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Truncated rhombicuboctahedron
The truncated rhombicuboctahedron is a polyhedron, constructed as a truncation of the rhombicuboctahedron. It has 50 faces consisting of 18 octagons, 8 hexagons, and 24 squares. It can fill space with the truncated cube, truncated tetrahedron and triangular prism as a truncated runcic cubic honeycomb.
Truncated rhombicuboctahedron
Schläfli symboltrr{4,3} = $tr{\begin{Bmatrix}4\\3\end{Bmatrix}}$
Conway notationtaaC
Faces50:
24 {4}
8 {6}
6+12 {8}
Edges144
Vertices96
Symmetry groupOh, [4,3], (*432) order 48
Rotation groupO, [4,3]+, (432), order 24
Dual polyhedronDisdyakis icositetrahedron
Propertiesconvex, zonohedron
Other names
• Truncated small rhombicuboctahedron
• Beveled cuboctahedron
Zonohedron
As a zonohedron, it can be constructed with all but 12 octagons as regular polygons. It has two sets of 48 vertices existing on two distances from its center.
It represents the Minkowski sum of a cube, a truncated octahedron, and a rhombic dodecahedron.
Excavated truncated rhombicuboctahedron
Excavated truncated rhombicuboctahedron
Faces148:
8 {3}
24+96+6 {4}
8 {6}
6 {8}
Edges312
Vertices144
Euler characteristic-20
Genus11
Symmetry groupOh, [4,3], (*432) order 48
The excavated truncated rhombicuboctahedron is a toroidal polyhedron, constructed from a truncated rhombicuboctahedron with its 12 irregular octagonal faces removed. It comprises a network of 6 square cupolae, 8 triangular cupolae, and 24 triangular prisms. [1] It has 148 faces (8 triangles, 126 squares, 8 hexagons, and 6 octagons), 312 edges, and 144 vertices. With Euler characteristic χ = f + v - e = -20, its genus (g = (2-χ)/2) is 11.
Without the triangular prisms, the toroidal polyhedron becomes a truncated cuboctahedron.
Excavated
Truncated rhombicuboctahedron Truncated cuboctahedron
Related polyhedra
The truncated cuboctahedron is similar, with all regular faces, and 4.6.8 vertex figure.
The triangle and squares of the rhombicuboctahedron can be independently rectified or truncated, creating four permutations of polyhedra. The partially truncated forms can be seen as edge contractions of the truncated form.
The truncated rhombicuboctahedron can be seen in sequence of rectification and truncation operations from the cuboctahedron. A further alternation step leads to the snub rhombicuboctahedron.
related polyhedra
Name r{4,3} rr{4,3} tr{4,3} Rectified
rrr{4,3}
Partially truncated Truncated
trr{4,3}
srCO
Conway aC aaC=eC taC=bC aaaC=eaC dXC dXdC taaC=baC saC
Image
VertFigs 3.4.3.4 3.4.4.4 4.6.8 4.4.4.4d and
3.4.4d.4
4.4.4.6i and
4.6.6i
4.6i.8 and
3.4.6i.4
4.8.8p and
4.6.8p
3.3.3.3.4 and
3.3.4.3.4
See also
• Expanded cuboctahedron
• Truncated rhombicosidodecahedron
References
1. "Prism Expansions".
• Eppstein, David (1996). "Zonohedra and zonotopes". Mathematica in Education and Research. 5 (4): 15–21.
• Coxeter Regular Polytopes, Third edition, (1973), Dover edition, ISBN 0-486-61480-8 (pp. 145–154 Chapter 8: Truncation)
• John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things 2008, ISBN 978-1-56881-220-5
External links
• George Hart's Conway interpreter: generates polyhedra in VRML, taking Conway notation as input
• Prism Expansions Toroid model
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Truncated square antiprism
The truncated square antiprism one in an infinite series of truncated antiprisms, constructed as a truncated square antiprism. It has 18 faces, 2 octagons, 8 hexagons, and 8 squares.
Truncated square antiprism
TypeTruncated antiprism
Schläfli symbolts{2,8}
tsr{4,2} or $ts{\begin{Bmatrix}4\\2\end{Bmatrix}}$
Conway notationtA4
Faces18: 2 {8}, 8 {6}, 8 {4}
Edges48
Vertices32
Symmetry groupD4d, [2+,8], (2*4), order 16
Rotation groupD4, [2,4]+, (224), order 8
Dual polyhedron
Propertiesconvex, zonohedron
Gyroelongated triamond square bicupola
If the hexagons are folded, it can be constructed by regular polygons. Or each folded hexagon can be replaced by two triamonds, adding 8 edges (56), and 4 faces (32). This form is called a gyroelongated triamond square bicupola.[1]
Related polyhedra
Truncated antiprisms
Symmetry D2d, [2+,4], (2*2) D3d, [2+,6], (2*3) D4d, [2+,8], (2*4) D5d, [2+,10], (2*5)
Antiprisms
s{2,4}
(v:4; e:8; f:6)
s{2,6}
(v:6; e:12; f:8)
s{2,8}
(v:8; e:16; f:10)
s{2,10}
(v:10; e:20; f:12)
Truncated
antiprisms
ts{2,4}
(v:16;e:24;f:10)
ts{2,6}
(v:24; e:36; f:14)
ts{2,8}
(v:32; e:48; f:18)
ts{2,10}
(v:40; e:60; f:22)
Snub square antiprism
Although it can't be made by all regular planar faces, its alternation is the Johnson solid, the snub square antiprism.
References
1. Convex Triamond Regular Polyhedra
• Snub Anti-Prisms
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Truncated square trapezohedron
In geometry, the square truncated trapezohedron is the second in an infinite series of truncated trapezohedra. It has 8 pentagon and 2 square faces.
Truncated square trapezohedron
TypeTruncated trapezohedron
Johnson solid dual
Faces8 pentagons,
2 squares
Edges24
Vertices16
Symmetry groupD4d, [2+,8], (2*4)
Dual polyhedronGyroelongated square bipyramid (J17)
Propertiesconvex
This polyhedron can be constructed by taking a tetragonal trapezohedron and truncating the polar axis vertices. The kite faces of the trapezohedron become pentagons.
The vertices exist as 4 squares in four parallel planes, with alternating orientation in the middle creating the pentagons.
A truncated trapezohedron has all valence-3 vertices. This means that the dual polyhedrona gyroelongated square dipyramid has all triangular faces.
It represents the dual polyhedron to the Johnson solid, gyroelongated square dipyramid (J17), with specific proportions:
Square truncated trapezohedron Net
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Compound of two truncated tetrahedra
This uniform polyhedron compound is a composition of two truncated tetrahedra, formed by truncating each of the tetrahedra in the stellated octahedron. It is related to the cantic cube construction of the truncated tetrahedron, as , which is one of the two dual positions represented in this compound.
Compound of two truncated tetrahedra
TypeUniform compound
IndexUC54
Schläfli symbola2{4,3}
Coxeter diagram + =
Polyhedra2 truncated tetrahedra
Faces8 triangles
8 hexagons
Edges36
Vertices24
Symmetry groupoctahedral (Oh)
[4,3]
Subgroup restricting
to one constituent
tetrahedral (Td)
[3,3]
The vertex arrangement is the same as a convex, but nonuniform rhombicuboctahedron having 12 rectangular faces.
References
• Skilling, John (1976), "Uniform Compounds of Uniform Polyhedra", Mathematical Proceedings of the Cambridge Philosophical Society, 79 (3): 447–457, doi:10.1017/S0305004100052440, MR 0397554.
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Truncated tesseractic honeycomb
In four-dimensional Euclidean geometry, the truncated tesseractic honeycomb is a uniform space-filling tessellation (or honeycomb) in Euclidean 4-space. It is constructed by a truncation of a tesseractic honeycomb creating truncated tesseracts, and adding new 16-cell facets at the original vertices.
Truncated tesseractic honeycomb
(No image)
TypeUniform 4-honeycomb
Schläfli symbolt{4,3,3,4}
t{4,3,31,1}
Coxeter-Dynkin diagram
4-face typetruncated tesseract
16-cell
Cell typeTruncated cube
Tetrahedron
Face type{3}, {8}
Vertex figureoctahedral pyramid
Coxeter group${\tilde {C}}_{4}$ = [4,3,3,4]
${\tilde {B}}_{4}$ = [4,3,31,1]
Dual
Propertiesvertex-transitive
Related honeycombs
The [4,3,3,4], , Coxeter group generates 31 permutations of uniform tessellations, 21 with distinct symmetry and 20 with distinct geometry. The expanded tesseractic honeycomb (also known as the stericated tesseractic honeycomb) is geometrically identical to the tesseractic honeycomb. Three of the symmetric honeycombs are shared in the [3,4,3,3] family. Two alternations (13) and (17), and the quarter tesseractic (2) are repeated in other families.
C4 honeycombs
Extended
symmetry
Extended
diagram
Order Honeycombs
[4,3,3,4]: ×1
1, 2, 3, 4,
5, 6, 7, 8,
9, 10, 11, 12,
13
[[4,3,3,4]] ×2 (1), (2), (13), 18
(6), 19, 20
[(3,3)[1+,4,3,3,4,1+]]
↔ [(3,3)[31,1,1,1]]
↔ [3,4,3,3]
↔
↔
×6
14, 15, 16, 17
See also
Regular and uniform honeycombs in 4-space:
• Tesseractic honeycomb
• Demitesseractic honeycomb
• 24-cell honeycomb
• Truncated 24-cell honeycomb
• Snub 24-cell honeycomb
• 5-cell honeycomb
• Truncated 5-cell honeycomb
• Omnitruncated 5-cell honeycomb
Notes
References
• Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6
• (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45] See p318
• George Olshevsky, Uniform Panoploid Tetracombs, Manuscript (2006) (Complete list of 11 convex uniform tilings, 28 convex uniform honeycombs, and 143 convex uniform tetracombs)
• Klitzing, Richard. "4D Euclidean tesselations#4D". o3o3o *b3x4x, x4x3o3o4o - tattit - O89
• Conway JH, Sloane NJH (1998). Sphere Packings, Lattices and Groups (3rd ed.). ISBN 0-387-98585-9.
Fundamental convex regular and uniform honeycombs in dimensions 2–9
Space Family ${\tilde {A}}_{n-1}$ ${\tilde {C}}_{n-1}$ ${\tilde {B}}_{n-1}$ ${\tilde {D}}_{n-1}$ ${\tilde {G}}_{2}$ / ${\tilde {F}}_{4}$ / ${\tilde {E}}_{n-1}$
E2 Uniform tiling {3[3]} δ3 hδ3 qδ3 Hexagonal
E3 Uniform convex honeycomb {3[4]} δ4 hδ4 qδ4
E4 Uniform 4-honeycomb {3[5]} δ5 hδ5 qδ5 24-cell honeycomb
E5 Uniform 5-honeycomb {3[6]} δ6 hδ6 qδ6
E6 Uniform 6-honeycomb {3[7]} δ7 hδ7 qδ7 222
E7 Uniform 7-honeycomb {3[8]} δ8 hδ8 qδ8 133 • 331
E8 Uniform 8-honeycomb {3[9]} δ9 hδ9 qδ9 152 • 251 • 521
E9 Uniform 9-honeycomb {3[10]} δ10 hδ10 qδ10
E10 Uniform 10-honeycomb {3[11]} δ11 hδ11 qδ11
En-1 Uniform (n-1)-honeycomb {3[n]} δn hδn qδn 1k2 • 2k1 • k21
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Truncated tetraapeirogonal tiling
In geometry, the truncated tetraapeirogonal tiling is a semiregular tiling of the hyperbolic plane. There are one square, one octagon, and one apeirogon on each vertex. It has Schläfli symbol of tr{∞,4}.
Truncated tetraapeirogonal tiling
Poincaré disk model of the hyperbolic plane
TypeHyperbolic uniform tiling
Vertex configuration4.8.∞
Schläfli symboltr{∞,4} or $t{\begin{Bmatrix}\infty \\4\end{Bmatrix}}$
Wythoff symbol2 ∞ 4 |
Coxeter diagram or
Symmetry group[∞,4], (*∞42)
DualOrder 4-infinite kisrhombille
PropertiesVertex-transitive
Related polyhedra and tilings
Paracompact uniform tilings in [∞,4] family
{∞,4} t{∞,4} r{∞,4} 2t{∞,4}=t{4,∞} 2r{∞,4}={4,∞} rr{∞,4} tr{∞,4}
Dual figures
V∞4 V4.∞.∞ V(4.∞)2 V8.8.∞ V4∞ V43.∞ V4.8.∞
Alternations
[1+,∞,4]
(*44∞)
[∞+,4]
(∞*2)
[∞,1+,4]
(*2∞2∞)
[∞,4+]
(4*∞)
[∞,4,1+]
(*∞∞2)
[(∞,4,2+)]
(2*2∞)
[∞,4]+
(∞42)
=
=
h{∞,4} s{∞,4} hr{∞,4} s{4,∞} h{4,∞} hrr{∞,4} s{∞,4}
Alternation duals
V(∞.4)4 V3.(3.∞)2 V(4.∞.4)2 V3.∞.(3.4)2 V∞∞ V∞.44 V3.3.4.3.∞
*n42 symmetry mutation of omnitruncated tilings: 4.8.2n
Symmetry
*n42
[n,4]
Spherical Euclidean Compact hyperbolic Paracomp.
*242
[2,4]
*342
[3,4]
*442
[4,4]
*542
[5,4]
*642
[6,4]
*742
[7,4]
*842
[8,4]...
*∞42
[∞,4]
Omnitruncated
figure
4.8.4
4.8.6
4.8.8
4.8.10
4.8.12
4.8.14
4.8.16
4.8.∞
Omnitruncated
duals
V4.8.4
V4.8.6
V4.8.8
V4.8.10
V4.8.12
V4.8.14
V4.8.16
V4.8.∞
*nn2 symmetry mutations of omnitruncated tilings: 4.2n.2n
Symmetry
*nn2
[n,n]
Spherical Euclidean Compact hyperbolic Paracomp.
*222
[2,2]
*332
[3,3]
*442
[4,4]
*552
[5,5]
*662
[6,6]
*772
[7,7]
*882
[8,8]...
*∞∞2
[∞,∞]
Figure
Config. 4.4.4 4.6.6 4.8.8 4.10.10 4.12.12 4.14.14 4.16.16 4.∞.∞
Dual
Config. V4.4.4 V4.6.6 V4.8.8 V4.10.10 V4.12.12 V4.14.14 V4.16.16 V4.∞.∞
Symmetry
The dual of this tiling represents the fundamental domains of [∞,4], (*∞42) symmetry. There are 15 small index subgroups constructed from [∞,4] by mirror removal and alternation. Mirrors can be removed if its branch orders are all even, and cuts neighboring branch orders in half. Removing two mirrors leaves a half-order gyration point where the removed mirrors met. In these images fundamental domains are alternately colored black and white, and mirrors exist on the boundaries between colors. The subgroup index-8 group, [1+,∞,1+,4,1+] (∞2∞2) is the commutator subgroup of [∞,4].
A larger subgroup is constructed as [∞,4*], index 8, as [∞,4+], (4*∞) with gyration points removed, becomes (*∞∞∞∞) or (*∞4), and another [∞*,4], index ∞ as [∞+,4], (∞*2) with gyration points removed as (*2∞). And their direct subgroups [∞,4*]+, [∞*,4]+, subgroup indices 16 and ∞ respectively, can be given in orbifold notation as (∞∞∞∞) and (2∞).
Small index subgroups of [∞,4], (*∞42)
Index 1 2 4
Diagram
Coxeter [∞,4]
[1+,∞,4]
=
[∞,4,1+]
=
[∞,1+,4]
=
[1+,∞,4,1+]
=
[∞+,4+]
Orbifold *∞42 *∞44 *∞∞2 *∞222 *∞2∞2 ∞2×
Semidirect subgroups
Diagram
Coxeter [∞,4+]
[∞+,4]
[(∞,4,2+)]
[1+,∞,1+,4]
= =
= =
[∞,1+,4,1+]
= =
= =
Orbifold 4*∞ ∞*2 2*∞2 ∞*22 2*∞∞
Direct subgroups
Index 2 4 8
Diagram
Coxeter [∞,4]+
=
[∞,4+]+
=
[∞+,4]+
=
[∞,1+,4]+
=
[∞+,4+]+ = [1+,∞,1+,4,1+]
= = =
Orbifold ∞42 ∞44 ∞∞2 ∞222 ∞2∞2
Radical subgroups
Index 8 ∞ 16 ∞
Diagram
Coxeter [∞,4*]
=
[∞*,4]
[∞,4*]+
=
[∞*,4]+
Orbifold *∞∞∞∞ *2∞ ∞∞∞∞ 2∞
See also
Wikimedia Commons has media related to Uniform tiling 4-8-i.
• Tilings of regular polygons
• List of uniform planar tilings
References
• John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 19, The Hyperbolic Archimedean Tessellations)
• "Chapter 10: Regular honeycombs in hyperbolic space". The Beauty of Geometry: Twelve Essays. Dover Publications. 1999. ISBN 0-486-40919-8. LCCN 99035678.
External links
• Weisstein, Eric W. "Hyperbolic tiling". MathWorld.
• Weisstein, Eric W. "Poincaré hyperbolic disk". MathWorld.
• Hyperbolic and Spherical Tiling Gallery
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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Truncated tetrahedral prism
In geometry, a truncated tetrahedral prism is a convex uniform polychoron (four-dimensional polytope). This polychoron has 10 polyhedral cells: 2 truncated tetrahedra connected by 4 triangular prisms and 4 hexagonal prisms. It has 24 faces: 8 triangular, 18 square, and 8 hexagons. It has 48 edges and 24 vertices.
Truncated tetrahedral prism
Schlegel diagram
TypePrismatic uniform polychoron
Uniform index49
Schläfli symbolt{3,3}×{}
Coxeter-Dynkin
Cells10:
2 3.6.6
4 3.4.4
4 4.4.6
Faces24:
8 {3} + 18 {4} + 8 {6}
Edges48
Vertices24
Vertex figure
Isosceles-triangular pyramid
Symmetry group[3,3,2], order 48
Propertiesconvex
It is one of 18 uniform polyhedral prisms created by using uniform prisms to connect pairs of parallel Platonic solids and Archimedean solids.
Net
Alternative names
1. Truncated-tetrahedral dyadic prism (Norman W. Johnson)
2. Tuttip (Jonathan Bowers: for truncated-tetrahedral prism)
3. Truncated tetrahedral hyperprism
External links
• 6. Convex uniform prismatic polychora - Model 49, George Olshevsky.
• Klitzing, Richard. "4D uniform polytopes (polychora) x x3x3o - tuttip".
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Truncated tetrahedron
In geometry, the truncated tetrahedron is an Archimedean solid. It has 4 regular hexagonal faces, 4 equilateral triangle faces, 12 vertices and 18 edges (of two types). It can be constructed by truncating all 4 vertices of a regular tetrahedron at one third of the original edge length.
