999), "Quadratic integers and Coxeter groups", Canadian Journal of Mathematics, 51 (6): 1307–1336, doi:10.4153/CJM-1999-060-6. See in particular the top of p. 1320.
5. Schattschneider, Doris; Senechal, Marjorie (2004), "Tilings", in Goodman, Jacob E.; O'Rourke, Joseph (eds.), Handbook of Discrete and Computational Geometry, Discrete Mathematics and Its Applications (2nd ed.), CRC Press, pp. 53–72, ISBN 9781420035315. See in particular the table on p. 62 relating IUC notation to orbifold notation.
6. Johnston, Bernard L.; Richman, Fred (1997), Numbers and Symmetry: An Introduction to Algebra, CRC Press, p. 159, ISBN 9780849303012.
Crystal systems
• Bravais lattice
• Crystallographic point group
Seven 3D systems
• triclinic (anorthic)
• monoclinic
• orthorhombic
• tetragonal
• trigonal & hexagonal
• cubic (isometric)
Four 2D systems
• oblique
• rectangular
• square
• hexagonal
Wikimedia Commons has media related to Square lattices.
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Square matrix
In mathematics, a square matrix is a matrix with the same number of rows and columns. An n-by-n matrix is known as a square matrix of order $n$. Any two square matrices of the same order can be added and multiplied.
Square matrices are often used to represent simple linear transformations, such as shearing or rotation. For example, if $R$ is a square matrix representing a rotation (rotation matrix) and $\mathbf {v} $ is a column vector describing the position of a point in space, the product $R\mathbf {v} $ yields another column vector describing the position of that point after that rotation. If $\mathbf {v} $ is a row vector, the same transformation can be obtained using $\mathbf {v} R^{\mathsf {T}}$, where $R^{\mathsf {T}}$ is the transpose of $R$.
Main diagonal
Main article: Main diagonal
The entries $a_{ii}$ (i = 1, ..., n) form the main diagonal of a square matrix. They lie on the imaginary line which runs from the top left corner to the bottom right corner of the matrix. For instance, the main diagonal of the 4×4 matrix above contains the elements a11 = 9, a22 = 11, a33 = 4, a44 = 10.
The diagonal of a square matrix from the top right to the bottom left corner is called antidiagonal or counterdiagonal.
Special kinds
NameExample with n = 3
Diagonal matrix${\begin{bmatrix}a_{11}&0&0\\0&a_{22}&0\\0&0&a_{33}\end{bmatrix}}$
Lower triangular matrix${\begin{bmatrix}a_{11}&0&0\\a_{21}&a_{22}&0\\a_{31}&a_{32}&a_{33}\end{bmatrix}}$
Upper triangular matrix${\begin{bmatrix}a_{11}&a_{12}&a_{13}\\0&a_{22}&a_{23}\\0&0&a_{33}\end{bmatrix}}$
Diagonal or triangular matrix
If all entries outside the main diagonal are zero, $A$ is called a diagonal matrix. If only all entries above (or below) the main diagonal are zero, $A$ is called an upper (or lower) triangular matrix.
Identity matrix
The identity matrix $I_{n}$ of size $n$ is the $n\times n$ matrix in which all the elements on the main diagonal are equal to 1 and all other elements are equal to 0, e.g.
$I_{1}={\begin{bmatrix}1\end{bmatrix}},\ I_{2}={\begin{bmatrix}1&0\\0&1\end{bmatrix}},\ \ldots ,\ I_{n}={\begin{bmatrix}1&0&\cdots &0\\0&1&\cdots &0\\\vdots &\vdots &\ddots &\vdots \\0&0&\cdots &1\end{bmatrix}}.$
It is a square matrix of order $n$, and also a special kind of diagonal matrix. It is called identity matrix because multiplication with it leaves a matrix unchanged:
$AI_{n}=I_{m}A=A$
for any m×n matrix $A$.
Invertible matrix and its inverse
A square matrix $A$ is called invertible or non-singular if there exists a matrix $B$ such that[1][2]
$AB=BA=I_{n}.$
If $B$ exists, it is unique and is called the inverse matrix of $A$, denoted $A^{-1}$.
Symmetric or skew-symmetric matrix
A square matrix $A$ that is equal to its transpose, i.e., $A^{\mathsf {T}}=A$, is a symmetric matrix. If instead $A^{\mathsf {T}}=-A$, then $A$ is called a skew-symmetric matrix.