Truncated tetrahedron
(Click here for rotating model)
TypeArchimedean solid
Uniform polyhedron
ElementsF = 8, E = 18, V = 12 (χ = 2)
Faces by sides4{3}+4{6}
Conway notationtT
Schläfli symbolst{3,3} = h2{4,3}
t0,1{3,3}
Wythoff symbol2 3 | 3
Coxeter diagram =
Symmetry groupTd, A3, [3,3], (*332), order 24
Rotation groupT, [3,3]+, (332), order 12
Dihedral angle3-6: 109°28′16″
6-6: 70°31′44″
ReferencesU02, C16, W6
PropertiesSemiregular convex
Colored faces
3.6.6
(Vertex figure)
Triakis tetrahedron
(dual polyhedron)
Net
A deeper truncation, removing a tetrahedron of half the original edge length from each vertex, is called rectification. The rectification of a tetrahedron produces an octahedron.[1]
A truncated tetrahedron is the Goldberg polyhedron GIII(1,1), containing triangular and hexagonal faces.
A truncated tetrahedron can be called a cantic cube, with Coxeter diagram, , having half of the vertices of the cantellated cube (rhombicuboctahedron), . There are two dual positions of this construction, and combining them creates the uniform compound of two truncated tetrahedra.
Area and volume
The area A and the volume V of a truncated tetrahedron of edge length a are:
${\begin{aligned}A&=7{\sqrt {3}}a^{2}&&\approx 12.124\,355\,65a^{2}\\V&={\tfrac {23}{12}}{\sqrt {2}}a^{3}&&\approx 2.710\,575\,995a^{3}.\end{aligned}}$
Densest packing
The densest packing of the Archimedean truncated tetrahedron is believed to be Φ = 207/208, as reported by two independent groups using Monte Carlo methods.[2][3] Although no mathematical proof exists that this is the best possible packing for the truncated tetrahedron, the high proximity to the unity and independency of the findings make it unlikely that an even denser packing is to be found. In fact, if the truncation of the corners is slightly smaller than that of an Archimedean truncated tetrahedron, this new shape can be used to completely fill space.[2]
Cartesian coordinates
Cartesian coordinates for the 12 vertices of a truncated tetrahedron centered at the origin, with edge length √8, are all permutations of (±1,±1,±3) with an even number of minus signs:
• (+3,+1,+1), (+1,+3,+1), (+1,+1,+3)
• (−3,−1,+1), (−1,−3,+1), (−1,−1,+3)
• (−3,+1,−1), (−1,+3,−1), (−1,+1,−3)
• (+3,−1,−1), (+1,−3,−1), (+1,−1,−3)
Orthogonal projection showing Cartesian coordinates inside its bounding box: (±3,±3,±3). The hexagonal faces of the truncated tetrahedra can be divided into six coplanar equilateral triangles. The four new vertices have Cartesian coordinates:
(−1,−1,−1), (−1,+1,+1),
(+1,−1,+1), (+1,+1,−1). As a solid, this can represent a 3D dissection, making four red octahedra and six yellow tetrahedra.
The set of vertex permutations (±1,±1,±3) with an odd number of minus signs forms a complementary truncated tetrahedron, and combined they form a uniform compound polyhedron.
Another simple construction exists in 4-space as cells of the truncated 16-cell, with vertices as coordinate permutation of:
(0,0,1,2)
Orthogonal projection
Orthogonal projection
Centered by Edge normal Face normal Edge Face
Wireframe
Wireframe
Dual
Projective
symmetry
[1] [1] [4] [3]
Spherical tiling
The truncated tetrahedron can also be represented as a spherical tiling, and projected onto the plane via a stereographic projection. This projection is conformal, preserving angles but not areas or lengths. Straight lines on the sphere are projected as circular arcs on the plane.
triangle-centered
hexagon-centered
Orthographic projection Stereographic projections
Friauf polyhedron
A lower symmetry version of the truncated tetrahedron (a truncated tetragonal disphenoid with order 8 D2d symmetry) is called a Friauf polyhedron in crystals such as complex metallic alloys. This form fits 5 Friauf polyhedra around an axis, giving a 72-degree dihedral angle on a subset of 6-6 edges.[4] It is named after J. B. Friauf and his 1927 paper "The crystal structure of the intermetallic compound MgCu2".[5]
Uses
Giant truncated tetrahedra were used for the "Man the Explorer" and "Man the Producer" theme pavilions in Expo 67. They were made of massive girders of steel bolted together in a geometric lattice. The truncated tetrahedra were interconnected with lattice steel platforms. All of these buildings were demolished after the end of Expo 67, as they had not been built to withstand the severity of the Montreal weather over the years. Their only remnants are in the Montreal city archives, the Public Archives Of Canada and the photo collections of tourists of the times.[6]
The Tetraminx puzzle has a truncated tetrahedral shape. This puzzle shows a dissection of a truncated tetrahedron into 4 octahedra and 6 tetrahedra. It contains 4 central planes of rotations.
Truncated tetrahedral graph
Truncated tetrahedral graph
3-fold symmetry
Vertices12[7]
Edges18
Radius3
Diameter3[7]
Girth3[7]
Automorphisms24 (S4)[7]
Chromatic number3[7]
Chromatic index3[7]
PropertiesHamiltonian, regular, 3-vertex-connected, planar graph
Table of graphs and parameters
In the mathematical field of graph theory, a truncated tetrahedral graph is an Archimedean graph, the graph of vertices and edges of the truncated tetrahedron, one of the Archimedean solids. It has 12 vertices and 18 edges.[8] It is a connected cubic graph,[9] and connected cubic transitive graph.[10]
Circular
Orthographic projections
4-fold symmetry
3-fold symmetry
Related polyhedra and tilings
Family of uniform tetrahedral polyhedra
Symmetry: [3,3], (*332) [3,3]+, (332)
{3,3} t{3,3} r{3,3} t{3,3} {3,3} rr{3,3} tr{3,3} sr{3,3}
Duals to uniform polyhedra
V3.3.3 V3.6.6 V3.3.3.3 V3.6.6 V3.3.3 V3.4.3.4 V4.6.6 V3.3.3.3.3
It is also a part of a sequence of cantic polyhedra and tilings with vertex configuration 3.6.n.6. In this wythoff construction the edges between the hexagons represent degenerate digons.
*n33 orbifold symmetries of cantic tilings: 3.6.n.6
Orbifold
*n32
Spherical Euclidean Hyperbolic Paracompact
*332 *333 *433 *533 *633... *∞33
Cantic figure
Vertex 3.6.2.6 3.6.3.6 3.6.4.6 3.6.5.6 3.6.6.6 3.6.∞.6
Symmetry mutations
This polyhedron is topologically related as a part of sequence of uniform truncated polyhedra with vertex configurations (3.2n.2n), and [n,3] Coxeter group symmetry.
*n32 symmetry mutation of truncated spherical tilings: t{n,3}
Symmetry
*n32
[n,3]
Spherical Euclid. Compact hyperb. Paraco.
*232
[2,3]
*332
[3,3]
*432
[4,3]
*532
[5,3]
*632
[6,3]
*732
[7,3]
*832
[8,3]...
*∞32
[∞,3]
Truncated
figures
Symbol t{2,3} t{3,3} t{4,3} t{5,3} t{6,3} t{7,3} t{8,3} t{∞,3}
Triakis
figures
Config. V3.4.4 V3.6.6 V3.8.8 V3.10.10 V3.12.12 V3.14.14 V3.16.16 V3.∞.∞
Examples
• drawing in De divina proportione (1509)
• drawing in Perspectiva Corporum Regularium (1568)
• crystal model
• photos from different perspectives (Matemateca)
• 4-sided die
• 12 permutations of $(4,2,0,0)$ (brown)
See also
• Quarter cubic honeycomb – Fills space using truncated tetrahedra and smaller tetrahedra
• Truncated 5-cell – Similar uniform polytope in 4-dimensions
• Truncated triakis tetrahedron
• Triakis truncated tetrahedron
• Octahedron – a rectified tetrahedron
References
1. Chisholm, Matt; Avnet, Jeremy (1997). "Truncated Trickery: Truncatering". theory.org. Retrieved 2013-09-02.
2. Damasceno, Pablo F.; Engel, Michael; Glotzer, Sharon C. (2012). "Crystalline Assemblies and Densest Packings of a Family of Truncated Tetrahedra and the Role of Directional Entropic Forces". ACS Nano. 6 (2012): 609–614. arXiv:1109.1323. doi:10.1021/nn204012y. PMID 22098586. S2CID 12785227.
3. Jiao, Yang; Torquato, Sal (Sep 2011). "A Packing of Truncated Tetrahedra that Nearly Fills All of Space". arXiv:1107.2300 [cond-mat.soft].
4. http://met.iisc.ernet.in/~lord/webfiles/clusters/polyclusters.pdf
5. Friauf, J. B. (1927). "The crystal structure of the intermetallic compound MgCu2". J. Am. Chem. Soc. 49: 3107–3114. doi:10.1021/ja01411a017.
6. "Expo 67 - Man the Producer - page 1".
7. An Atlas of Graphs, page=172, C105
8. An Atlas of Graphs, page 267, truncated tetrahedral graph
9. An Atlas of Graphs, page 130, connected cubic graphs, 12 vertices, C105
10. An Atlas of Graphs, page 161, connected cubic transitive graphs, 12 vertices, Ct11
• Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. ISBN 0-486-23729-X. (Section 3-9)
• Read, R. C.; Wilson, R. J. (1998), An Atlas of Graphs, Oxford University Press
Wikimedia Commons has media related to Truncated tetrahedron.
External links
• Eric W. Weisstein, Truncated tetrahedron (Archimedean solid) at MathWorld.
• Weisstein, Eric W. "Truncated tetrahedral graph". MathWorld.
• Klitzing, Richard. "3D convex uniform polyhedra x3x3o - tut".
• Editable printable net of a truncated tetrahedron with interactive 3D view
• The Uniform Polyhedra
• Virtual Reality Polyhedra The Encyclopedia of Polyhedra
Archimedean solids
Tetrahedron
(Seed)
Tetrahedron
(Dual)
Cube
(Seed)
Octahedron
(Dual)
Dodecahedron
(Seed)
Icosahedron
(Dual)
Truncated tetrahedron
(Truncate)
Truncated tetrahedron
(Zip)
Truncated cube
(Truncate)
Truncated octahedron
(Zip)
Truncated dodecahedron
(Truncate)
Truncated icosahedron
(Zip)
Tetratetrahedron
(Ambo)
Cuboctahedron
(Ambo)
Icosidodecahedron
(Ambo)
Rhombitetratetrahedron
(Expand)
Truncated tetratetrahedron
(Bevel)
Rhombicuboctahedron
(Expand)
Truncated cuboctahedron
(Bevel)
Rhombicosidodecahedron
(Expand)
Truncated icosidodecahedron
(Bevel)
Snub tetrahedron
(Snub)
Snub cube
(Snub)
Snub dodecahedron
(Snub)
Catalan duals
Tetrahedron
(Dual)
Tetrahedron
(Seed)
Octahedron
(Dual)
Cube
(Seed)
Icosahedron
(Dual)
Dodecahedron
(Seed)
Triakis tetrahedron
(Needle)
Triakis tetrahedron
(Kis)
Triakis octahedron
(Needle)
Tetrakis hexahedron
(Kis)
Triakis icosahedron
(Needle)
Pentakis dodecahedron
(Kis)
Rhombic hexahedron
(Join)
Rhombic dodecahedron
(Join)
Rhombic triacontahedron
(Join)
Deltoidal dodecahedron
(Ortho)
Disdyakis hexahedron
(Meta)
Deltoidal icositetrahedron
(Ortho)
Disdyakis dodecahedron
(Meta)
Deltoidal hexecontahedron
(Ortho)
Disdyakis triacontahedron
(Meta)
Pentagonal dodecahedron
(Gyro)
Pentagonal icositetrahedron
(Gyro)
Pentagonal hexecontahedron
(Gyro)
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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Wikipedia
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Truncated tetraheptagonal tiling
In geometry, the truncated tetraheptagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of tr{4,7}.
Truncated tetraheptagonal tiling
Poincaré disk model of the hyperbolic plane
TypeHyperbolic uniform tiling
Vertex configuration4.8.14
Schläfli symboltr{7,4} or $t{\begin{Bmatrix}7\\4\end{Bmatrix}}$
Wythoff symbol2 7 4 |
Coxeter diagram
Symmetry group[7,4], (*742)
DualOrder-4-7 kisrhombille tiling
PropertiesVertex-transitive
Images
Poincaré disk projection, centered on 14-gon:
Symmetry
The dual to this tiling represents the fundamental domains of [7,4] (*742) symmetry. There are 3 small index subgroups constructed from [7,4] by mirror removal and alternation. In these images fundamental domains are alternately colored black and white, and mirrors exist on the boundaries between colors.
Small index subgroups of [7,4] (*742)
Index 1 2 14
Diagram
Coxeter
(orbifold)
[7,4] =
(*742)
[7,4,1+] = =
(*772)
[7+,4] =
(7*2)
[7*,4] =
(*2222222)
Index 2 4 28
Diagram
Coxeter
(orbifold)
[7,4]+ =
(742)
[7+,4]+ = =
(772)
[7*,4]+ =
(2222222)
Related polyhedra and tiling
Uniform heptagonal/square tilings
Symmetry: [7,4], (*742) [7,4]+, (742) [7+,4], (7*2) [7,4,1+], (*772)
{7,4} t{7,4} r{7,4} 2t{7,4}=t{4,7} 2r{7,4}={4,7} rr{7,4} tr{7,4} sr{7,4} s{7,4} h{4,7}
Uniform duals
V74 V4.14.14 V4.7.4.7 V7.8.8 V47 V4.4.7.4 V4.8.14 V3.3.4.3.7 V3.3.7.3.7 V77
*n42 symmetry mutation of omnitruncated tilings: 4.8.2n
Symmetry
*n42
[n,4]
Spherical Euclidean Compact hyperbolic Paracomp.
*242
[2,4]
*342
[3,4]
*442
[4,4]
*542
[5,4]
*642
[6,4]
*742
[7,4]
*842
[8,4]...
*∞42
[∞,4]
Omnitruncated
figure
4.8.4
4.8.6
4.8.8
4.8.10
4.8.12
4.8.14
4.8.16
4.8.∞
Omnitruncated
duals
V4.8.4
V4.8.6
V4.8.8
V4.8.10
V4.8.12
V4.8.14
V4.8.16
V4.8.∞
*nn2 symmetry mutations of omnitruncated tilings: 4.2n.2n
Symmetry
*nn2
[n,n]
Spherical Euclidean Compact hyperbolic Paracomp.
*222
[2,2]
*332
[3,3]
*442
[4,4]
*552
[5,5]
*662
[6,6]
*772
[7,7]
*882
[8,8]...
*∞∞2
[∞,∞]
Figure
Config. 4.4.4 4.6.6 4.8.8 4.10.10 4.12.12 4.14.14 4.16.16 4.∞.∞
Dual
Config. V4.4.4 V4.6.6 V4.8.8 V4.10.10 V4.12.12 V4.14.14 V4.16.16 V4.∞.∞
References
• John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 19, The Hyperbolic Archimedean Tessellations)
• "Chapter 10: Regular honeycombs in hyperbolic space". The Beauty of Geometry: Twelve Essays. Dover Publications. 1999. ISBN 0-486-40919-8. LCCN 99035678.
See also
Wikimedia Commons has media related to Uniform tiling 4-8-14.
• Uniform tilings in hyperbolic plane
• List of regular polytopes
External links
• Weisstein, Eric W. "Hyperbolic tiling". MathWorld.
• Weisstein, Eric W. "Poincaré hyperbolic disk". MathWorld.
• Hyperbolic and Spherical Tiling Gallery
• KaleidoTile 3: Educational software to create spherical, planar and hyperbolic tilings
• Hyperbolic Planar Tessellations, Don Hatch
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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Truncated tetrahexagonal tiling
In geometry, the truncated tetrahexagonal tiling is a semiregular tiling of the hyperbolic plane. There are one square, one octagon, and one dodecagon on each vertex. It has Schläfli symbol of tr{6,4}.
Truncated tetrahexagonal tiling
Poincaré disk model of the hyperbolic plane
TypeHyperbolic uniform tiling
Vertex configuration4.8.12
Schläfli symboltr{6,4} or $t{\begin{Bmatrix}6\\4\end{Bmatrix}}$
Wythoff symbol2 6 4 |
Coxeter diagram or
Symmetry group[6,4], (*642)
DualOrder-4-6 kisrhombille tiling
PropertiesVertex-transitive
Dual tiling
The dual tiling is called an order-4-6 kisrhombille tiling, made as a complete bisection of the order-4 hexagonal tiling, here with triangles shown in alternating colors. This tiling represents the fundamental triangular domains of [6,4] (*642) symmetry.
Related polyhedra and tilings
*n42 symmetry mutation of omnitruncated tilings: 4.8.2n
Symmetry
*n42
[n,4]
Spherical Euclidean Compact hyperbolic Paracomp.
*242
[2,4]
*342
[3,4]
*442
[4,4]
*542
[5,4]
*642
[6,4]
*742
[7,4]
*842
[8,4]...
*∞42
[∞,4]
Omnitruncated
figure
4.8.4
4.8.6
4.8.8
4.8.10
4.8.12
4.8.14
4.8.16
4.8.∞
Omnitruncated
duals
V4.8.4
V4.8.6
V4.8.8
V4.8.10
V4.8.12
V4.8.14
V4.8.16
V4.8.∞
*nn2 symmetry mutations of omnitruncated tilings: 4.2n.2n
Symmetry
*nn2
[n,n]
Spherical Euclidean Compact hyperbolic Paracomp.
*222
[2,2]
*332
[3,3]
*442
[4,4]
*552
[5,5]
*662
[6,6]
*772
[7,7]
*882
[8,8]...
*∞∞2
[∞,∞]
Figure
Config. 4.4.4 4.6.6 4.8.8 4.10.10 4.12.12 4.14.14 4.16.16 4.∞.∞
Dual
Config. V4.4.4 V4.6.6 V4.8.8 V4.10.10 V4.12.12 V4.14.14 V4.16.16 V4.∞.∞
From a Wythoff construction there are fourteen hyperbolic uniform tilings that can be based from the regular order-4 hexagonal tiling.
Drawing the tiles colored as red on the original faces, yellow at the original vertices, and blue along the original edges, there are 7 forms with full [6,4] symmetry, and 7 with subsymmetry.