For a complex square matrix $A$, often the appropriate analogue of the transpose is the conjugate transpose $A^{*}$, defined as the transpose of the complex conjugate of $A$. A complex square matrix $A$ satisfying $A^{*}=A$ is called a Hermitian matrix. If instead $A^{*}=-A$, then $A$ is called a skew-Hermitian matrix.
By the spectral theorem, real symmetric (or complex Hermitian) matrices have an orthogonal (or unitary) eigenbasis; i.e., every vector is expressible as a linear combination of eigenvectors. In both cases, all eigenvalues are real.[3]
Definite matrix
Positive definiteIndefinite
${\begin{bmatrix}1/4&0\\0&1\\\end{bmatrix}}$ ${\begin{bmatrix}1/4&0\\0&-1/4\end{bmatrix}}$
Q(x,y) = 1/4 x2 + y2 Q(x,y) = 1/4 x2 − 1/4 y2
Points such that Q(x, y) = 1
(Ellipse).
Points such that Q(x, y) = 1
(Hyperbola).
A symmetric n×n-matrix is called positive-definite (respectively negative-definite; indefinite), if for all nonzero vectors $x\in \mathbb {R} ^{n}$ the associated quadratic form given by
$Q(\mathbf {x} )=\mathbf {x} ^{\mathsf {T}}A\mathbf {x} $
takes only positive values (respectively only negative values; both some negative and some positive values).[4] If the quadratic form takes only non-negative (respectively only non-positive) values, the symmetric matrix is called positive-semidefinite (respectively negative-semidefinite); hence the matrix is indefinite precisely when it is neither positive-semidefinite nor negative-semidefinite.
A symmetric matrix is positive-definite if and only if all its eigenvalues are positive.[5] The table at the right shows two possibilities for 2×2 matrices.
Allowing as input two different vectors instead yields the bilinear form associated to A:[6]
$B_{A}(\mathbf {x} ,\mathbf {y} )=\mathbf {x} ^{\mathsf {T}}A\mathbf {y} .$
Orthogonal matrix
An orthogonal matrix is a square matrix with real entries whose columns and rows are orthogonal unit vectors (i.e., orthonormal vectors). Equivalently, a matrix A is orthogonal if its transpose is equal to its inverse:
$A^{\textsf {T}}=A^{-1},$
which entails
$A^{\textsf {T}}A=AA^{\textsf {T}}=I,$
where I is the identity matrix.
An orthogonal matrix A is necessarily invertible (with inverse A−1 = AT), unitary (A−1 = A*), and normal (A*A = AA*). The determinant of any orthogonal matrix is either +1 or −1. The special orthogonal group $\operatorname {SO} (n)$ consists of the n × n orthogonal matrices with determinant +1.
The complex analogue of an orthogonal matrix is a unitary matrix.
Normal matrix
A real or complex square matrix $A$ is called normal if $A^{*}A=AA^{*}$. If a real square matrix is symmetric, skew-symmetric, or orthogonal, then it is normal. If a complex square matrix is Hermitian, skew-Hermitian, or unitary, then it is normal. Normal matrices are of interest mainly because they include the types of matrices just listed and form the broadest class of matrices for which the spectral theorem holds.[7]
Operations
Trace
The trace, tr(A) of a square matrix A is the sum of its diagonal entries. While matrix multiplication is not commutative, the trace of the product of two matrices is independent of the order of the factors:
$\operatorname {tr} (AB)=\operatorname {tr} (BA).$
This is immediate from the definition of matrix multiplication:
$\operatorname {tr} (AB)=\sum _{i=1}^{m}\sum _{j=1}^{n}A_{ij}B_{ji}=\operatorname {tr} (BA).$
Also, the trace of a matrix is equal to that of its transpose, i.e.,
$\operatorname {tr} (A)=\operatorname {tr} (A^{\mathrm {T} }).$
Determinant
Main article: Determinant
The determinant $\det(A)$ or $|A|$ of a square matrix $A$ is a number encoding certain properties of the matrix. A matrix is invertible if and only if its determinant is nonzero. Its absolute value equals the area (in $\mathbb {R} ^{2}$) or volume (in $\mathbb {R} ^{3}$) of the image of the unit square (or cube), while its sign corresponds to the orientation of the corresponding linear map: the determinant is positive if and only if the orientation is preserved.