Uniform tetrahexagonal tilings
Symmetry: [6,4], (*642)
(with [6,6] (*662), [(4,3,3)] (*443) , [∞,3,∞] (*3222) index 2 subsymmetries)
(And [(∞,3,∞,3)] (*3232) index 4 subsymmetry)
=
=
=
=
=
=
=
=
=
=
=
=
{6,4} t{6,4} r{6,4} t{4,6} {4,6} rr{6,4} tr{6,4}
Uniform duals
V64 V4.12.12 V(4.6)2 V6.8.8 V46 V4.4.4.6 V4.8.12
Alternations
[1+,6,4]
(*443)
[6+,4]
(6*2)
[6,1+,4]
(*3222)
[6,4+]
(4*3)
[6,4,1+]
(*662)
[(6,4,2+)]
(2*32)
[6,4]+
(642)
=
=
=
=
=
=
h{6,4} s{6,4} hr{6,4} s{4,6} h{4,6} hrr{6,4} sr{6,4}
Symmetry
The dual of the tiling represents the fundamental domains of (*642) orbifold symmetry. From [6,4] symmetry, there are 15 small index subgroup by mirror removal and alternation operators. Mirrors can be removed if its branch orders are all even, and cuts neighboring branch orders in half. Removing two mirrors leaves a half-order gyration point where the removed mirrors met. In these images unique mirrors are colored red, green, and blue, and alternately colored triangles show the location of gyration points. The [6+,4+], (32×) subgroup has narrow lines representing glide reflections. The subgroup index-8 group, [1+,6,1+,4,1+] (3232) is the commutator subgroup of [6,4].
Larger subgroup constructed as [6,4*], removing the gyration points of [6,4+], (3*22), index 6 becomes (*3333), and [6*,4], removing the gyration points of [6+,4], (2*33), index 12 as (*222222). Finally their direct subgroups [6,4*]+, [6*,4]+, subgroup indices 12 and 24 respectively, can be given in orbifold notation as (3333) and (222222).
Small index subgroups of [6,4]
Index 1 2 4
Diagram
Coxeter [6,4]
= =
[1+,6,4]
=
[6,4,1+]
= =
[6,1+,4]
=
[1+,6,4,1+]
=
[6+,4+]
Generators {0,1,2}{1,010,2}{0,1,212}{0,101,2,121}{1,010,212,20102}{012,021}
Orbifold *642 *443 *662 *3222 *3232 32×
Semidirect subgroups
Diagram
Coxeter [6,4+]
[6+,4]
[(6,4,2+)]
[6,1+,4,1+]
= =
= =
[1+,6,1+,4]
= =
= =
Generators {0,12}{01,2}{1,02}{0,101,1212}{0101,2,121}
Orbifold 4*3 6*2 2*32 2*33 3*22
Direct subgroups
Index 2 4 8
Diagram
Coxeter [6,4]+
=
[6,4+]+
=
[6+,4]+
=
[(6,4,2+)]+
=
[6+,4+]+ = [1+,6,1+,4,1+]
= = =
Generators {01,12} {(01)2,12} {01,(12)2} {02,(01)2,(12)2} {(01)2,(12)2,2(01)22}
Orbifold 642 443 662 3222 3232
Radical subgroups
Index 8 12 16 24
Diagram
Coxeter [6,4*]
=
[6*,4]
[6,4*]+
=
[6*,4]+
Orbifold *3333 *222222 3333 222222
See also
Wikimedia Commons has media related to Uniform tiling 4-8-12.
• Tilings of regular polygons
• List of uniform planar tilings
References
• John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 19, The Hyperbolic Archimedean Tessellations)
• "Chapter 10: Regular honeycombs in hyperbolic space". The Beauty of Geometry: Twelve Essays. Dover Publications. 1999. ISBN 0-486-40919-8. LCCN 99035678.
External links
• Weisstein, Eric W. "Hyperbolic tiling". MathWorld.
• Weisstein, Eric W. "Poincaré hyperbolic disk". MathWorld.
• Hyperbolic and Spherical Tiling Gallery
• KaleidoTile 3: Educational software to create spherical, planar and hyperbolic tilings
• Hyperbolic Planar Tessellations, Don Hatch
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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Truncated tetrakis cube
The truncated tetrakis cube, or more precisely an order-6 truncated tetrakis cube or hexatruncated tetrakis cube, is a convex polyhedron with 32 faces: 24 sets of 3 bilateral symmetry pentagons arranged in an octahedral arrangement, with 8 regular hexagons in the gaps.
Truncated tetrakis cube
Hexatruncated tetrakis cube
Conway notationt6kC = dk6tO
Faces8 hexagons
24 pentagons
Edges84
Vertices54
DualHexakis truncated octahedron
Vertex configuration6 (5.5.5.5)
48 (5.5.6)
Symmetry groupOh
Propertiesconvex
Construction
It is constructed from taking a tetrakis cube by truncating the order-6 vertices. This creates 4 regular hexagon faces, and leaves 12 mirror-symmetric pentagons.
tetrakis cube
Hexakis truncated octahedron
The dual of the order-6 truncated triakis tetrahedron is called a hexakis truncated octahedron. It is constructed by a truncated octahedron with hexagonal pyramids augmented.
Truncated octahedron
hexakis truncated octahedron
See also
• Truncated triakis tetrahedron
• Truncated triakis octahedron
• Truncated triakis icosahedron
External links
• George Hart's Polyhedron generator - "t6kC" (Conway polyhedron notation)
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Wikipedia
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Truncated tetraoctagonal tiling
In geometry, the truncated tetraoctagonal tiling is a semiregular tiling of the hyperbolic plane. There are one square, one octagon, and one hexakaidecagon on each vertex. It has Schläfli symbol of tr{8,4}.
Truncated tetraoctagonal tiling
Poincaré disk model of the hyperbolic plane
TypeHyperbolic uniform tiling
Vertex configuration4.8.16
Schläfli symboltr{8,4} or $t{\begin{Bmatrix}8\\4\end{Bmatrix}}$
Wythoff symbol2 8 4 |
Coxeter diagram or
Symmetry group[8,4], (*842)
DualOrder-4-8 kisrhombille tiling
PropertiesVertex-transitive
Dual tiling
The dual tiling is called an order-4-8 kisrhombille tiling, made as a complete bisection of the order-4 octagonal tiling, here with triangles are shown with alternating colors. This tiling represents the fundamental triangular domains of [8,4] (*842) symmetry.
Symmetry
There are 15 subgroups constructed from [8,4] by mirror removal and alternation. Mirrors can be removed if its branch orders are all even, and cuts neighboring branch orders in half. Removing two mirrors leaves a half-order gyration point where the removed mirrors met. In these images fundamental domains are alternately colored black and white, and mirrors exist on the boundaries between colors. The subgroup index-8 group, [1+,8,1+,4,1+] (4242) is the commutator subgroup of [8,4].
A larger subgroup is constructed as [8,4*], index 8, as [8,4+], (4*4) with gyration points removed, becomes (*4444) or (*44), and another [8*,4], index 16 as [8+,4], (8*2) with gyration points removed as (*22222222) or (*28). And their direct subgroups [8,4*]+, [8*,4]+, subgroup indices 16 and 32 respectively, can be given in orbifold notation as (4444) and (22222222).
Small index subgroups of [8,4] (*842)
Index 1 2 4
Diagram
Coxeter [8,4]
=
[1+,8,4]
=
[8,4,1+]
= =
[8,1+,4]
=
[1+,8,4,1+]
=
[8+,4+]
Orbifold *842 *444 *882 *4222 *4242 42×
Semidirect subgroups
Diagram
Coxeter [8,4+]
[8+,4]
[(8,4,2+)]
[8,1+,4,1+]
= =
= =
[1+,8,1+,4]
= =
= =
Orbifold 4*4 8*2 2*42 2*44 4*22
Direct subgroups
Index 2 4 8
Diagram
Coxeter [8,4]+
=
[8,4+]+
=
[8+,4]+
=
[8,1+,4]+
=
[8+,4+]+ = [1+,8,1+,4,1+]
= = =
Orbifold 842 444 882 4222 4242
Radical subgroups
Index 8 16 32
Diagram
Coxeter [8,4*]
=
[8*,4]
[8,4*]+
=
[8*,4]+
Orbifold *4444 *22222222 4444 22222222
Related polyhedra and tilings
From a Wythoff construction there are fourteen hyperbolic uniform tilings that can be based from the regular order-4 octagonal tiling.
Drawing the tiles colored as red on the original faces, yellow at the original vertices, and blue along the original edges, there are 7 forms with full [8,4] symmetry, and 7 with subsymmetry.
Uniform octagonal/square tilings
[8,4], (*842)
(with [8,8] (*882), [(4,4,4)] (*444) , [∞,4,∞] (*4222) index 2 subsymmetries)
(And [(∞,4,∞,4)] (*4242) index 4 subsymmetry)
=
=
=
=
=
=
=
=
=
=
=
{8,4} t{8,4}
r{8,4} 2t{8,4}=t{4,8} 2r{8,4}={4,8} rr{8,4} tr{8,4}
Uniform duals
V84 V4.16.16 V(4.8)2 V8.8.8 V48 V4.4.4.8 V4.8.16
Alternations
[1+,8,4]
(*444)
[8+,4]
(8*2)
[8,1+,4]
(*4222)
[8,4+]
(4*4)
[8,4,1+]
(*882)
[(8,4,2+)]
(2*42)
[8,4]+
(842)
=
=
=
=
=
=
h{8,4} s{8,4} hr{8,4} s{4,8} h{4,8} hrr{8,4} sr{8,4}
Alternation duals
V(4.4)4 V3.(3.8)2 V(4.4.4)2 V(3.4)3 V88 V4.44 V3.3.4.3.8
*n42 symmetry mutation of omnitruncated tilings: 4.8.2n
Symmetry
*n42
[n,4]
Spherical Euclidean Compact hyperbolic Paracomp.
*242
[2,4]
*342
[3,4]
*442
[4,4]
*542
[5,4]
*642
[6,4]
*742
[7,4]
*842
[8,4]...
*∞42
[∞,4]
Omnitruncated
figure
4.8.4
4.8.6
4.8.8
4.8.10
4.8.12
4.8.14
4.8.16
4.8.∞
Omnitruncated
duals
V4.8.4
V4.8.6
V4.8.8
V4.8.10
V4.8.12
V4.8.14
V4.8.16
V4.8.∞
*nn2 symmetry mutations of omnitruncated tilings: 4.2n.2n
Symmetry
*nn2
[n,n]
Spherical Euclidean Compact hyperbolic Paracomp.
*222
[2,2]
*332
[3,3]
*442
[4,4]
*552
[5,5]
*662
[6,6]
*772
[7,7]
*882
[8,8]...
*∞∞2
[∞,∞]
Figure
Config. 4.4.4 4.6.6 4.8.8 4.10.10 4.12.12 4.14.14 4.16.16 4.∞.∞
Dual
Config. V4.4.4 V4.6.6 V4.8.8 V4.10.10 V4.12.12 V4.14.14 V4.16.16 V4.∞.∞
See also
Wikimedia Commons has media related to Uniform tiling 4-8-16.
• Tilings of regular polygons
• List of uniform planar tilings
References
• John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 19, The Hyperbolic Archimedean Tessellations)
• "Chapter 10: Regular honeycombs in hyperbolic space". The Beauty of Geometry: Twelve Essays. Dover Publications. 1999. ISBN 0-486-40919-8. LCCN 99035678.
External links
• Weisstein, Eric W. "Hyperbolic tiling". MathWorld.
• Weisstein, Eric W. "Poincaré hyperbolic disk". MathWorld.
• Hyperbolic and Spherical Tiling Gallery
• KaleidoTile 3: Educational software to create spherical, planar and hyperbolic tilings
• Hyperbolic Planar Tessellations, Don Hatch
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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Truncated tetrapentagonal tiling
In geometry, the truncated tetrapentagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t0,1,2{4,5} or tr{4,5}.
Truncated tetrapentagonal tiling
Poincaré disk model of the hyperbolic plane
TypeHyperbolic uniform tiling
Vertex configuration4.8.10
Schläfli symboltr{5,4} or $t{\begin{Bmatrix}5\\4\end{Bmatrix}}$
Wythoff symbol2 5 4 |
Coxeter diagram or
Symmetry group[5,4], (*542)
DualOrder-4-5 kisrhombille tiling
PropertiesVertex-transitive
Symmetry
There are four small index subgroup constructed from [5,4] by mirror removal and alternation. In these images fundamental domains are alternately colored black and white, and mirrors exist on the boundaries between colors.
A radical subgroup is constructed [5*,4], index 10, as [5+,4], (5*2) with gyration points removed, becoming orbifold (*22222), and its direct subgroup [5*,4]+, index 20, becomes orbifold (22222).
Small index subgroups of [5,4]
Index 1 2 10
Diagram
Coxeter
(orbifold)
[5,4] =
(*542)
[5,4,1+] = =
(*552)
[5+,4] =
(5*2)
[5*,4] =
(*22222)
Direct subgroups
Index 2 4 20
Diagram
Coxeter
(orbifold)
[5,4]+ =
(542)
[5+,4]+ = =
(552)
[5*,4]+ =
(22222)
Related polyhedra and tiling
*n42 symmetry mutation of omnitruncated tilings: 4.8.2n
Symmetry
*n42
[n,4]
Spherical Euclidean Compact hyperbolic Paracomp.
*242
[2,4]
*342
[3,4]
*442
[4,4]
*542
[5,4]
*642
[6,4]
*742
[7,4]
*842
[8,4]...
*∞42
[∞,4]
Omnitruncated
figure
4.8.4
4.8.6
4.8.8
4.8.10
4.8.12
4.8.14
4.8.16
4.8.∞
Omnitruncated
duals
V4.8.4
V4.8.6
V4.8.8
V4.8.10
V4.8.12
V4.8.14
V4.8.16
V4.8.∞
*nn2 symmetry mutations of omnitruncated tilings: 4.2n.2n
Symmetry
*nn2
[n,n]
Spherical Euclidean Compact hyperbolic Paracomp.
*222
[2,2]
*332
[3,3]
*442
[4,4]
*552
[5,5]
*662
[6,6]
*772
[7,7]
*882
[8,8]...
*∞∞2
[∞,∞]
Figure
Config. 4.4.4 4.6.6 4.8.8 4.10.10 4.12.12 4.14.14 4.16.16 4.∞.∞
Dual
Config. V4.4.4 V4.6.6 V4.8.8 V4.10.10 V4.12.12 V4.14.14 V4.16.16 V4.∞.∞
Uniform pentagonal/square tilings
Symmetry: [5,4], (*542) [5,4]+, (542) [5+,4], (5*2) [5,4,1+], (*552)
{5,4} t{5,4} r{5,4} 2t{5,4}=t{4,5} 2r{5,4}={4,5} rr{5,4} tr{5,4} sr{5,4} s{5,4} h{4,5}
Uniform duals
V54 V4.10.10 V4.5.4.5 V5.8.8 V45 V4.4.5.4 V4.8.10 V3.3.4.3.5 V3.3.5.3.5 V55
See also
Wikimedia Commons has media related to Uniform tiling 4-8-10.
• Uniform tilings in hyperbolic plane
• List of regular polytopes
References
• John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 19, The Hyperbolic Archimedean Tessellations)
• Coxeter, H. S. M. (1999). "Chapter 10: Regular honeycombs in hyperbolic space". The Beauty of Geometry: Twelve Essays. Dover Publications. ISBN 0-486-40919-8. LCCN 99035678.
External links
• Weisstein, Eric W. "Hyperbolic tiling". MathWorld.
• Weisstein, Eric W. "Poincaré hyperbolic disk". MathWorld.
• Hyperbolic and Spherical Tiling Gallery
• KaleidoTile 3: Educational software to create spherical, planar and hyperbolic tilings
• Hyperbolic Planar Tessellations, Don Hatch
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
|
Wikipedia
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Truncated trapezohedron
In geometry, an n-gonal truncated trapezohedron is a polyhedron formed by a n-gonal trapezohedron with n-gonal pyramids truncated from its two polar axis vertices. If the polar vertices are completely truncated (diminished), a trapezohedron becomes an antiprism.
Set of n-gonal truncated trapezohedra
Example: pentagonal truncated trapezohedron (regular dodecahedron)
Faces2 n-sided polygons,
2n pentagons
Edges6n
Vertices4n
Conway notationt4dA4
t5dA5
t6dA6
Symmetry groupDnd, [2+,2n], (2*n), order 4n
Rotation groupDn, [2,n]+, (22n), order 2n
Dual polyhedrongyroelongated bipyramids
Propertiesconvex
The vertices exist as 4 n-gons in four parallel planes, with alternating orientation in the middle creating the pentagons.
The regular dodecahedron is the most common polyhedron in this class, being a Platonic solid, with 12 congruent pentagonal faces.
A truncated trapezohedron has all vertices with 3 faces. This means that the dual polyhedra, the set of gyroelongated dipyramids, have all triangular faces. For example, the icosahedron is the dual of the dodecahedron.
Forms
• Triangular truncated trapezohedron (Dürer's solid) – 6 pentagons, 2 triangles, dual gyroelongated triangular dipyramid
• Truncated square trapezohedron – 8 pentagons, 2 squares, dual gyroelongated square dipyramid
• Truncated pentagonal trapezohedron or regular dodecahedron – 12 pentagonal faces, dual icosahedron
• Truncated hexagonal trapezohedron – 12 pentagons, 2 hexagons, dual gyroelongated hexagonal dipyramid
• ...
• Truncated n-gonal trapezohedron – 2n pentagons, 2 n-gons, dual gyroelongated dipyramids
See also
• Diminished trapezohedron
External links
• Conway Notation for Polyhedra Try: "tndAn", where n=4,5,6... example "t5dA5" is a dodecahedron.
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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Truncated triakis icosahedron
The truncated triakis icosahedron, or more precisely an order-10 truncated triakis icosahedron, is a convex polyhedron with 72 faces: 10 sets of 3 pentagons arranged in an icosahedral arrangement, with 12 decagons in the gaps.
Truncated triakis icosahedron
Conway notationt10kI = dk10tD
Faces12 decagons
60 pentagons
Edges210
Vertices140
DualDecakis truncated dodecahedron
Vertex configuration12 (5.5.5)
60 (5.5.10)
Symmetry groupIh
Propertiesconvex
Net
Triakis icosahedron
It is constructed from taking a triakis icosahedron by truncating the order-10 vertices. This creates 12 regular decagon faces, and leaves 60 mirror-symmetric pentagons.
Triakis icosahedron
Decakis truncated dodecahedron
The dual of the truncated triakis icosahedron is called a decakis truncated dodecahedron. It can be seen as a truncated dodecahedron with decagonal pyramids augmented to the faces.
Truncated dodecahedron
Decakis truncated dodecahedron
Net
See also
• Truncated triakis tetrahedron
• Truncated triakis octahedron
• Truncated tetrakis cube
External links
• George Hart's Polyhedron generator - "t10kI" (Conway polyhedron notation)
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Truncated triakis octahedron
The truncated triakis octahedron, or more precisely an order-8 truncated triakis octahedron, is a convex polyhedron with 30 faces: 8 sets of 3 pentagons arranged in an octahedral arrangement, with 6 octagons in the gaps.
Truncated triakis octahedron
Conway notationt8kO = dk8tC
Faces6 octagons
24 pentagons
Edges84
Vertices56
DualOctakis truncated cube
Vertex configuration8 (5.5.5)
48 (5.5.8)
Symmetry groupOh
Propertiesconvex
Net
Triakis octahedron
It is constructed from taking a triakis octahedron by truncating the order-8 vertices. This creates 6 regular octagon faces, and leaves 24 mirror-symmetric pentagons.
Triakis octahedron
Octakis truncated cube
The dual of the order-8 truncated triakis octahedron is called a octakis truncated cube. It can be seen as a truncated cube with octagonal pyramids augmented to the faces.
Truncated cube
Octakis truncated cube
Net
See also
• Truncated triakis tetrahedron
• Truncated tetrakis cube
• Truncated triakis icosahedron
External links
• George Hart's Polyhedron generator - "t8kO" (Conway polyhedron notation)
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Truncated triangular trapezohedron
In geometry, the truncated triangular trapezohedron is the first in an infinite series of truncated trapezohedra. It has 6 pentagon and 2 triangle faces.
Truncated triangular trapezohedron
TypeTruncated trapezohedron
Faces6 pentagons,
2 triangles
Edges18
Vertices12
Symmetry groupD3d, [2+,6], (2*3)
Dual polyhedronGyroelongated triangular bipyramid
Propertiesconvex
Geometry
This polyhedron can be constructed by truncating two opposite vertices of a cube, of a trigonal trapezohedron (a convex polyhedron with six congruent rhombus sides, formed by stretching or shrinking a cube along one of its long diagonals), or of a rhombohedron or parallelepiped (less symmetric polyhedra that still have the same combinatorial structure as a cube). In the case of a cube, or of a trigonal trapezohedron where the two truncated vertices are the ones on the stretching axes, the resulting shape has three-fold rotational symmetry.
Dürer's solid
This polyhedron is sometimes called Dürer's solid, from its appearance in Albrecht Dürer's 1514 engraving Melencolia I. The graph formed by its edges and vertices is called the Dürer graph.
The shape of the solid depicted by Dürer is a subject of some academic debate.[1] According to Lynch (1982), the hypothesis that the shape is a misdrawn truncated cube was promoted by Strauss (1972); however most sources agree that it is the truncation of a rhombohedron. Despite this agreement, the exact geometry of this rhombohedron is the subject of several contradictory theories:
• Richter (1957) claims that the rhombi of the rhombohedron from which this shape is formed have 5:6 as the ratio between their short and long diagonals, from which the acute angles of the rhombi would be approximately 80°.
• Schröder (1980) and Lynch (1982) instead conclude that the ratio is √3:2 and that the angle is approximately 82°.
• MacGillavry (1981) measures features of the drawing and finds that the angle is approximately 79°. She and a later author, Wolf von Engelhardt (see Hideko 2009) argue that this choice of angle comes from its physical occurrence in calcite crystals.
• Schreiber (1999) argues based on the writings of Dürer that all vertices of Dürer's solid lie on a common sphere, and further claims that the rhombus angles are 72°. Hideko (2009) lists several other scholars who also favor the 72° theory, beginning with Paul Grodzinski in 1955. He argues that this theory is motivated less by analysis of the actual drawing, and more by aesthetic principles relating to regular pentagons and the golden ratio.
• Weitzel (2004) analyzes a 1510 sketch by Dürer of the same solid, from which he confirms Schreiber's hypothesis that the shape has a circumsphere but with rhombus angles of approximately 79.5°.
• Hideko (2009) argues that the shape is intended to depict a solution to the famous geometric problem of doubling the cube, which Dürer also wrote about in 1525. He therefore concludes that (before the corners are cut off) the shape is a cube stretched along its long diagonal. More specifically, he argues that Dürer drew an actual cube, with the long diagonal parallel to the perspective plane, and then enlarged his drawing by some factor in the direction of the long diagonal; the result would be the same as if he had drawn the elongated solid. The enlargement factor that is relevant for doubling the cube is 21/3 ≈ 1.253, but Hideko derives a different enlargement factor that fits the drawing better, 1.277, in a more complicated way.
• Futamura, Frantz & Crannell (2014) classify the proposed solutions to this problem by two parameters: the acute angle and the level of cutting, called the cross ratio. Their estimate of the cross ratio is close to MacGillavry's, and has a numerical value close to the golden ratio. Based on this they posit that the acute angle is $2\arctan(\varphi /2)\approx 78^{\circ }$ and that the cross ratio is exactly $\varphi $.
See also
• Chamfered tetrahedron, another shape formed by truncating a subset of the vertices of a cube
Notes
1. See Weitzel (2004) and Ziegler (2014), from which much of the following history is drawn.
References
• Lynch, Terence (1982), "The geometric body in Dürer's engraving Melencolia I", Journal of the Warburg and Courtauld Institutes, The Warburg Institute, 45: 226–232, doi:10.2307/750979, JSTOR 750979.
• MacGillavry, C. (1981), "The polyhedron in A. Dürers Melencolia I", Nederl. Akad. Wetensch. Proc. Ser. B, 84: 287–294. As cited by Weitzel (2004).
• Richter, D. H. (1957), "Perspektive und Proportionen in Albrecht Dürers "Melancholie"", Z. Vermessungswesen, 82: 284–288 and 350–357. As cited by Weitzel (2004).
• Schreiber, Peter (1999), "A new hypothesis on Dürer's enigmatic polyhedron in his copper engraving "Melencolia I"", Historia Mathematica, 26 (4): 369–377, doi:10.1006/hmat.1999.2245.
• Schröder, E. (1980), Dürer, Kunst und Geometrie, Dürers künstlerisches Schaffen aus der Sicht seiner "Underweysung", Basel{{citation}}: CS1 maint: location missing publisher (link). As cited by Weitzel (2004).
• Strauss, Walter L. (1972), The Complete Engravings of Dürer, New York, p. 168, ISBN 0-486-22851-7{{citation}}: CS1 maint: location missing publisher (link). As cited by Lynch (1982).
• Weber, P. (1900), Beiträge zu Dürers Weltanschauung—Eine Studie über die drei Stiche Ritter, Tod und Teufel, Melancholie und Hieronymus im Gehäus, Strassburg{{citation}}: CS1 maint: location missing publisher (link). As cited by Weitzel (2004).
• Weitzel, Hans (2004), "A further hypothesis on the polyhedron of A. Dürer's engraving Melencolia I", Historia Mathematica, 31 (1): 11–14, doi:10.1016/S0315-0860(03)00029-6.
• Hideko, Ishizu (2009), "Another solution to the polyhedron in Dürer's Melencolia: A visual demonstration of the Delian problem" (PDF), Aesthetics, The Japanese Society for Aesthetics, 13: 179–194.
• Ziegler, Günter M. (December 3, 2014), "Dürer's polyhedron: 5 theories that explain Melencolia's crazy cube", Alex Bellos's Adventures in Numberland, The Guardian.
• Futamura, F.; Frantz, M.; Crannell, A. (2014), "The cross ratio as a shape parameter for Dürer's solid", Journal of Mathematics and the Arts, 8 (3–4): 111–119, arXiv:1405.6481, doi:10.1080/17513472.2014.974483, S2CID 120958490.
External links
• Weisstein, Eric W., "Dürer's Solid", MathWorld
• How to build Dürer's Polyhedron - by DUPLICON (in German)
• Open-source 3D models of Dürer's Solid
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Truncated triapeirogonal tiling
In geometry, the truncated triapeirogonal tiling is a uniform tiling of the hyperbolic plane with a Schläfli symbol of tr{∞,3}.
Truncated triapeirogonal tiling
Poincaré disk model of the hyperbolic plane
TypeHyperbolic uniform tiling
Vertex configuration4.6.∞
Schläfli symboltr{∞,3} or $t{\begin{Bmatrix}\infty \\3\end{Bmatrix}}$
Wythoff symbol2 ∞ 3 |
Coxeter diagram or
Symmetry group[∞,3], (*∞32)
DualOrder 3-infinite kisrhombille
PropertiesVertex-transitive
Symmetry
The dual of this tiling represents the fundamental domains of [∞,3], *∞32 symmetry. There are 3 small index subgroup constructed from [∞,3] by mirror removal and alternation. In these images fundamental domains are alternately colored black and white, and mirrors exist on the boundaries between colors.
A special index 4 reflective subgroup, is [(∞,∞,3)], (*∞∞3), and its direct subgroup [(∞,∞,3)]+, (∞∞3), and semidirect subgroup [(∞,∞,3+)], (3*∞).[1] Given [∞,3] with generating mirrors {0,1,2}, then its index 4 subgroup has generators {0,121,212}.
An index 6 subgroup constructed as [∞,3*], becomes [(∞,∞,∞)], (*∞∞∞).
Small index subgroups of [∞,3], (*∞32)
Index 1 2 3 4 6 8 12 24
Diagrams
Coxeter
(orbifold)
[∞,3]
=
(*∞32)
[1+,∞,3]
=
(*∞33)
[∞,3+]
(3*∞)
[∞,∞]
(*∞∞2)
[(∞,∞,3)]
(*∞∞3)
[∞,3*]
=
(*∞3)
[∞,1+,∞]
(*(∞2)2)
[(∞,1+,∞,3)]
(*(∞3)2)
[1+,∞,∞,1+]
(*∞4)
[(∞,∞,3*)]
(*∞6)
Direct subgroups
Index 2 4 6 8 12 16 24 48
Diagrams
Coxeter
(orbifold)
[∞,3]+
=
(∞32)
[∞,3+]+
=
(∞33)
[∞,∞]+
(∞∞2)
[(∞,∞,3)]+
(∞∞3)
[∞,3*]+
=
(∞3)
[∞,1+,∞]+
(∞2)2
[(∞,1+,∞,3)]+
(∞3)2
[1+,∞,∞,1+]+
(∞4)
[(∞,∞,3*)]+
(∞6)
Related polyhedra and tiling
Paracompact uniform tilings in [∞,3] family
Symmetry: [∞,3], (*∞32) [∞,3]+
(∞32)
[1+,∞,3]
(*∞33)
[∞,3+]
(3*∞)
=
=
=
=
or
=
or
=
{∞,3} t{∞,3} r{∞,3} t{3,∞} {3,∞} rr{∞,3} tr{∞,3} sr{∞,3} h{∞,3} h2{∞,3} s{3,∞}
Uniform duals
V∞3 V3.∞.∞ V(3.∞)2 V6.6.∞ V3∞ V4.3.4.∞ V4.6.∞ V3.3.3.3.∞ V(3.∞)3 V3.3.3.3.3.∞
This tiling can be considered a member of a sequence of uniform patterns with vertex figure (4.6.2p) and Coxeter-Dynkin diagram . For p < 6, the members of the sequence are omnitruncated polyhedra (zonohedrons), shown below as spherical tilings. For p > 6, they are tilings of the hyperbolic plane, starting with the truncated triheptagonal tiling.
*n32 symmetry mutation of omnitruncated tilings: 4.6.2n
Sym.
*n32
[n,3]
Spherical Euclid. Compact hyperb. Paraco. Noncompact hyperbolic
*232
[2,3]
*332
[3,3]
*432
[4,3]
*532
[5,3]
*632
[6,3]
*732
[7,3]
*832
[8,3]
*∞32
[∞,3]
[12i,3]
[9i,3]
[6i,3]
[3i,3]
Figures
Config. 4.6.4 4.6.6 4.6.8 4.6.10 4.6.12 4.6.14 4.6.16 4.6.∞ 4.6.24i 4.6.18i 4.6.12i 4.6.6i
Duals
Config. V4.6.4 V4.6.6 V4.6.8 V4.6.10 V4.6.12 V4.6.14 V4.6.16 V4.6.∞ V4.6.24i V4.6.18i V4.6.12i V4.6.6i
See also
Wikimedia Commons has media related to Uniform tiling 4-6-i.
• List of uniform planar tilings
• Tilings of regular polygons
• Uniform tilings in hyperbolic plane
References
1. Norman W. Johnson and Asia Ivic Weiss, Quadratic Integers and Coxeter Groups, Can. J. Math. Vol. 51 (6), 1999 pp. 1307–1336
• John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 19, The Hyperbolic Archimedean Tessellations)
• "Chapter 10: Regular honeycombs in hyperbolic space". The Beauty of Geometry: Twelve Essays. Dover Publications. 1999. ISBN 0-486-40919-8. LCCN 99035678.
External links
• Weisstein, Eric W. "Hyperbolic tiling". MathWorld.
• Weisstein, Eric W. "Poincaré hyperbolic disk". 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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Truncated triheptagonal tiling
In geometry, the truncated triheptagonal tiling is a semiregular tiling of the hyperbolic plane. There is one square, one hexagon, and one tetradecagon (14-sides) on each vertex. It has Schläfli symbol of tr{7,3}.
Truncated triheptagonal tiling
Poincaré disk model of the hyperbolic plane
TypeHyperbolic uniform tiling
Vertex configuration4.6.14
Schläfli symboltr{7,3} or $t{\begin{Bmatrix}7\\3\end{Bmatrix}}$
Wythoff symbol2 7 3 |
Coxeter diagram or
Symmetry group[7,3], (*732)
DualOrder 3-7 kisrhombille
PropertiesVertex-transitive
Uniform colorings
There is only one uniform coloring of a truncated triheptagonal tiling. (Naming the colors by indices around a vertex: 123.)
Symmetry
Each triangle in this dual tiling, order 3-7 kisrhombille, represent a fundamental domain of the Wythoff construction for the symmetry group [7,3].
The dual tiling is called an order-3 bisected heptagonal tiling, made as a complete bisection of the heptagonal tiling, here shown with triangles with alternating colors.
Related polyhedra and tilings
This tiling can be considered a member of a sequence of uniform patterns with vertex figure (4.6.2p) and Coxeter-Dynkin diagram . For p < 6, the members of the sequence are omnitruncated polyhedra (zonohedrons), shown below as spherical tilings. For p > 6, they are tilings of the hyperbolic plane, starting with the truncated triheptagonal tiling.
*n32 symmetry mutation of omnitruncated tilings: 4.6.2n
Sym.
*n32
[n,3]
Spherical Euclid. Compact hyperb. Paraco. Noncompact hyperbolic
*232
[2,3]
*332
[3,3]
*432
[4,3]
*532
[5,3]
*632
[6,3]
*732
[7,3]
*832
[8,3]
*∞32
[∞,3]
[12i,3]
[9i,3]
[6i,3]
[3i,3]
Figures
Config. 4.6.4 4.6.6 4.6.8 4.6.10 4.6.12 4.6.14 4.6.16 4.6.∞ 4.6.24i 4.6.18i 4.6.12i 4.6.6i
Duals
Config. V4.6.4 V4.6.6 V4.6.8 V4.6.10 V4.6.12 V4.6.14 V4.6.16 V4.6.∞ V4.6.24i V4.6.18i V4.6.12i V4.6.6i
From a Wythoff construction there are eight hyperbolic uniform tilings that can be based from the regular heptagonal tiling.
Drawing the tiles colored as red on the original faces, yellow at the original vertices, and blue along the original edges, there are 8 forms.
Uniform heptagonal/triangular tilings
Symmetry: [7,3], (*732) [7,3]+, (732)
{7,3} t{7,3} r{7,3} t{3,7} {3,7} rr{7,3} tr{7,3} sr{7,3}
Uniform duals
V73 V3.14.14 V3.7.3.7 V6.6.7 V37 V3.4.7.4 V4.6.14 V3.3.3.3.7
See also
Wikimedia Commons has media related to Uniform tiling 4-6-14.
• Tilings of regular polygons
• List of uniform planar tilings
References
• John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 19, The Hyperbolic Archimedean Tessellations)
• "Chapter 10: Regular honeycombs in hyperbolic space". The Beauty of Geometry: Twelve Essays. Dover Publications. 1999. ISBN 0-486-40919-8. LCCN 99035678.
External links
• Weisstein, Eric W. "Hyperbolic tiling". MathWorld.
• Weisstein, Eric W. "Poincaré hyperbolic disk". MathWorld.
• Hyperbolic and Spherical Tiling Gallery
• KaleidoTile 3: Educational software to create spherical, planar and hyperbolic tilings
• Hyperbolic Planar Tessellations, Don Hatch
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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Truncation (statistics)
In statistics, truncation results in values that are limited above or below, resulting in a truncated sample.[1] A random variable $y$ is said to be truncated from below if, for some threshold value $c$, the exact value of $y$ is known for all cases $y>c$, but unknown for all cases $y\leq c$. Similarly, truncation from above means the exact value of $y$ is known in cases where $y<c$, but unknown when $y\geq c$.[2]
Truncation is similar to but distinct from the concept of statistical censoring. A truncated sample can be thought of as being equivalent to an underlying sample with all values outside the bounds entirely omitted, with not even a count of those omitted being kept. With statistical censoring, a note would be recorded documenting which bound (upper or lower) had been exceeded and the value of that bound. With truncated sampling, no note is recorded.
Applications
Usually the values that insurance adjusters receive are either left-truncated, right-censored, or both. For example, if policyholders are subject to a policy limit u, then any loss amounts that are actually above u are reported to the insurance company as being exactly u because u is the amount the insurance company pays. The insurer knows that the actual loss is greater than u but they don't know what it is. On the other hand, left truncation occurs when policyholders are subject to a deductible. If policyholders are subject to a deductible d, any loss amount that is less than d will not even be reported to the insurance company. If there is a claim on a policy limit of u and a deductible of d, any loss amount that is greater than u will be reported to the insurance company as a loss of $u-d$ because that is the amount the insurance company has to pay. Therefore, insurance loss data is left-truncated because the insurance company doesn't know if there are values below the deductible d because policyholders won't make a claim. The insurance loss is also right-censored if the loss is greater than u because u is the most the insurance company will pay. Thus, it only knows that your claim is greater than u, not the exact claim amount.
Probability distributions
Truncation can be applied to any probability distribution. This will usually lead to a new distribution, not one within the same family. Thus, if a random variable X has F(x) as its distribution function, the new random variable Y defined as having the distribution of X truncated to the semi-open interval (a, b] has the distribution function
$F_{Y}(y)={\frac {F(y)-F(a)}{F(b)-F(a)}}\,$
for y in the interval (a, b], and 0 or 1 otherwise. If truncation were to the closed interval [a, b], the distribution function would be
$F_{Y}(y)={\frac {F(y)-F(a-)}{F(b)-F(a-)}}\,$
for y in the interval [a, b], and 0 or 1 otherwise.
Data analysis
The analysis of data where observations are treated as being from truncated versions of standard distributions can be undertaken using maximum likelihood, where the likelihood would be derived from the distribution or density of the truncated distribution. This involves taking account of the factor ${F(b)-F(a)}$ in the modified density function which will depend on the parameters of the original distribution.