The determinant of 2×2 matrices is given by
$\det {\begin{bmatrix}a&b\\c&d\end{bmatrix}}=ad-bc.$
The determinant of 3×3 matrices involves 6 terms (rule of Sarrus). The more lengthy Leibniz formula generalizes these two formulae to all dimensions.[8]
The determinant of a product of square matrices equals the product of their determinants:[9]
$\det(AB)=\det(A)\cdot \det(B)$
Adding a multiple of any row to another row, or a multiple of any column to another column, does not change the determinant. Interchanging two rows or two columns affects the determinant by multiplying it by −1.[10] Using these operations, any matrix can be transformed to a lower (or upper) triangular matrix, and for such matrices the determinant equals the product of the entries on the main diagonal; this provides a method to calculate the determinant of any matrix. Finally, the Laplace expansion expresses the determinant in terms of minors, i.e., determinants of smaller matrices.[11] This expansion can be used for a recursive definition of determinants (taking as starting case the determinant of a 1×1 matrix, which is its unique entry, or even the determinant of a 0×0 matrix, which is 1), that can be seen to be equivalent to the Leibniz formula. Determinants can be used to solve linear systems using Cramer's rule, where the division of the determinants of two related square matrices equates to the value of each of the system's variables.[12]
Eigenvalues and eigenvectors
Main article: Eigenvalues and eigenvectors
A number λ and a non-zero vector $\mathbf {v} $ satisfying
$A\mathbf {v} =\lambda \mathbf {v} $
are called an eigenvalue and an eigenvector of $A$, respectively.[13][14] The number λ is an eigenvalue of an n×n-matrix A if and only if A − λIn is not invertible, which is equivalent to[15]
$\det(A-\lambda I)=0.$
The polynomial pA in an indeterminate X given by evaluation of the determinant det(XIn − A) is called the characteristic polynomial of A. It is a monic polynomial of degree n. Therefore the polynomial equation pA(λ) = 0 has at most n different solutions, i.e., eigenvalues of the matrix.[16] They may be complex even if the entries of A are real. According to the Cayley–Hamilton theorem, pA(A) = 0, that is, the result of substituting the matrix itself into its own characteristic polynomial yields the zero matrix.
See also
• Cartan matrix
Notes
1. Brown 1991, Definition I.2.28
2. Brown 1991, Definition I.5.13
3. Horn & Johnson 1985, Theorem 2.5.6
4. Horn & Johnson 1985, Chapter 7
5. Horn & Johnson 1985, Theorem 7.2.1
6. Horn & Johnson 1985, Example 4.0.6, p. 169
7. Artin, Algebra, 2nd edition, Pearson, 2018, section 8.6.
8. Brown 1991, Definition III.2.1
9. Brown 1991, Theorem III.2.12
10. Brown 1991, Corollary III.2.16
11. Mirsky 1990, Theorem 1.4.1
12. Brown 1991, Theorem III.3.18
13. Eigen means "own" in German and in Dutch.
14. Brown 1991, Definition III.4.1
15. Brown 1991, Definition III.4.9
16. Brown 1991, Corollary III.4.10
References
• Brown, William C. (1991), Matrices and vector spaces, New York, NY: Marcel Dekker, ISBN 978-0-8247-8419-5
• Horn, Roger A.; Johnson, Charles R. (1985), Matrix Analysis, Cambridge University Press, ISBN 978-0-521-38632-6
• Mirsky, Leonid (1990), An Introduction to Linear Algebra, Courier Dover Publications, ISBN 978-0-486-66434-7
External links
• Media related to Square matrices at Wikimedia Commons
Linear algebra
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Graph power
In graph theory, a branch of mathematics, the kth power Gk of an undirected graph G is another graph that has the same set of vertices, but in which two vertices are adjacent when their distance in G is at most k. Powers of graphs are referred to using terminology similar to that of exponentiation of numbers: G2 is called the square of G, G3 is called the cube of G, etc.[1]
Graph powers should be distinguished from the products of a graph with itself, which (unlike powers) generally have many more vertices than the original graph.