In practice, if the fraction truncated is very small the effect of truncation might be ignored when analysing data. For example, it is common to use a normal distribution to model data whose values can only be positive but for which the typical range of values is well away from zero. In such cases, a truncated or censored version of the normal distribution may formally be preferable (although there would be alternatives); there would be very little change in results from the more complicated analysis. However, software is readily available for maximum-likelihood estimation of even moderately complicated models, such as regression models, for truncated data.[3]
In econometrics, truncated dependent variables are variables for which observations cannot be made for certain values in some range.[4] Regression models with such dependent variables require special care that properly recognizes the truncated nature of the variable. Estimation of such truncated regression model can be done in parametric,[5][6][7] or semi- and non-parametric frameworks.[8][9]
See also
• Censoring (statistics)
• Trimmed estimator
• Truncated distribution
• Truncated mean
References
1. Dodge, Y. (2003) The Oxford Dictionary of Statistical Terms. OUP. ISBN 0-19-920613-9
2. Breen, Richard (1996). Regression Models : Censored, Sample Selected, or Truncated Data. Quantitative Applications in the Social Sciences. Vol. 111. Thousand Oaks: Sage. pp. 2–4. ISBN 0-8039-5710-6.
3. Wolynetz, M. S. (1979). "Maximum Likelihood Estimation in a Linear Model from Confined and Censored Normal Data". Journal of the Royal Statistical Society. Series C. 28 (2): 195–206. doi:10.2307/2346749. JSTOR 2346749.
4. "Truncated Dependent Variables". About.com. Retrieved 2008-03-22.
5. Amemiya, T. (1973). "Regression Analysis When the Dependent Variable is Truncated Normal". Econometrica. 41 (6): 997–1016. doi:10.2307/1914031. JSTOR 1914031.
6. Heckman, James (1976). "The Common Structure of Statistical Models of Truncation, Sample Selection and Limited Dependent Variables and a Simple Estimator for Such Models". Annals of Economic and Social Measurement. 5 (4): 475–492.
7. Vancak, V.; Goldberg, Y.; Bar-Lev, S. K.; Boukai, B. (2015). "Continuous statistical models: With or without truncation parameters?". Mathematical Methods of Statistics. 24 (1): 55–73. doi:10.3103/S1066530715010044. hdl:1805/7048. S2CID 255455365.{{cite journal}}: CS1 maint: multiple names: authors list (link)
8. Lewbel, A.; Linton, O. (2002). "Nonparametric Censored and Truncated Regression". Econometrica. 70 (2): 765–779. doi:10.1111/1468-0262.00304. JSTOR 2692291. S2CID 120113700.
9. Park, B. U.; Simar, L.; Zelenyuk, V. (2008). "Local Likelihood Estimation of Truncated Regression and its Partial Derivatives: Theory and Application" (PDF). Journal of Econometrics. 146 (1): 185–198. doi:10.1016/j.jeconom.2008.08.007. S2CID 55496460.
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Wikipedia
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Truncation error
In numerical analysis and scientific computing, truncation error is an error caused by approximating a mathematical process.[1][2]
Examples
Infinite series
A summation series for $e^{x}$ is given by an infinite series such as
$e^{x}=1+x+{\frac {x^{2}}{2!}}+{\frac {x^{3}}{3!}}+{\frac {x^{4}}{4!}}+\cdots $
In reality, we can only use a finite number of these terms as it would take an infinite amount of computational time to make use of all of them. So let's suppose we use only three terms of the series, then
$e^{x}\approx 1+x+{\frac {x^{2}}{2!}}$
In this case, the truncation error is ${\frac {x^{3}}{3!}}+{\frac {x^{4}}{4!}}+\cdots $
Example A:
Given the following infinite series, find the truncation error for x = 0.75 if only the first three terms of the series are used.
$S=1+x+x^{2}+x^{3}+\cdots ,\qquad \left|x\right|<1.$
Solution
Using only first three terms of the series gives
${\begin{aligned}S_{3}&=\left(1+x+x^{2}\right)_{x=0.75}\\&=1+0.75+\left(0.75\right)^{2}\\&=2.3125\end{aligned}}$
The sum of an infinite geometrical series
$S=a+ar+ar^{2}+ar^{3}+\cdots ,\ r<1$
is given by
$S={\frac {a}{1-r}}$
For our series, a = 1 and r = 0.75, to give
$S={\frac {1}{1-0.75}}=4$
The truncation error hence is
$\mathrm {TE} =4-2.3125=1.6875$
Differentiation
The definition of the exact first derivative of the function is given by
$f'(x)=\lim _{h\to 0}{\frac {f(x+h)-f(x)}{h}}$
However, if we are calculating the derivative numerically, $h$ has to be finite. The error caused by choosing $h$ to be finite is a truncation error in the mathematical process of differentiation.
Example A:
Find the truncation in calculating the first derivative of $f(x)=5x^{3}$ at $x=7$ using a step size of $h=0.25$
Solution:
The first derivative of $f(x)=5x^{3}$ is
$f'(x)=15x^{2},$
and at $x=7$,
$f'(7)=735.$
The approximate value is given by
$f'(7)={\frac {f(7+0.25)-f(7)}{0.25}}=761.5625$
The truncation error hence is
$\mathrm {TE} =735-761.5625=-26.5625$
Integration
The definition of the exact integral of a function $f(x)$ from $a$ to $b$ is given as follows.
Let $f:[a,b]\to \mathbb {R} $ be a function defined on a closed interval $[a,b]$ of the real numbers, $\mathbb {R} $, and
$P=\left\{[x_{0},x_{1}],[x_{1},x_{2}],\dots ,[x_{n-1},x_{n}]\right\},$
be a partition of I, where
$a=x_{0}<x_{1}<x_{2}<\cdots <x_{n}=b.$
$\int _{a}^{b}f(x)\,dx=\sum _{i=1}^{n}f(x_{i}^{*})\,\Delta x_{i}$
where $\Delta x_{i}=x_{i}-x_{i-1}$ and $x_{i}^{*}\in [x_{i-1},x_{i}]$.
This implies that we are finding the area under the curve using infinite rectangles. However, if we are calculating the integral numerically, we can only use a finite number of rectangles. The error caused by choosing a finite number of rectangles as opposed to an infinite number of them is a truncation error in the mathematical process of integration.
Example A.
For the integral
$\int _{3}^{9}x^{2}{dx}$
find the truncation error if a two-segment left-hand Riemann sum is used with equal width of segments.
Solution
We have the exact value as
${\begin{aligned}\int _{3}^{9}{x^{2}{dx}}&=\left[{\frac {x^{3}}{3}}\right]_{3}^{9}\\&=\left[{\frac {9^{3}-3^{3}}{3}}\right]\\&=234\end{aligned}}$
Using two rectangles of equal width to approximate the area (see Figure 2) under the curve, the approximate value of the integral
${\begin{aligned}\int _{3}^{9}x^{2}\,dx&\approx \left.\left(x^{2}\right)\right|_{x=3}(6-3)+\left.\left(x^{2}\right)\right|_{x=6}(9-6)\\&=(3^{2})3+(6^{2})3\\&=27+108\\&=135\end{aligned}}$
${\begin{aligned}{\text{Truncation Error}}&={\text{Exact Value}}-{\text{Approximate Value}}\\&=234-135\\&=99.\end{aligned}}$
Occasionally, by mistake, round-off error (the consequence of using finite precision floating point numbers on computers), is also called truncation error, especially if the number is rounded by chopping. That is not the correct use of "truncation error"; however calling it truncating a number may be acceptable.
Addition
Truncation error can cause $(A+B)+C\neq A+(B+C)$ within a computer when $A=-10^{25},B=10^{25},C=1$ because $(A+B)+C=(0)+C=1$ (like it should), while $A+(B+C)=A+(B)=0$. Here, $A+(B+C)$ has a truncation error equal to 1. This truncation error occurs because computers do not store the least significant digits of an extremely large integer.
See also
• Quantization error
References
1. Atkinson, Kendall E. (1989). An Introduction to Numerical Analysis (2nd ed.). New York: Wiley. p. 20. ISBN 978-0-471-62489-9. OCLC 803318878.
2. Stoer, Josef; Bulirsch, Roland (2002), Introduction to Numerical Analysis (3rd ed.), Princeton, N.J.: Recording for the Blind & Dyslexic, OCLC 50556273, retrieved 2022-02-08
• Atkinson, Kendall E. (1989), An Introduction to Numerical Analysis (2nd ed.), New York: John Wiley & Sons, p. 20, ISBN 978-0-471-50023-0
• Stoer, Josef; Bulirsch, Roland (2002), Introduction to Numerical Analysis (3rd ed.), Berlin, New York: Springer-Verlag, p. 1, ISBN 978-0-387-95452-3.
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Wikipedia
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Truncus (mathematics)
In analytic geometry, a truncus is a curve in the Cartesian plane consisting of all points (x,y) satisfying an equation of the form
$f(x)={a \over (x+b)^{2}}+c$
where a, b, and c are given constants. The two asymptotes of a truncus are parallel to the coordinate axes. The basic truncus y = 1 / x2 has asymptotes at x = 0 and y = 0, and every other truncus can be obtained from this one through a combination of translations and dilations.
For the general truncus form above, the constant a dilates the graph by a factor of a from the x-axis; that is, the graph is stretched vertically when a > 1 and compressed vertically when 0 < a < 1. When a < 0 the graph is reflected in the x-axis as well as being stretched vertically. The constant b translates the graph horizontally left b units when b > 0, or right when b < 0. The constant c translates the graph vertically up c units when c > 0 or down when c < 0. The asymptotes of a truncus are found at x = -b (for the vertical asymptote) and y = c (for the horizontal asymptote).
This function is more commonly known as a reciprocal squared function, particularly the basic example $1/x^{2}$.[1]
See also
• Rational functions
• Multiplicative inverse
References
1. https://cool4ed.calstate.edu/bitstream/handle/10211.3/206113/appcalc.pdf?sequence=1 pg27
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Wikipedia
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Tautology (logic)
In mathematical logic, a tautology (from Greek: ταυτολογία) is a formula or assertion that is true in every possible interpretation. An example is "x=y or x≠y". Similarly, "either the ball is green, or the ball is not green" is always true, regardless of the colour of the ball.
The philosopher Ludwig Wittgenstein first applied the term to redundancies of propositional logic in 1921, borrowing from rhetoric, where a tautology is a repetitive statement. In logic, a formula is satisfiable if it is true under at least one interpretation, and thus a tautology is a formula whose negation is unsatisfiable. In other words, it cannot be false. It cannot be untrue.
Unsatisfiable statements, both through negation and affirmation, are known formally as contradictions. A formula that is neither a tautology nor a contradiction is said to be logically contingent.
Such a formula can be made either true or false based on the values assigned to its propositional variables. The double turnstile notation $\vDash S$ is used to indicate that S is a tautology. Tautology is sometimes symbolized by "Vpq", and contradiction by "Opq". The tee symbol $\top $ is sometimes used to denote an arbitrary tautology, with the dual symbol $\bot $ (falsum) representing an arbitrary contradiction; in any symbolism, a tautology may be substituted for the truth value "true", as symbolized, for instance, by "1".[1]
Tautologies are a key concept in propositional logic, where a tautology is defined as a propositional formula that is true under any possible Boolean valuation of its propositional variables.[2] A key property of tautologies in propositional logic is that an effective method exists for testing whether a given formula is always satisfied (equiv., whether its negation is unsatisfiable).
The definition of tautology can be extended to sentences in predicate logic, which may contain quantifiers—a feature absent from sentences of propositional logic.[3] Indeed, in propositional logic, there is no distinction between a tautology and a logically valid formula. In the context of predicate logic, many authors define a tautology to be a sentence that can be obtained by taking a tautology of propositional logic, and uniformly replacing each propositional variable by a first-order formula (one formula per propositional variable). The set of such formulas is a proper subset of the set of logically valid sentences of predicate logic (i.e., sentences that are true in every model).
History
The word tautology was used by the ancient Greeks to describe a statement that was asserted to be true merely by virtue of saying the same thing twice, a pejorative meaning that is still used for rhetorical tautologies. Between 1800 and 1940, the word gained new meaning in logic, and is currently used in mathematical logic to denote a certain type of propositional formula, without the pejorative connotations it originally possessed.
In 1800, Immanuel Kant wrote in his book Logic:
The identity of concepts in analytical judgments can be either explicit (explicita) or non-explicit (implicita). In the former case analytic propositions are tautological.
Here, analytic proposition refers to an analytic truth, a statement in natural language that is true solely because of the terms involved.
In 1884, Gottlob Frege proposed in his Grundlagen that a truth is analytic exactly if it can be derived using logic. However, he maintained a distinction between analytic truths (i.e., truths based only on the meanings of their terms) and tautologies (i.e., statements devoid of content).
In his Tractatus Logico-Philosophicus in 1921, Ludwig Wittgenstein proposed that statements that can be deduced by logical deduction are tautological (empty of meaning), as well as being analytic truths. Henri Poincaré had made similar remarks in Science and Hypothesis in 1905. Although Bertrand Russell at first argued against these remarks by Wittgenstein and Poincaré, claiming that mathematical truths were not only non-tautologous but were synthetic, he later spoke in favor of them in 1918:
Everything that is a proposition of logic has got to be in some sense or the other like a tautology. It has got to be something that has some peculiar quality, which I do not know how to define, that belongs to logical propositions but not to others.
Here, logical proposition refers to a proposition that is provable using the laws of logic.
During the 1930s, the formalization of the semantics of propositional logic in terms of truth assignments was developed. The term "tautology" began to be applied to those propositional formulas that are true regardless of the truth or falsity of their propositional variables. Some early books on logic (such as Symbolic Logic by C. I. Lewis and Langford, 1932) used the term for any proposition (in any formal logic) that is universally valid. It is common in presentations after this (such as Stephen Kleene 1967 and Herbert Enderton 2002) to use tautology to refer to a logically valid propositional formula, but to maintain a distinction between "tautology" and "logically valid" in the context of first-order logic (see below).
Background
Main article: Propositional logic
Propositional logic begins with propositional variables, atomic units that represent concrete propositions. A formula consists of propositional variables connected by logical connectives, built up in such a way that the truth of the overall formula can be deduced from the truth or falsity of each variable. A valuation is a function that assigns each propositional variable to either T (for truth) or F (for falsity). So by using the propositional variables A and B, the binary connectives $\lor $ and $\land $ representing disjunction and conjunction respectively, and the unary connective $\lnot $ representing negation, the following formula can be obtained:$(A\land B)\lor (\lnot A)\lor (\lnot B)$.
A valuation here must assign to each of A and B either T or F. But no matter how this assignment is made, the overall formula will come out true. For if the first conjunction $(A\land B)$ is not satisfied by a particular valuation, then one of A and B is assigned F, which will make one of the following disjunct to be assigned T.
Definition and examples
A formula of propositional logic is a tautology if the formula itself is always true, regardless of which valuation is used for the propositional variables. There are infinitely many tautologies. Examples include:
• $(A\lor \lnot A)$ ("A or not A"), the law of excluded middle. This formula has only one propositional variable, A. Any valuation for this formula must, by definition, assign A one of the truth values true or false, and assign $\lnot $A the other truth value. For instance, "The cat is black or the cat is not black".
• $(A\to B)\Leftrightarrow (\lnot B\to \lnot A)$ ("if A implies B, then not-B implies not-A", and vice versa), which expresses the law of contraposition. For instance, "If it's a book, it is blue; if it's not blue, it's not a book."
• $((\lnot A\to B)\land (\lnot A\to \lnot B))\to A$ ("if not-A implies both B and its negation not-B, then not-A must be false, then A must be true"), which is the principle known as reductio ad absurdum. For instance, "If it's not blue, it's a book, if it's not blue, it's also not a book, so it is blue."
• $\lnot (A\land B)\Leftrightarrow (\lnot A\lor \lnot B)$ ("if not both A and B, then not-A or not-B", and vice versa), which is known as De Morgan's law. "If it is not both blue and a book, then it's either not a book or it's not blue"
• $((A\to B)\land (B\to C))\to (A\to C)$ ("if A implies B and B implies C, then A implies C"), which is the principle known as syllogism. "If it's a book, then it's blue and if it's blue, then it's on that shelf, then if it's a book, it's on that shelf."
• $((A\lor B)\land (A\to C)\land (B\to C))\to C$ ("if at least one of A or B is true, and each implies C, then C must be true as well"), which is the principle known as proof by cases. "Books and blue things are on that shelf. If it's either a book or it's blue, it's on that shelf."
A minimal tautology is a tautology that is not the instance of a shorter tautology.
• $(A\lor B)\to (A\lor B)$ is a tautology, but not a minimal one, because it is an instantiation of $C\to C$.
Verifying tautologies
The problem of determining whether a formula is a tautology is fundamental in propositional logic. If there are n variables occurring in a formula then there are 2n distinct valuations for the formula. Therefore, the task of determining whether or not the formula is a tautology is a finite and mechanical one: one needs only to evaluate the truth value of the formula under each of its possible valuations. One algorithmic method for verifying that every valuation makes the formula to be true is to make a truth table that includes every possible valuation.[2]
For example, consider the formula
$((A\land B)\to C)\Leftrightarrow (A\to (B\to C)).$
There are 8 possible valuations for the propositional variables A, B, C, represented by the first three columns of the following table. The remaining columns show the truth of subformulas of the formula above, culminating in a column showing the truth value of the original formula under each valuation.
$A$$B$$C$$A\land B$$(A\land B)\to C$$B\to C$$A\to (B\to C)$$((A\land B)\to C)\Leftrightarrow (A\to (B\to C))$
TTTTTTTT
TTFTFFFT
TFTFTTTT
TFFFTTTT
FTTFTTTT
FTFFTFTT
FFTFTTTT
FFFFTTTT
Because each row of the final column shows T, the sentence in question is verified to be a tautology.
It is also possible to define a deductive system (i.e., proof system) for propositional logic, as a simpler variant of the deductive systems employed for first-order logic (see Kleene 1967, Sec 1.9 for one such system). A proof of a tautology in an appropriate deduction system may be much shorter than a complete truth table (a formula with n propositional variables requires a truth table with 2n lines, which quickly becomes infeasible as n increases). Proof systems are also required for the study of intuitionistic propositional logic, in which the method of truth tables cannot be employed because the law of the excluded middle is not assumed.
Tautological implication
A formula R is said to tautologically imply a formula S if every valuation that causes R to be true also causes S to be true. This situation is denoted $R\models S$. It is equivalent to the formula $R\to S$ being a tautology (Kleene 1967 p. 27).