Properties
If a graph has diameter d, then its d-th power is the complete graph.[2] If a graph family has bounded clique-width, then so do its d-th powers for any fixed d.[3]
Coloring
Graph coloring on the square of a graph may be used to assign frequencies to the participants of wireless communication networks so that no two participants interfere with each other at any of their common neighbors,[4] and to find graph drawings with high angular resolution.[5]
Both the chromatic number and the degeneracy of the kth power of a planar graph of maximum degree Δ are O(Δ⌊k/2⌋), where the degeneracy bound shows that a greedy coloring algorithm may be used to color the graph with this many colors.[4] For the special case of a square of a planar graph, Wegner conjectured in 1977 that the chromatic number of the square of a planar graph is at most max(Δ + 5, 3Δ/2 + 1), and it is known that the chromatic number is at most 5Δ/3 + O(1).[6][7] More generally, for any graph with degeneracy d and maximum degree Δ, the degeneracy of the square of the graph is O(dΔ), so many types of sparse graph other than the planar graphs also have squares whose chromatic number is proportional to Δ.
Although the chromatic number of the square of a nonplanar graph with maximum degree Δ may be proportional to Δ2 in the worst case, it is smaller for graphs of high girth, being bounded by O(Δ2 / log Δ) in this case.[8]
Determining the minimum number of colors needed to color the square of a graph is NP-hard, even in the planar case.[9]
Hamiltonicity
The cube of every connected graph necessarily contains a Hamiltonian cycle.[10] It is not necessarily the case that the square of a connected graph is Hamiltonian, and it is NP-complete to determine whether the square is Hamiltonian.[11] Nevertheless, by Fleischner's theorem, the square of a 2-vertex-connected graph is always Hamiltonian.[12]
Computational complexity
The kth power of a graph with n vertices and m edges may be computed in time O(mn) by performing a breadth first search starting from each vertex to determine the distances to all other vertices, or slightly faster using more sophisticated algorithms.[13] Alternatively, If A is an adjacency matrix for the graph, modified to have nonzero entries on its main diagonal, then the nonzero entries of Ak give the adjacency matrix of the kth power of the graph,[14] from which it follows that constructing kth powers may be performed in an amount of time that is within a logarithmic factor of the time for matrix multiplication.
The kth powers of trees can be recognized in time linear in the size of the input graph. [15]
Given a graph, deciding whether it is the square of another graph is NP-complete. [16] Moreover, it is NP-complete to determine whether a graph is a kth power of another graph, for a given number k ≥ 2, or whether it is a kth power of a bipartite graph, for k > 2.[17]
Induced subgraphs
The half-square of a bipartite graph G is the subgraph of G2 induced by one side of the bipartition of G. Map graphs are the half-squares of planar graphs,[18] and halved cube graphs are the half-squares of hypercube graphs.[19]
Leaf powers are the subgraphs of powers of trees induced by the leaves of the tree. A k-leaf power is a leaf power whose exponent is k.[20]
References
1. Bondy, Adrian; Murty, U. S. R. (2008), Graph Theory, Graduate Texts in Mathematics, vol. 244, Springer, p. 82, ISBN 9781846289699.
2. Weisstein, Eric W. "Graph Power". MathWorld.
3. Todinca, Ioan (2003), "Coloring powers of graphs of bounded clique-width", Graph-theoretic concepts in computer science, Lecture Notes in Comput. Sci., vol. 2880, Springer, Berlin, pp. 370–382, doi:10.1007/978-3-540-39890-5_32, MR 2080095.
4. Agnarsson, Geir; Halldórsson, Magnús M. (2000), "Coloring powers of planar graphs", Proceedings of the Eleventh Annual ACM-SIAM Symposium on Discrete Algorithms (SODA '00), San Francisco, California, USA, pp. 654–662{{citation}}: CS1 maint: location missing publisher (link).
5. Formann, M.; Hagerup, T.; Haralambides, J.; Kaufmann, M.; Leighton, F. T.; Symvonis, A.; Welzl, E.; Woeginger, G. (1993), "Drawing graphs in the plane with high resolution", SIAM Journal on Computing, 22 (5): 1035–1052, doi:10.1137/0222063, MR 1237161.
6. Kramer, Florica; Kramer, Horst (2008), "A survey on the distance-colouring of graphs", Discrete Mathematics, 308 (2–3): 422–426, doi:10.1016/j.disc.2006.11.059, MR 2378044.
7. Molloy, Michael; Salavatipour, Mohammad R. (2005), "A bound on the chromatic number of the square of a planar graph", Journal of Combinatorial Theory, Series B, 94 (2): 189–213, doi:10.1016/j.jctb.2004.12.005, hdl:1807/9473, MR 2145512.
8. Alon, Noga; Mohar, Bojan (2002), "The chromatic number of graph powers", Combinatorics, Probability and Computing, 11 (1): 1–10, doi:10.1017/S0963548301004965, MR 1888178, S2CID 2706926.
9. Agnarsson & Halldórsson (2000) list publications proving NP-hardness for general graphs by McCormick (1983) and Lin and Skiena (1995), and for planar graphs by Ramanathan and Lloyd (1992, 1993).
10. Bondy & Murty (2008), p. 105.
11. Underground, Polly (1978), "On graphs with Hamiltonian squares", Discrete Mathematics, 21 (3): 323, doi:10.1016/0012-365X(78)90164-4, MR 0522906.
12. Diestel, Reinhard (2012), "10. Hamiltonian cycles", Graph Theory (PDF) (corrected 4th electronic ed.).
13. Chan, Timothy M. (2012), "All-pairs shortest paths for unweighted undirected graphs in $o(mn)$ time", ACM Transactions on Algorithms, 8 (4): A34:1–A34:17, doi:10.1145/2344422.2344424, MR 2981912, S2CID 1212001
14. Hammack, Richard; Imrich, Wilfried; Klavžar, Sandi (2011), Handbook of Product Graphs, Discrete Mathematics and Its Applications (2nd ed.), CRC Press, p. 94, ISBN 9781439813058.
15. Chang, Maw-Shang; Ko, Ming-Tat; Lu, Hsueh-I (2015), "Linear-Time Algorithms for Tree Root Problems", Algorithmica, 71 (2): 471–495, doi:10.1007/s00453-013-9815-y, S2CID 253971732.
16. Motwani, R.; Sudan, M. (1994), "Computing roots of graphs is hard", Discrete Applied Mathematics, 54: 81–88, doi:10.1016/0166-218x(94)00023-9.
17. Le, Van Bang; Nguyen, Ngoc Tuy (2010), "Hardness results and efficient algorithms for graph powers", Graph-Theoretic Concepts in Computer Science: 35th International Workshop, WG 2009, Montpellier, France, June 24-26, 2009, Revised Papers, Lecture Notes in Computer Science, vol. 5911, Berlin: Springer, pp. 238–249, doi:10.1007/978-3-642-11409-0_21, ISBN 978-3-642-11408-3, MR 2587715.
18. Chen, Zhi-Zhong; Grigni, Michelangelo; Papadimitriou, Christos H. (2002), "Map graphs", Journal of the ACM, 49 (2): 127–138, arXiv:cs/9910013, doi:10.1145/506147.506148, MR 2147819, S2CID 2657838.
19. Shpectorov, S. V. (1993), "On scale embeddings of graphs into hypercubes", European Journal of Combinatorics, 14 (2): 117–130, doi:10.1006/eujc.1993.1016, MR 1206617.
20. Nishimura, N.; Ragde, P.; Thilikos, D.M. (2002), "On graph powers for leaf-labeled trees", Journal of Algorithms, 42: 69–108, doi:10.1006/jagm.2001.1195.
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Square orthobicupola
In geometry, the square orthobicupola is one of the Johnson solids (J28). As the name suggests, it can be constructed by joining two square cupolae (J4) along their octagonal bases, matching like faces. A 45-degree rotation of one cupola before the joining yields a square gyrobicupola (J29).
Square orthobicupola
TypeJohnson
J27 – J28 – J29
Faces8 triangles
2+8 squares
Edges32
Vertices16
Vertex configuration8(32.42)
8(3.43)
Symmetry groupD4h
Dual polyhedron-
Propertiesconvex
Net
A Johnson solid is one of 92 strictly convex polyhedra that is composed of regular polygon faces but are not uniform polyhedra (that is, they are not Platonic solids, Archimedean solids, prisms, or antiprisms). They were named by Norman Johnson, who first listed these polyhedra in 1966.[1]
The square orthobicupola is the second in an infinite set of orthobicupolae.
The square orthobicupola can be elongated by the insertion of an octagonal prism between its two cupolae to yield a rhombicuboctahedron, or collapsed by the removal of an irregular hexagonal prism to yield an elongated square dipyramid (J15), which itself is merely an elongated octahedron.