For example, let $S$ be $A\land (B\lor \lnot B)$. Then $S$ is not a tautology, because any valuation that makes $A$ false will make $S$ false. But any valuation that makes $A$ true will make $S$ true, because $B\lor \lnot B$ is a tautology. Let $R$ be the formula $A\land C$. Then $R\models S$, because any valuation satisfying $R$ will make $A$ true—and thus makes $S$ true.
It follows from the definition that if a formula $R$ is a contradiction, then $R$ tautologically implies every formula, because there is no truth valuation that causes $R$ to be true, and so the definition of tautological implication is trivially satisfied. Similarly, if $S$ is a tautology, then $S$ is tautologically implied by every formula.
Substitution
Main article: Substitution instance
There is a general procedure, the substitution rule, that allows additional tautologies to be constructed from a given tautology (Kleene 1967 sec. 3). Suppose that S is a tautology and for each propositional variable A in S a fixed sentence SA is chosen. Then the sentence obtained by replacing each variable A in S with the corresponding sentence SA is also a tautology.
For example, let S be the tautology
$(A\land B)\lor \lnot A\lor \lnot B$.
Let SA be $C\lor D$ and let SB be $C\to E$.
It follows from the substitution rule that the sentence
$((C\lor D)\land (C\to E))\lor \lnot (C\lor D)\lor \lnot (C\to E)$
Semantic completeness and soundness
An axiomatic system is complete if every tautology is a theorem (derivable from axioms). An axiomatic system is sound if every theorem is a tautology.
Efficient verification and the Boolean satisfiability problem
The problem of constructing practical algorithms to determine whether sentences with large numbers of propositional variables are tautologies is an area of contemporary research in the area of automated theorem proving.
The method of truth tables illustrated above is provably correct – the truth table for a tautology will end in a column with only T, while the truth table for a sentence that is not a tautology will contain a row whose final column is F, and the valuation corresponding to that row is a valuation that does not satisfy the sentence being tested. This method for verifying tautologies is an effective procedure, which means that given unlimited computational resources it can always be used to mechanistically determine whether a sentence is a tautology. This means, in particular, the set of tautologies over a fixed finite or countable alphabet is a decidable set.
As an efficient procedure, however, truth tables are constrained by the fact that the number of valuations that must be checked increases as 2k, where k is the number of variables in the formula. This exponential growth in the computation length renders the truth table method useless for formulas with thousands of propositional variables, as contemporary computing hardware cannot execute the algorithm in a feasible time period.
The problem of determining whether there is any valuation that makes a formula true is the Boolean satisfiability problem; the problem of checking tautologies is equivalent to this problem, because verifying that a sentence S is a tautology is equivalent to verifying that there is no valuation satisfying $\lnot S$. It is known that the Boolean satisfiability problem is NP complete, and widely believed that there is no polynomial-time algorithm that can perform it. Consequently, tautology is co-NP-complete. Current research focuses on finding algorithms that perform well on special classes of formulas, or terminate quickly on average even though some inputs may cause them to take much longer.
Tautologies versus validities in first-order logic
The fundamental definition of a tautology is in the context of propositional logic. The definition can be extended, however, to sentences in first-order logic.[4] These sentences may contain quantifiers, unlike sentences of propositional logic. In the context of first-order logic, a distinction is maintained between logical validities, sentences that are true in every model, and tautologies (or, tautological validities), which are a proper subset of the first-order logical validities. In the context of propositional logic, these two terms coincide.
A tautology in first-order logic is a sentence that can be obtained by taking a tautology of propositional logic and uniformly replacing each propositional variable by a first-order formula (one formula per propositional variable). For example, because $A\lor \lnot A$ is a tautology of propositional logic, $(\forall x(x=x))\lor (\lnot \forall x(x=x))$ is a tautology in first order logic. Similarly, in a first-order language with a unary relation symbols R,S,T, the following sentence is a tautology:
$(((\exists xRx)\land \lnot (\exists xSx))\to \forall xTx)\Leftrightarrow ((\exists xRx)\to ((\lnot \exists xSx)\to \forall xTx)).$
It is obtained by replacing $A$ with $\exists xRx$, $B$ with $\lnot \exists xSx$, and $C$ with $\forall xTx$ in the propositional tautology $((A\land B)\to C)\Leftrightarrow (A\to (B\to C))$.
Not all logical validities are tautologies in first-order logic. For example, the sentence
$(\forall xRx)\to \lnot \exists x\lnot Rx$
is true in any first-order interpretation, but it corresponds to the propositional sentence $A\to B$ which is not a tautology of propositional logic.
See also
Normal forms
• Algebraic normal form
• Conjunctive normal form
• Disjunctive normal form
• Logic optimization
Related logical topics
• Boolean algebra
• Boolean domain
• Boolean function
• Contradiction
• False (logic)
• Syllogism
• List of logic symbols
• Logic synthesis
• Logical consequence
• Logical graph
• Logical truth
• Vacuous truth
References
1. Weisstein, Eric W. "Tautology". mathworld.wolfram.com. Retrieved 2020-08-14.
2. "tautology | Definition & Facts". Encyclopedia Britannica. Retrieved 2020-08-14.
3. "Tautology (logic)". wikipedia.org.
4. "New Members". Naval Engineers Journal. 114 (1): 17–18. January 2002. doi:10.1111/j.1559-3584.2002.tb00103.x. ISSN 0028-1425.
Further reading
• Bocheński, J. M. (1959) Précis of Mathematical Logic, translated from the French and German editions by Otto Bird, Dordrecht, South Holland: D. Reidel.
• Enderton, H. B. (2002) A Mathematical Introduction to Logic, Harcourt/Academic Press, ISBN 0-12-238452-0.
• Kleene, S. C. (1967) Mathematical Logic, reprinted 2002, Dover Publications, ISBN 0-486-42533-9.
• Reichenbach, H. (1947). Elements of Symbolic Logic, reprinted 1980, Dover, ISBN 0-486-24004-5
• Wittgenstein, L. (1921). "Logisch-philosophiche Abhandlung", Annalen der Naturphilosophie (Leipzig), v. 14, pp. 185–262, reprinted in English translation as Tractatus logico-philosophicus, New York City and London, 1922.
External links
• "Tautology", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
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Solution set
In mathematics, a solution set is the set of values that satisfy a given set of equations or inequalities.
For example, for a set $\{f_{i}\}$ of polynomials over a ring $R$, the solution set is the subset of $R$ on which the polynomials all vanish (evaluate to 0), formally
$\{x\in R:\forall i\in I,f_{i}(x)=0\}$
The feasible region of a constrained optimization problem is the solution set of the constraints.
Examples
1. The solution set of the single equation $x=0$ is the set {0}.
2. For any non-zero polynomial $f$ over the complex numbers in one variable, the solution set is made up of finitely many points.
3. However, for a complex polynomial in more than one variable the solution set has no isolated points.
Remarks
In algebraic geometry, solution sets are called algebraic sets if there are no inequalities. Over the reals, and with inequalities, there are called semialgebraic sets.
Other meanings
More generally, the solution set to an arbitrary collection E of relations (Ei) (i varying in some index set I) for a collection of unknowns ${(x_{j})}_{j\in J}$, supposed to take values in respective spaces ${(X_{j})}_{j\in J}$, is the set S of all solutions to the relations E, where a solution $x^{(k)}$ is a family of values $ {\left(x_{j}^{(k)}\right)}_{j\in J}\in \prod _{j\in J}X_{j}$ such that substituting ${\left(x_{j}\right)}_{j\in J}$ by $x^{(k)}$ in the collection E makes all relations "true".
(Instead of relations depending on unknowns, one should speak more correctly of predicates, the collection E is their logical conjunction, and the solution set is the inverse image of the boolean value true by the associated boolean-valued function.)
The above meaning is a special case of this one, if the set of polynomials fi if interpreted as the set of equations fi(x)=0.
Examples
• The solution set for E = { x+y = 0 } with respect to $(x,y)\in \mathbb {R} ^{2}$ is S = { (a,−a) : a ∈ R }.
• The solution set for E = { x+y = 0 } with respect to $x\in \mathbb {R} $ is S = { −y }. (Here, y is not "declared" as an unknown, and thus to be seen as a parameter on which the equation, and therefore the solution set, depends.)
• The solution set for $E=\{{\sqrt {x}}\leq 4\}$ with respect to $x\in \mathbb {R} $ is the interval S = [0,2] (since ${\sqrt {x}}$ is undefined for negative values of x).
• The solution set for $E=\{e^{ix}=1\}$ with respect to $x\in \mathbb {C} $ is S = 2πZ (see Euler's identity).
See also
• Equation solving
• Extraneous and missing solutions
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Wikipedia
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Subobject classifier
In category theory, a subobject classifier is a special object Ω of a category such that, intuitively, the subobjects of any object X in the category correspond to the morphisms from X to Ω. In typical examples, that morphism assigns "true" to the elements of the subobject and "false" to the other elements of X. Therefore, a subobject classifier is also known as a "truth value object" and the concept is widely used in the categorical description of logic. Note however that subobject classifiers are often much more complicated than the simple binary logic truth values {true, false}.
Introductory example
As an example, the set Ω = {0,1} is a subobject classifier in the category of sets and functions: to every subset A of S defined by the inclusion function j : A → S we can assign the function χA from S to Ω that maps precisely the elements of A to 1 (see characteristic function). Every function from S to Ω arises in this fashion from precisely one subset A.
To be clearer, consider a subset A of S (A ⊆ S), where S is a set. The notion of being a subset can be expressed mathematically using the so-called characteristic function χA : S → {0,1}, which is defined as follows:
$\chi _{A}(x)={\begin{cases}0,&{\mbox{if }}x\notin A\\1,&{\mbox{if }}x\in A\end{cases}}$
(Here we interpret 1 as true and 0 as false.) The role of the characteristic function is to determine which elements belong to the subset A. In fact, χA is true precisely on the elements of A.
In this way, the collection of all subsets of S and the collection of all maps from S to Ω = {0,1} are isomorphic.
To categorize this notion, recall that, in category theory, a subobject is actually a pair consisting of an object and a monic arrow (interpreted as the inclusion into another object). Accordingly, true refers to the element 1, which is selected by the arrow: true: {0} → {0, 1} that maps 0 to 1. The subset A of S can now be defined as the pullback of true along the characteristic function χA, shown on the following diagram:
Defined that way, χ is a morphism SubC(S) → HomC(S, Ω). By definition, Ω is a subobject classifier if this morphism is an isomorphism.
Definition
For the general definition, we start with a category C that has a terminal object, which we denote by 1. The object Ω of C is a subobject classifier for C if there exists a morphism
1 → Ω
with the following property:
For each monomorphism j: U → X there is a unique morphism χ j: X → Ω such that the following commutative diagram
is a pullback diagram—that is, U is the limit of the diagram:
The morphism χ j is then called the classifying morphism for the subobject represented by j.
Further examples
Sheaves of sets
The category of sheaves of sets on a topological space X has a subobject classifier Ω which can be described as follows: For any open set U of X, Ω(U) is the set of all open subsets of U. The terminal object is the sheaf 1 which assigns the singleton {*} to every open set U of X. The morphism η:1 → Ω is given by the family of maps ηU : 1(U) → Ω(U) defined by ηU(*)=U for every open set U of X. Given a sheaf F on X and a sub-sheaf j: G → F, the classifying morphism χ j : F → Ω is given by the family of maps χ j,U : F(U) → Ω(U), where χ j,U(x) is the union of all open sets V of U such that the restriction of x to V (in the sense of sheaves) is contained in jV(G(V)).
Roughly speaking an assertion inside this topos is variably true or false, and its truth value from the viewpoint of an open subset U is the open subset of U where the assertion is true.
Presheaves
Given a small category $C$, the category of presheaves $\mathrm {Set} ^{C^{op}}$ (i.e. the functor category consisting of all contravariant functors from $C$ to $\mathrm {Set} $) has a subobject classifer given by the functor sending any $c\in C$ to the set of sieves on $c$. The classifying morphisms are constructed quite similarly to the ones in the sheaves-of-sets example above.
Elementary topoi
Both examples above are subsumed by the following general fact: every elementary topos, defined as a category with finite limits and power objects, necessarily has a subobject classifier.[1] The two examples above are Grothendieck topoi, and every Grothendieck topos is an elementary topos.
Related concepts
A quasitopos has an object that is almost a subobject classifier; it only classifies strong subobjects.
Notes
1. Pedicchio & Tholen (2004) p.8
References
• Artin, Michael; Alexander Grothendieck; Jean-Louis Verdier (1964). Séminaire de Géometrie Algébrique IV. Springer-Verlag.
• Barr, Michael; Charles Wells (1985). Toposes, Triples and Theories. Springer-Verlag. ISBN 0-387-96115-1.
• Bell, John (1988). Toposes and Local Set Theories: an Introduction. Oxford: Oxford University Press.
• Goldblatt, Robert (1983). Topoi: The Categorial Analysis of Logic. North-Holland, Reprinted by Dover Publications, Inc (2006). ISBN 0-444-85207-7.
• Johnstone, Peter (2002). Sketches of an Elephant: A Topos Theory Compendium. Oxford: Oxford University Press.
• Johnstone, Peter (1977). Topos Theory. Academic Press. ISBN 0-12-387850-0.
• Mac Lane, Saunders (1998). Categories for the Working Mathematician. Graduate Texts in Mathematics. Vol. 5 (2nd ed.). New York, NY: Springer-Verlag. ISBN 0-387-98403-8. Zbl 0906.18001.
• Mac Lane, Saunders; Ieke Moerdijk (1992). Sheaves in Geometry and Logic: a First Introduction to Topos Theory. Springer-Verlag. ISBN 0-387-97710-4.
• McLarty, Colin (1992). Elementary Categories, Elementary Toposes. Oxford: Oxford University Press. ISBN 0-19-853392-6.
• Pedicchio, Maria Cristina; Tholen, Walter, eds. (2004). Categorical foundations. Special topics in order, topology, algebra, and sheaf theory. Encyclopedia of Mathematics and Its Applications. Vol. 97. Cambridge: Cambridge University Press. ISBN 0-521-83414-7. Zbl 1034.18001.
• Taylor, Paul (1999). Practical Foundations of Mathematics. Cambridge: Cambridge University Press. ISBN 0-521-63107-6.
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Wikipedia
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Truthful cake-cutting
Truthful cake-cutting is the study of algorithms for fair cake-cutting that are also truthful mechanisms, i.e., they incentivize the participants to reveal their true valuations to the various parts of the cake.
The classic divide and choose procedure for cake-cutting is not truthful: if the cutter knows the chooser's preferences, he can get much more than 1/2 by acting strategically. For example, suppose the cutter values a piece by its size while the chooser values a piece by the amount of chocolate in it. So the cutter can cut the cake into two pieces with almost the same amount of chocolate, such that the smaller piece has slightly more chocolate. Then, the chooser will take the smaller piece and the cutter will win the larger piece, which may be worth much more than 1/2 (depending on how the chocolate is distributed).
Randomized mechanisms
There is a trivial randomized truthful mechanism for fair cake-cutting: select a single agent uniformly at random, and give him/her the entire cake. This mechanism is trivially truthful because it asks no questions. Moreover, it is fair in expectation: the expected value of each partner is exactly 1/n. However, the resulting allocation is not fair. The challenge is to develop truthful mechanisms that are fair ex-post and not just ex-ante. Several such mechanisms have been developed.
Exact division mechanism
An exact division (aka consensus division) is a partition of the cake into n pieces such that each agent values each piece at exactly 1/n. The existence of such a division is a corollary of the Dubins–Spanier convexity theorem. Moreover, there exists such a division with at most $n(n-1)^{2}$ cuts; this is a corollary of the Stromquist–Woodall theorem and the necklace splitting theorem.
In general, an exact division cannot be found by a finite algorithm. However, it can be found in some special cases, for example when all agents have piecewise-linear valuations. Suppose we have a non-truthful algorithm (or oracle) for finding an exact division. It can be used to construct a randomized mechanism that is truthful in expectation.[1][2] The randomized mechanism is a direct-revelation mechanism - it starts by asking all agents to reveal their entire value-measures:
1. Ask the agents to report their value measures.
2. Use the existing algorithm/oracle to generate an exact division.
3. Perform a random permutation on the consensus partition and give each partner one of the pieces.
Here, the expected value of each agent is always 1/n regardless of the reported value function. Hence, the mechanism is truthful – no agent can gain anything from lying. Moreover, a truthful partner is guaranteed a value of exactly 1/n with probability 1 (not only in expectation). Hence the partners have an incentive to reveal their true value functions.
Super-proportional mechanism
A super-proportional division is a cake-division in which each agent receives strictly more than 1/n by their own value measures. Such a division is known to exist if and only if there are at least two agents that have different valuations to at least one piece of the cake. Any deterministic mechanism that always returns a proportional division, and always returns a super-proportional division when it exists, cannot be truthful.
Mossel and Tamuz present a super-proportional randomized mechanism that is truthful in expectation:[1]
1. Pick a division from a certain distribution D over divisions.
2. Ask each agent to evaluate his/her piece.
3. If all n evaluations are more than 1/n, then implement the allocation and finish.
4. Otherwise, use the exact-division mechanism.
The distribution D in step 1 should be chosen such that, regardless of the agents' valuations, there is a positive probability that a super-proportional division be selected iff it exists. Then, in step 2 it is optimal for each agent to report the true value: reporting a lower value either has no effect or might cause the agent's value to drop from super-proportional to just proportional (in step 4); reporting a higher value either has no effect or might cause the agent's value to drop from proportional to less than 1/n (in step 3).
Approximate exact division using queries
Suppose that, rather than directly revealing their valuations, the agents reveal their values indirectly by answering mark and eval queries (as in the Robertson-Webb model).
Branzei and Miltersen[3] show that the exact-division mechanism can be "discretized" and executed in the query model. This yields, for any $\epsilon >0$, a randomized query-based protocol, that asks at most $O(n^{2}/\epsilon )$ queries, is truthful in expectation, and allocates each agent a piece of value between $1/n-\epsilon $ and $1/n+\epsilon $, by the valuations of all agents.
On the other hand, they prove that, in any deterministic truthful query-based protocol, if all agents value all parts of the cake positively, there is at least one agent who gets the empty piece. This implies that, if there are only two agents, then at least one agent is a "dictator" and gets the entire cake. Obviously, any such mechanism cannot be envy-free.
Randomized mechanism for piecewise-constant valuations
Suppose all agents have piecewise-constant valuations. This means that, for each agent, the cake is partitioned into finitely many subsets, and the agent's value density in each subset is constant. For this case, Aziz and Ye present a randomized algorithm that is more economically-efficient: Constrained Serial Dictatorship is truthful in expectation, robust proportional, and satisfies a property called unanimity: if each agent's most preferred 1/n length of the cake is disjoint from other agents, then each agent gets their most preferred 1/n length of the cake. This is a weak form of efficiency that is not satisfied by the mechanisms based on exact division. When there are only two agents, it is also polynomial-time and robust envy-free.[4]
Deterministic mechanisms: piecewise-constant valuations
For deterministic mechanisms, the results are mostly negative, even when all agents have piecewise-constant valuations.
Kurokawa, Lai and Procaccia prove that there is no deterministic, truthful and envy-free mechanism that requires a bounded number of Robertson-Webb queries.[5]
Aziz and Ye prove that there is no deterministic truthful mechanism that satisfies either one of the following properties:[4]
• Proportional and Pareto-optimal;
• Robust-proportional and non-wasteful ("non-wasteful" means that no piece is allocated to an agent who does not want it; it is weaker than Pareto-optimality).
Menon and Larson introduce the notion of ε-truthfulness, which means that no agent gains more than a fraction ε from misreporting, where ε is a positive constant independent of the agents' valuations. They prove that no deterministic mechanism satisfies either one of the following properties:[6]
• ε-truthful, approximately-proportional and non-wasteful (for approximation constants at most 1/n);
• Truthful, approximately-proportional and connected (for approximation constant at most 1/n).
They present a minor modification to the Even–Paz protocol and prove that it is ε-truthful with ε = 1 - 3/(2n) when n is even, and ε = 1 - 3/(2n) + 1/n2 when n is odd.
Bei, Chen, Huzhang, Tao and Wu prove that there is no deterministic, truthful and envy-free mechanism, even in the direct-revelation model, that satisfies either one of the following additional properties:[7]
• Connected pieces;
• Non-wasteful;
• Position oblivious - the allocation of a cake-part is based only on the agents' valuations of that part, and not on its relative position on the cake.
Note that these impossibility results hold with or without free disposal.
On the positive side, in a replicate economy, where each agent is replicated k times, there are envy-free mechanisms in which truth-telling is a Nash equilibrium:[7]
• With connectivity requirement, in any envy-free mechanism, truth-telling converges to a Nash equilibrium when k approaches infinity;
• Without connectivity requirement, in the mechanism that allocates each homogeneous sub-interval equally among all agents, truth-telling is a Nash equilibrium already when k ≥ 2.
Tao improves the previous impossibility result by Bei, Chen, Huzhang, Tao and Wu and shows that there is no deterministic, truthful and proportional mechanism, even in the direct-revelation model, and even when all of the followings hold:[8]
• There are only two agents;
• Agents are hungry: each agent's valuation is positive (i.e., cannot be 0);
• The mechanism is allowed to leave some part of the cake unallocated.
It is open whether this impossibility result extends to three or more agents.
On the positive side, Tao presents two algorithms that attain a weaker notion called "proportional risk-averse truthfulness" (PRAT). It means that, in any profitable deviation for agent i, there exist valuations of the other agents, for which i gets less than his proportional share. This property is stronger than "risk-averse truthfulness", which means that, in any profitable deviation for i, there exist valuations of the other agents, for which i gets less than his value in a truthful reporting. He presents an algorithm that is PRAT and envy-free, and an algorithm that is PRAT, proportional and connected.[8]
Piecewise-uniform valuations
Suppose all agents have piecewise-uniform valuations. This means that, for each agent, there is a subset of the cake that is desirable for the agent, and the agent's value for each piece is just the amount of desirable cake that it contains. For example, suppose some parts of the cake are covered by a uniform layer of chocolate, while other parts are not. An agent who values each piece only by the amount of chocolate it contains has a piecewise-uniform valuation. This is a special case of piecewise-constant valuations. Several truthful algorithms have been developed for this special case.
Chen, Lai, Parkes and Procaccia present a direct-revelation mechanism that is deterministic, proportional, envy-free, Pareto-optimal, and polynomial-time.[2] It works for any number of agents. Here is an illustration of the CLPP mechanism for two agents (where the cake is an interval).
1. Ask each agent to report his/her desired intervals.
2. Each sub-interval, that is desired by no agent, is discarded.
3. Each sub-interval, that is desired by exactly one agent, is allocated to that agent.
4. The sub-intervals, that are desired by both agents, are allocated such that both agents get an equal total length.
Now, if an agent says that he wants an interval that he actually does not want, then he may get more useless cake in step 3 and less useful cake in step 4. If he says that he does not want an interval that he actually wants, then he gets less useful cake in step 3 and more useful cake in step 4, however, the amount given in step 4 is shared with the other agent, so all in all, the lying agent is at a loss. The mechanism can be generalized to any number of agents.
The CLPP mechanism relies on the free disposal assumption, i.e., the ability to discard pieces that are not desired by any agent.
Note: Aziz and Ye[4] presented two mechanisms that extend the CLPP mechanism to piecewise-constant valuations - Constrained Cake Eating Algorithm and Market Equilibrium Algorithm. However, both these extensions are no longer truthful when the valuations are not piecewise-uniform.
Maya and Nisan show that the CLPP mechanism is unique in the following sense.[9] Consider the special case of two agents with piecewise-uniform valuations, where the cake is [0,1], Alice wants only the subinterval [0,a] for some a<1, and Bob desires only the subinterval [1-b,1] for some b<1. Consider only non-wasteful mechanisms - mechanisms that allocate each piece desired by at least one player to a player who wants it. Each such mechanism must give Alice a subset [0,c] for some c<1 and Bob a subset [1-d,1] for some d<1. In this model:
• A non-wasteful determininstic mechanism is truthful iff, for some parameter t in [0,1], it gives Alice the interval [0, min(a, max(1-b,t))] and Bob the interval [1-min(b,max(1-a,1-t)),1]
• Such mechanism is envy-free iff t=1/2; in this case it is equivalent to the CLPP mechanism
They also show that, even for 2 agents, any truthful mechanism achieves at most 0.93 of the optimal social welfare.
Li, Zhang and Zhang show that the CLPP mechanism works well even when there are externalities (i.e., some agents derive some benefit from the value given to others), as long as the externalities are sufficiently small. On the other hand, if the externalities (either positive or negative) are large, no truthful non-wasteful and position independent mechanism exists.[10]
Alijani, Farhadi, Ghodsi, Seddighin and Tajik present several mechanisms for special cases of piecewise-uniform valuations:[11]
• The expansion process handles piecewise-uniform valuations where each agent has a single desired interval, and moreover, the agents' desired intervals satisfy an ordering property. It is polynomial-time, truthful, envy-free, and guarantees connected pieces.
• The expansion process with unlocking handles piecewise-uniform valuations where each agent has a single desired interval, but without the ordering requirement. It is polynomial-time, truthful, envy-free, and not necessarily connected, but it makes at most 2n-2 cuts.
Bei, Huzhang and Suksompong present a mechanism for two agents with piecewise-uniform valuations, that has the same properties of CLPP (truthful, deterministic, proportional, envy-free, Pareto-optimal and runs in polynomial time), but guarantees that the entire cake is allocated:[12]
1. Find the smallest x in [0,1] such that Alice's desired length in [0,x] equals Bob's desired length in [x,1].
2. Give Alice the intervals in [0,x] valued by Alice and the intervals in [x,1] not valued by Bob; give the remainder to Bob.
The BHS mechanism works both for cake-cutting and for chore division (where the agents' valuations are negative). Note that BHS does not satisfy some natural desirable properties:
• It does not guarantee connected pieces, for example when Alice wants [0,1] and Bob wants [0,0.5], then x=0.25, Alice gets [0,0.25] and [0.5,1], and Bob gets [0.25,0.5].
• It is not anonymous (see symmetric fair cake-cutting): if Alice wants [0,1] and Bob wants [0,0.5], then Alice gets a desired length of 0.75 and Bob gets 0.25, but if the valuations are switched (Alice wants [0,0.5] and Bob wants [0,1]), then x=0.5 and both agents get desired length 0.5.
• It is not position oblivious: if Alice wants [0,0.5] and Bob wants [0,1] then both agents get value 0.5, but if Alice's desired interval moves to [0.5,1] then x=0.75 and Alice gets 0.25 and Bob gets 0.75.
This is not a problem with the specific mechanism: it is provably impossible to have a truthful and envy-free mechanism that allocates the entire cake and guarantees any of these three properties, even for two agents with piecewise-uniform valuations.[12]
The BHS mechanism was extended to any number of agents, but only for a special case of piecewise-uniform valuations, in which each agent desires only a single interval of the form [0, xi].
Ianovsky[13] proves that no truthful mechanism can attain a utilitarian-optimal cake-cutting, even when all agents have piecewise-uniform valuations. Moreover, no truthful mechanism can attain an allocation with utilitarian welfare at least as large as any other mechanism. However, there is a simple truthful mechanism (denoted Lex Order) that is non-wasteful: give to agent 1 all pieces that he likes; then, give to agent 2 all pieces that he likes and were not yet given to agent 1; etc. A variant of this mechanism is the Length Game, in which the agents are renamed by the total length of their desired intervals, such that the agent with the shortest interval is called 1, the agent with the next-shortest interval is called 2, etc. This is not a truthful mechanism, however:
• If all agents are truthful, then it produces a utilitarian-optimal allocation.
• If the agents are strategic, then all its well-behaved Nash equilibria are Pareto-efficient and envy-free, and yield the same payoffs as the CLPP mechanism.
Summary of truthful mechanisms and impossibility results
Name Type Deterministic? #agents(n) Valuations[14] Chores?[15] Run time All?[16] PO?[17] EF?[18] Anon?[19] Conn?[20] Pos.Ob.?[21] No Waste?[22]
Exact division[1][2]DirectNoManyGeneralYes Unbounded[23]YesNoYesYesNo ? ?
Super-proportional[1] DirectNoManyGeneralYes UnboundedYesNoNoYesNo ? ?
Discrete exact division[3] QueriesNoMany GeneralYes $O(n^{2}/\epsilon )$YesNo $\epsilon $-EFYesNo ? ?
Constrained Serial Dictatorship[4] DirectNoMany PWC ? ?No Unanimity Prop. ?No ? ?
CLPP[2] DirectYesManyPWUNo PolynomialNoYesYesYesNo ?Yes
BHS 1, 2 DirectYes2PWUYes PolynomialYesYesYesNoNoNoYes
BHS 3, 4 DirectYesMany PWU1Yes PolynomialYesYes Yes (for cakes) ? ? ?Yes
Expansion[11] DirectYesMany PWU1+order ? Polynomial ? ?Yes ?Yes ? ?
Expansion+ Unlocking DirectYesMany PWU1 ? Polynomial ? ?Yes ? 2n-2 cuts ? ?
IMPOSSIBLE COMBINATIONS:
[BM][3] QueriesYes 2+ Any
[BHS][12] DirectYes 2+ PWU Yes YesYes
[BHS] DirectYes 2+ PWU Yes Yes Yes
[BHS] DirectYes 2+ PWU Yes Yes Yes
[T][8] DirectYes 2+ PWC Yes(even for Prop.)
[BCHTW][7] DirectYes 2+ PWC Yes Yes Yes
[BCHTW] DirectYes 2+ PWC Yes Yes Yes
[BCHTW] DirectYes 2+ PWC Yes Yes Yes
[BCHTW] SequentialYes 2+ PWC Yes Yes
See also
• Strategic fair division
• Truthful resource allocation
References
1. Mossel, Elchanan; Tamuz, Omer (2010). "Truthful Fair Division". Algorithmic Game Theory. Lecture Notes in Computer Science. Springer Berlin Heidelberg. 6386: 288–299. arXiv:1003.5480. Bibcode:2010LNCS.6386..288M. doi:10.1007/978-3-642-16170-4_25. ISBN 9783642161704. S2CID 11732339.
2. Chen, Yiling; Lai, John K.; Parkes, David C.; Procaccia, Ariel D. (2013-01-01). "Truth, justice, and cake cutting". Games and Economic Behavior. 77 (1): 284–297. doi:10.1016/j.geb.2012.10.009. ISSN 0899-8256. S2CID 2096977.
3. Brânzei, Simina; Miltersen, Peter Bro (2015-06-22). "A Dictatorship Theorem for Cake Cutting". Twenty-Fourth International Joint Conference on Artificial Intelligence.
4. Aziz, Haris; Ye, Chun (2014). Liu, Tie-Yan; Qi, Qi; Ye, Yinyu (eds.). "Cake Cutting Algorithms for Piecewise Constant and Piecewise Uniform Valuations". Web and Internet Economics. Lecture Notes in Computer Science. Springer International Publishing. 8877: 1–14. doi:10.1007/978-3-319-13129-0_1. ISBN 9783319131290. S2CID 18365892.
5. Kurokawa, David; Lai, John K.; Procaccia, Ariel D. (2013-06-30). "How to Cut a Cake Before the Party Ends". Twenty-Seventh AAAI Conference on Artificial Intelligence. 27: 555–561. doi:10.1609/aaai.v27i1.8629. S2CID 12638556.
6. Menon, Vijay; Larson, Kate (2017-05-17). "Deterministic, Strategyproof, and Fair Cake Cutting". arXiv:1705.06306 [cs.GT].
7. Bei, Xiaohui; Chen, Ning; Huzhang, Guangda; Tao, Biaoshuai; Wu, Jiajun (2017). "Cake Cutting: Envy and Truth". Proceedings of the 26th International Joint Conference on Artificial Intelligence. IJCAI'17. AAAI Press: 3625–3631. ISBN 9780999241103.
8. Tao, Biaoshuai (2022-07-13). "On Existence of Truthful Fair Cake Cutting Mechanisms". Proceedings of the 23rd ACM Conference on Economics and Computation. pp. 404–434. arXiv:2104.07387. doi:10.1145/3490486.3538321. ISBN 9781450391504. S2CID 233241229.
9. Maya, Avishay; Nisan, Noam (2012). Goldberg, Paul W. (ed.). "Incentive Compatible Two Player Cake Cutting". Internet and Network Economics. Lecture Notes in Computer Science. Springer Berlin Heidelberg. 7695: 170–183. arXiv:1210.0155. Bibcode:2012arXiv1210.0155M. doi:10.1007/978-3-642-35311-6_13. ISBN 9783642353116. S2CID 1927798.
10. Li, Minming; Zhang, Jialin; Zhang, Qiang (2015-06-22). "Truthful Cake Cutting Mechanisms with Externalities: Do Not Make Them Care for Others Too Much!". Twenty-Fourth International Joint Conference on Artificial Intelligence.
11. Alijani, Reza; Farhadi, Majid; Ghodsi, Mohammad; Seddighin, Masoud; Tajik, Ahmad S. (2017-02-10). "Envy-Free Mechanisms with Minimum Number of Cuts". Thirty-First AAAI Conference on Artificial Intelligence. 31. doi:10.1609/aaai.v31i1.10584. S2CID 789550.
12. Bei, Xiaohui; Huzhang, Guangda; Suksompong, Warut (2020). "Truthful fair division without free disposal". Social Choice and Welfare. 55 (3): 523–545. arXiv:1804.06923. doi:10.1007/s00355-020-01256-0. PMC 7497335. PMID 33005068.
13. Ianovski, Egor (2012-03-01). "Cake Cutting Mechanisms". arXiv:1203.0100 [cs.GT].
14. PWC = piecewise-constant, PWU = piecewise-uniform, PWU1 = piecewise-uniform with a single desired interval.
15. Whether the algorithm can handle also cakes with negative utilities (chores).
16. Whether the entire cake is divided, with no disposal.
17. Whether the resulting allocation is always Pareto optimal.
18. Whether the resulting allocation is always envy-free.
19. Whether the mechanismn is anonymous.
20. Whether the resulting pieces are always connected.
21. Whether the mechanism is position oblivious.
22. Whether the algorithm guarantees non-wastefulness.
23. The run-time is dominated by calculating an exact division. In general it is unbounded, but in special cases it may be polynomial.
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Wikipedia
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Strategic fair division
Strategic fair division is the branch of fair division in which the participants are assumed to hide their preferences and act strategically in order to maximize their own utility, rather than playing sincerely according to their true preferences.
To illustrate the difference between strategic fair division and classic fair division, consider the divide and choose procedure for dividing a cake among two agents. In classic fair division, it is assumed that the cutter cuts the cake into two pieces that are equal in his eyes, and thus he always gets a piece that he values at exactly 1/2 of the total cake value. However, if the cutter knows the chooser's preferences, he can get much more than 1/2 by acting strategically. For example, suppose the cutter values a piece by its size while the chooser values a piece by the amount of chocolate in it. So the cutter can cut the cake into two pieces with almost the same amount of chocolate, such that the smaller piece has slightly more chocolate. Then, the chooser will take the smaller piece and the cutter will win the larger piece, which may be worth much more than 1/2 (depending on how the chocolate is distributed).
The research in strategic fair division has two main branches.
One branch is related to game theory and studies the equilibria in games created by fair division algorithms:
• The Nash equilibrium of the Dubins-Spanier moving-knife protocol;[1]
• The Nash equilibrium and subgame-perfect equilibrium of generalized-cut-and-choose protocols;[2]
• The equilibria of envy-free protocols for allocating an indivisible good with monetary compensations.[3]
• The price of anarchy of Nash equilibria of two mechanisms for homogeneous-resource allocation: the Fisher market game and the Trading Post game.[4]
The other branch is related to mechanism design and aims to find truthful mechanisms for fair division, in particular:
• Truthful cake-cutting;
• Truthful resource allocation;
• Truthful fair division of rooms and rent.
References
1. Brânzei, Simina; Miltersen, Peter Bro (2013). "Equilibrium Analysis in Cake Cutting". Proceedings of the 2013 International Conference on Autonomous Agents and Multi-agent Systems. AAMAS '13. Richland, SC: International Foundation for Autonomous Agents and Multiagent Systems: 327–334. ISBN 9781450319935.
2. Brânzei, Simina; Caragiannis, Ioannis; Kurokawa, David; Procaccia, Ariel D. (2016-02-21). "An Algorithmic Framework for Strategic Fair Division". Thirtieth AAAI Conference on Artificial Intelligence. 30. arXiv:1307.2225. doi:10.1609/aaai.v30i1.10042. S2CID 7226490.
3. Tadenuma, Koichi; Thomson, William (1995-05-01). "Games of Fair Division". Games and Economic Behavior. 9 (2): 191–204. doi:10.1006/game.1995.1015. ISSN 0899-8256.
4. Brânzei, Simina; Gkatzelis, Vasilis; Mehta, Ruta (2016-07-06). "Nash Social Welfare Approximation for Strategic Agents". arXiv:1607.01569 [cs.GT].
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Wikipedia
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Truth table
A truth table is a mathematical table used in logic—specifically in connection with Boolean algebra, boolean functions, and propositional calculus—which sets out the functional values of logical expressions on each of their functional arguments, that is, for each combination of values taken by their logical variables.[1] In particular, truth tables can be used to show whether a propositional expression is true for all legitimate input values, that is, logically valid.
A truth table has one column for each input variable (for example, A and B), and one final column showing all of the possible results of the logical operation that the table represents (for example, A XOR B). Each row of the truth table contains one possible configuration of the input variables (for instance, A=true, B=false), and the result of the operation for those values.