It can be constructed from the disphenocingulum (J90) by replacing the band of up-and-down triangles by a band of rectangles, while fixing two opposite sphenos.
Related polyhedra and honeycombs
The square orthobicupola forms space-filling honeycombs with tetrahedra; with cubes and cuboctahedra; with tetrahedra and cubes; with square pyramids, tetrahedra and various combinations of cubes, elongated square pyramids and/or elongated square bipyramids.[2]
References
1. Johnson, Norman W. (1966), "Convex polyhedra with regular faces", Canadian Journal of Mathematics, 18: 169–200, doi:10.4153/cjm-1966-021-8, MR 0185507, Zbl 0132.14603.
2. "J28 honeycomb".
External links
• Eric W. Weisstein, Square orthobicupola (Johnson solid) at MathWorld.
Johnson solids
Pyramids, cupolae and rotundae
• square pyramid
• pentagonal pyramid
• triangular cupola
• square cupola
• pentagonal cupola
• pentagonal rotunda
Modified pyramids
• elongated triangular pyramid
• elongated square pyramid
• elongated pentagonal pyramid
• gyroelongated square pyramid
• gyroelongated pentagonal pyramid
• triangular bipyramid
• pentagonal bipyramid
• elongated triangular bipyramid
• elongated square bipyramid
• elongated pentagonal bipyramid
• gyroelongated square bipyramid
Modified cupolae and rotundae
• elongated triangular cupola
• elongated square cupola
• elongated pentagonal cupola
• elongated pentagonal rotunda
• gyroelongated triangular cupola
• gyroelongated square cupola
• gyroelongated pentagonal cupola
• gyroelongated pentagonal rotunda
• gyrobifastigium
• triangular orthobicupola
• square orthobicupola
• square gyrobicupola
• pentagonal orthobicupola
• pentagonal gyrobicupola
• pentagonal orthocupolarotunda
• pentagonal gyrocupolarotunda
• pentagonal orthobirotunda
• elongated triangular orthobicupola
• elongated triangular gyrobicupola
• elongated square gyrobicupola
• elongated pentagonal orthobicupola
• elongated pentagonal gyrobicupola
• elongated pentagonal orthocupolarotunda
• elongated pentagonal gyrocupolarotunda
• elongated pentagonal orthobirotunda
• elongated pentagonal gyrobirotunda
• gyroelongated triangular bicupola
• gyroelongated square bicupola
• gyroelongated pentagonal bicupola
• gyroelongated pentagonal cupolarotunda
• gyroelongated pentagonal birotunda
Augmented prisms
• augmented triangular prism
• biaugmented triangular prism
• triaugmented triangular prism
• augmented pentagonal prism
• biaugmented pentagonal prism
• augmented hexagonal prism
• parabiaugmented hexagonal prism
• metabiaugmented hexagonal prism
• triaugmented hexagonal prism
Modified Platonic solids
• augmented dodecahedron
• parabiaugmented dodecahedron
• metabiaugmented dodecahedron
• triaugmented dodecahedron
• metabidiminished icosahedron
• tridiminished icosahedron
• augmented tridiminished icosahedron
Modified Archimedean solids
• augmented truncated tetrahedron
• augmented truncated cube
• biaugmented truncated cube
• augmented truncated dodecahedron
• parabiaugmented truncated dodecahedron
• metabiaugmented truncated dodecahedron
• triaugmented truncated dodecahedron
• gyrate rhombicosidodecahedron
• parabigyrate rhombicosidodecahedron
• metabigyrate rhombicosidodecahedron
• trigyrate rhombicosidodecahedron
• diminished rhombicosidodecahedron
• paragyrate diminished rhombicosidodecahedron
• metagyrate diminished rhombicosidodecahedron
• bigyrate diminished rhombicosidodecahedron
• parabidiminished rhombicosidodecahedron
• metabidiminished rhombicosidodecahedron
• gyrate bidiminished rhombicosidodecahedron
• tridiminished rhombicosidodecahedron
Elementary solids
• snub disphenoid
• snub square antiprism
• sphenocorona
• augmented sphenocorona
• sphenomegacorona
• hebesphenomegacorona
• disphenocingulum
• bilunabirotunda
• triangular hebesphenorotunda
(See also List of Johnson solids, a sortable table)
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