A truth table is a structured representation that presents all possible combinations of truth values for the input variables of a Boolean function and their corresponding output values. A function f from A to F is a special relation, a subset of A×F, which simply means that f can be listed as a list of input-output pairs. Clearly, for the Boolean functions, the output belongs to a binary set, i.e. F = {0, 1}. For an n-ary Boolean function, the inputs come from a domain that is itself a Cartesian product of binary sets corresponding to the input Boolean variables. For example for a binary function, f(A, B), the domain of f is A×B, which can be listed as: A×B = {(A = 0, B = 0), (A = 0, B = 1), (A = 1, B = 0), (A = 1, B = 1)}. Each element in the domain represents a combination of input values for the variables A and B. These combinations now can be combined with the output of the function corresponding to that combination, thus forming the set of input-output pairs as a special relation that is subset of A×F. For a relation to be a function, the special requirement is that each element of the domain of the function must be mapped to one and only one member of the codomain. Thus, the function f itself can be listed as: f = {((0, 0), f0), ((0, 1), f1), ((1, 0), f2), ((1, 1), f3)}, where f0, f1, f2, and f3 are each Boolean, 0 or 1, values as members of the codomain {0, 1}, as the outputs corresponding to the member of the domain, respectively. Rather than a list (set) given above, the truth table then presents these input-output pairs in a tabular format, in which each row corresponds to a member of the domain paired with its corresponding output value, 0 or 1. Of course, for the Boolean functions, we do not have to list all the members of the domain with their images in the codomain; we can simply list the mappings that map the member to “1”, because all the others will have to be mapped to “0” automatically (that leads us to the minterms idea).
Ludwig Wittgenstein is generally credited with inventing and popularizing the truth table in his Tractatus Logico-Philosophicus, which was completed in 1918 and published in 1921.[2] Such a system was also independently proposed in 1921 by Emil Leon Post.[3]
Nullary operations
There are 2 nullary operations:
• Always true
• Never true, unary falsum
Logical true
The output value is always true, because this operator has zero operands and therefore no input values
p T
TT
FT
Logical false
The output value is never true: that is, always false, because this operantor has zero operands and therefore no input values
p F
TF
FF
Unary operations
There are 2 unary operations:
• Unary identity
• Unary negation
Logical identity
Logical identity is an operation on one logical value p, for which the output value remains p.
The truth table for the logical identity operator is as follows:
p p
TT
FF
Logical negation
Logical negation is an operation on one logical value, typically the value of a proposition, that produces a value of true if its operand is false and a value of false if its operand is true.
The truth table for NOT p (also written as ¬p, Np, Fpq, or ~p) is as follows:
p ¬p
TF
FT
Binary operations
There are 16 possible truth functions of two binary variables:
Truth table for all binary logical operators
Here is an extended truth table giving definitions of all sixteen possible truth functions of two Boolean variables P and Q:[note 1]
pq F0 NOR1 ↚2 ¬p3 NIMPLY4 ¬q5 XOR6 NAND7 AND8 XNOR9 q10IMPLY11p12←13OR14T15
TT FFFFFFFFTTTTTTTT
TF FFFFTTTTFFFFTTTT
FT FFTTFFTTFFTTFFTT
FF FTFTFTFTFTFTFTFT
Com ✓✓✓✓✓✓✓✓
Assoc ✓✓✓✓✓✓✓✓
Adj F0NOR1↛4¬q5NIMPLY2¬p3XOR6NAND7AND8XNOR9p12IMPLY13q10→11OR14T15
Neg T15OR14←13p12IMPLY11q10XNOR9AND8NAND7XOR6¬q5NIMPLY4¬p3↚2NOR1F0
Dual T15NAND7→11¬p3←13¬q5XNOR9NOR1OR14XOR6q10↚2p12↛4AND8F0
L id FFTTT,FTF
R id FFTTT,FTF
where
T = true.
F = false.
The superscripts 0 to 15 is the number resulting from reading the four truth values as a binary number with F = 0 and T = 1.
The Com row indicates whether an operator, op, is commutative - P op Q = Q op P.
The Assoc row indicates whether an operator, op, is associative - (P op Q) op R = P op (Q op R).
The Adj row shows the operator op2 such that P op Q = Q op2 P.
The Neg row shows the operator op2 such that P op Q = ¬(P op2 Q).
The Dual row shows the dual operation obtained by interchanging T with F, and AND with OR.
The L id row shows the operator's left identities if it has any - values I such that I op Q = Q.
The R id row shows the operator's right identities if it has any - values I such that P op I = P.[note 2]
The four combinations of input values for p, q, are read by row from the table above. The output function for each p, q combination, can be read, by row, from the table.
Key:
The following table is oriented by column, rather than by row. There are four columns rather than four rows, to display the four combinations of p, q, as input.
p: T T F F
q: T F T F
There are 16 rows in this key, one row for each binary function of the two binary variables, p, q. For example, in row 2 of this Key, the value of Converse nonimplication ('$\nleftarrow $') is solely T, for the column denoted by the unique combination p=F, q=T; while in row 2, the value of that '$\nleftarrow $' operation is F for the three remaining columns of p, q. The output row for $\nleftarrow $ is thus
2: F F T F
and the 16-row[4] key is
[4]operatorOperation name
0(F F F F)(p, q)⊥false, OpqContradiction
1(F F F T)(p, q)NORp ↓ q, XpqLogical NOR
2(F F T F)(p, q)↚p ↚ q, MpqConverse nonimplication
3(F F T T)(p, q)¬p, ~p¬p, Np, FpqNegation
4(F T F F)(p, q)↛p ↛ q, LpqMaterial nonimplication
5(F T F T)(p, q)¬q, ~q¬q, Nq, GpqNegation
6(F T T F)(p, q)XORp ⊕ q, JpqExclusive disjunction
7(F T T T)(p, q)NANDp ↑ q, DpqLogical NAND
8(T F F F)(p, q)ANDp ∧ q, KpqLogical conjunction
9(T F F T)(p, q)XNORp If and only if q, EpqLogical biconditional
10(T F T F)(p, q)qq, HpqProjection function
11(T F T T)(p, q)p → qif p then q, CpqMaterial implication
12(T T F F)(p, q)pp, IpqProjection function
13(T T F T)(p, q)p ← qp if q, BpqConverse implication
14(T T T F)(p, q)ORp ∨ q, ApqLogical disjunction
15(T T T T)(p, q)⊤true, VpqTautology
Logical operators can also be visualized using Venn diagrams.
Logical conjunction (AND)
Logical conjunction is an operation on two logical values, typically the values of two propositions, that produces a value of true if both of its operands are true.
The truth table for p AND q (also written as p ∧ q, Kpq, p & q, or p $\cdot $ q) is as follows:
p q p ∧ q
TTT
TFF
FTF
FFF
In ordinary language terms, if both p and q are true, then the conjunction p ∧ q is true. For all other assignments of logical values to p and to q the conjunction p ∧ q is false.
It can also be said that if p, then p ∧ q is q, otherwise p ∧ q is p.
Logical disjunction (OR)
Logical disjunction is an operation on two logical values, typically the values of two propositions, that produces a value of true if at least one of its operands is true.
The truth table for p OR q (also written as p ∨ q, Apq, p || q, or p + q) is as follows:
p q p ∨ q
TTT
TFT
FTT
FFF
Stated in English, if p, then p ∨ q is p, otherwise p ∨ q is q.
Logical implication
Logical implication and the material conditional are both associated with an operation on two logical values, typically the values of two propositions, which produces a value of false if the first operand is true and the second operand is false, and a value of true otherwise.
The truth table associated with the logical implication p implies q (symbolized as p ⇒ q, or more rarely Cpq) is as follows:
p q p ⇒ q
TTT
TFF
FTT
FFT
The truth table associated with the material conditional if p then q (symbolized as p → q) is as follows:
p q p → q
TTT
TFF
FTT
FFT
It may also be useful to note that p ⇒ q and p → q are equivalent to ¬p ∨ q.
Logical equality
Logical equality (also known as biconditional or exclusive nor) is an operation on two logical values, typically the values of two propositions, that produces a value of true if both operands are false or both operands are true.
The truth table for p XNOR q (also written as p ↔ q, Epq, p = q, or p ≡ q) is as follows:
p q p ↔ q
TTT
TFF
FTF
FFT
So p EQ q is true if p and q have the same truth value (both true or both false), and false if they have different truth values.
Exclusive disjunction
Exclusive disjunction is an operation on two logical values, typically the values of two propositions, that produces a value of true if one but not both of its operands is true.
The truth table for p XOR q (also written as Jpq, or p ⊕ q) is as follows:
p q p ⊕ q
TTF
TFT
FTT
FFF
For two propositions, XOR can also be written as (p ∧ ¬q) ∨ (¬p ∧ q).
Logical NAND
The logical NAND is an operation on two logical values, typically the values of two propositions, that produces a value of false if both of its operands are true. In other words, it produces a value of true if at least one of its operands is false.
The truth table for p NAND q (also written as p ↑ q, Dpq, or p | q) is as follows:
p q p ↑ q
TTF
TFT
FTT
FFT
It is frequently useful to express a logical operation as a compound operation, that is, as an operation that is built up or composed from other operations. Many such compositions are possible, depending on the operations that are taken as basic or "primitive" and the operations that are taken as composite or "derivative".
In the case of logical NAND, it is clearly expressible as a compound of NOT and AND.
The negation of a conjunction: ¬(p ∧ q), and the disjunction of negations: (¬p) ∨ (¬q) can be tabulated as follows:
p q p ∧ q ¬(p ∧ q) ¬p ¬q (¬p) ∨ (¬q)
TTTFFFF
TFFTFTT
FTFTTFT
FFFTTTT
Logical NOR
The logical NOR is an operation on two logical values, typically the values of two propositions, that produces a value of true if both of its operands are false. In other words, it produces a value of false if at least one of its operands is true. ↓ is also known as the Peirce arrow after its inventor, Charles Sanders Peirce, and is a Sole sufficient operator.
The truth table for p NOR q (also written as p ↓ q, or Xpq) is as follows:
p q p ↓ q
TTF
TFF
FTF
FFT
The negation of a disjunction ¬(p ∨ q), and the conjunction of negations (¬p) ∧ (¬q) can be tabulated as follows:
p q p ∨ q ¬(p ∨ q) ¬p ¬q (¬p) ∧ (¬q)
TTTFFFF
TFTFFTF
FTTFTFF
FFFTTTT
Inspection of the tabular derivations for NAND and NOR, under each assignment of logical values to the functional arguments p and q, produces the identical patterns of functional values for ¬(p ∧ q) as for (¬p) ∨ (¬q), and for ¬(p ∨ q) as for (¬p) ∧ (¬q). Thus the first and second expressions in each pair are logically equivalent, and may be substituted for each other in all contexts that pertain solely to their logical values.
This equivalence is one of De Morgan's laws.
Size of truth tables
If there are n input variables then there are 2n possible combinations of their truth values. A given function may produce true or false for each combination so the number of different functions of n variables is the double exponential 22n.
n2n22n
012
124
2416
38256
41665,536
5324,294,967,296≈ 4.3×109
66418,446,744,073,709,551,616≈ 1.8×1019
7128340,282,366,920,938,463,463,374,607,431,768,211,456≈ 3.4×1038
8256115,792,089,237,316,195,423,570,985,008,687,907,853,269,984,665,640,564,039,457,584,007,913,129,639,936≈ 1.2×1077
Truth tables for functions of three or more variables are rarely given.
Applications
Truth tables can be used to prove many other logical equivalences. For example, consider the following truth table:
$(p\Rightarrow q)\equiv (\lnot p\lor q)$
$p$ $q$ $\lnot p$ $\lnot p\lor q$ $p\Rightarrow q$
TTFTT
TFFFF
FTTTT
FFTTT
This demonstrates the fact that $p\Rightarrow q$ is logically equivalent to $\lnot p\lor q$.
Truth table for most commonly used logical operators
Here is a truth table that gives definitions of the 7 most commonly used out of the 16 possible truth functions of two Boolean variables P and Q:
PQ$P\land Q$$P\lor Q$$P\ {\underline {\lor }}\ Q$$P\ {\underline {\land }}\ Q$$P\Rightarrow Q$$P\Leftarrow Q$$P\Leftrightarrow Q$
TTTTFTTTT
TFFTTFFTF
FTFTTFTFF
FFFFFTTTT
PQ$P\land Q$$P\lor Q$$P\ {\underline {\lor }}\ Q$$P\ {\underline {\land }}\ Q$$P\Rightarrow Q$$P\Leftarrow Q$$P\Leftrightarrow Q$
AND
(conjunction)
OR
(disjunction)
XOR
(exclusive or)
XNOR
(exclusive nor)
conditional
"if-then"
conditional
"then-if"
biconditional
"if-and-only-if"
where T means true and F means false
Condensed truth tables for binary operators
For binary operators, a condensed form of truth table is also used, where the row headings and the column headings specify the operands and the table cells specify the result. For example, Boolean logic uses this condensed truth table notation:
∧FT
F FF
T FT
∨FT
F FT
T TT
This notation is useful especially if the operations are commutative, although one can additionally specify that the rows are the first operand and the columns are the second operand. This condensed notation is particularly useful in discussing multi-valued extensions of logic, as it significantly cuts down on combinatoric explosion of the number of rows otherwise needed. It also provides for quickly recognizable characteristic "shape" of the distribution of the values in the table which can assist the reader in grasping the rules more quickly.
Truth tables in digital logic
Truth tables are also used to specify the function of hardware look-up tables (LUTs) in digital logic circuitry. For an n-input LUT, the truth table will have 2^n values (or rows in the above tabular format), completely specifying a boolean function for the LUT. By representing each boolean value as a bit in a binary number, truth table values can be efficiently encoded as integer values in electronic design automation (EDA) software. For example, a 32-bit integer can encode the truth table for a LUT with up to 5 inputs.
When using an integer representation of a truth table, the output value of the LUT can be obtained by calculating a bit index k based on the input values of the LUT, in which case the LUT's output value is the kth bit of the integer. For example, to evaluate the output value of a LUT given an array of n boolean input values, the bit index of the truth table's output value can be computed as follows: if the ith input is true, let $V_{i}=1$, else let $V_{i}=0$. Then the kth bit of the binary representation of the truth table is the LUT's output value, where $k=V_{0}\times 2^{0}+V_{1}\times 2^{1}+V_{2}\times 2^{2}+\dots +V_{n}\times 2^{n}$.
Truth tables are a simple and straightforward way to encode boolean functions, however given the exponential growth in size as the number of inputs increase, they are not suitable for functions with a large number of inputs. Other representations which are more memory efficient are text equations and binary decision diagrams.
Applications of truth tables in digital electronics
In digital electronics and computer science (fields of applied logic engineering and mathematics), truth tables can be used to reduce basic boolean operations to simple correlations of inputs to outputs, without the use of logic gates or code. For example, a binary addition can be represented with the truth table:
Binary addition
$A$ $B$ $C$ $R$
T T T F
T F F T
F T F T
F F F F
where A is the first operand, B is the second operand, C is the carry digit, and R is the result.
This truth table is read left to right:
• Value pair (A,B) equals value pair (C,R).
• Or for this example, A plus B equal result R, with the Carry C.
Note that this table does not describe the logic operations necessary to implement this operation, rather it simply specifies the function of inputs to output values.
With respect to the result, this example may be arithmetically viewed as modulo 2 binary addition, and as logically equivalent to the exclusive-or (exclusive disjunction) binary logic operation.
In this case it can be used for only very simple inputs and outputs, such as 1s and 0s. However, if the number of types of values one can have on the inputs increases, the size of the truth table will increase.
For instance, in an addition operation, one needs two operands, A and B. Each can have one of two values, zero or one. The number of combinations of these two values is 2×2, or four. So the result is four possible outputs of C and R. If one were to use base 3, the size would increase to 3×3, or nine possible outputs.
The first "addition" example above is called a half-adder. A full-adder is when the carry from the previous operation is provided as input to the next adder. Thus, a truth table of eight rows would be needed to describe a full adder's logic:
A B C* | C R
0 0 0 | 0 0
0 1 0 | 0 1
1 0 0 | 0 1
1 1 0 | 1 0
0 0 1 | 0 1
0 1 1 | 1 0
1 0 1 | 1 0
1 1 1 | 1 1
Same as previous, but..
C* = Carry from previous adder
History
Irving Anellis's research shows that C.S. Peirce appears to be the earliest logician (in 1883) to devise a truth table matrix.[5]
From the summary of Peirce's paper:
In 1997, John Shosky discovered, on the verso of a page of the typed transcript of Bertrand Russell's 1912 lecture on "The Philosophy of Logical Atomism" truth table matrices. The matrix for negation is Russell's, alongside of which is the matrix for material implication in the hand of Ludwig Wittgenstein. It is shown that an unpublished manuscript identified as composed by Peirce in 1893 includes a truth table matrix that is equivalent to the matrix for material implication discovered by John Shosky. An unpublished manuscript by Peirce identified as having been composed in 1883–84 in connection with the composition of Peirce's "On the Algebra of Logic: A Contribution to the Philosophy of Notation" that appeared in the American Journal of Mathematics in 1885 includes an example of an indirect truth table for the conditional.
See also
• Boolean domain
• Boolean-valued function
• Espresso heuristic logic minimizer
• Excitation table
• State-transition table
• First-order logic
• Functional completeness
• Karnaugh maps
• Logic gate
• Logical connective
• Logical graph
• Mathematical table
• Method of analytic tableaux
• Propositional calculus
• Truth function
• Decision table
Notes
1. Information about notation may be found in (Bocheński 1959), (Enderton 2001), and (Quine 1982).
2. The operators here with equal left and right identities (XOR, AND, XNOR, and OR) are also commutative monoids because they are also associative. While this distinction may be irrelevant in a simple discussion of logic, it can be quite important in more advanced mathematics. For example, in category theory an enriched category is described as a base category enriched over a monoid, and any of these operators can be used for enrichment.
References
1. Enderton 2001
2. von Wright, Georg Henrik (1955). "Ludwig Wittgenstein, A Biographical Sketch". The Philosophical Review. 64 (4): 527–545 (p. 532, note 9). doi:10.2307/2182631. JSTOR 2182631.
3. Post, Emil (July 1921). "Introduction to a general theory of elementary propositions". American Journal of Mathematics. 43 (3): 163–185. doi:10.2307/2370324. hdl:2027/uiuo.ark:/13960/t9j450f7q. JSTOR 2370324.
4. Wittgenstein, Ludwig (1922). "Proposition 5.101" (PDF). Tractatus Logico-Philosophicus.
5. Anellis, Irving H. (2012). "Peirce's Truth-functional Analysis and the Origin of the Truth Table". History and Philosophy of Logic. 33: 87–97. doi:10.1080/01445340.2011.621702. S2CID 170654885.
Works cited
• Bocheński, Józef Maria (1959). A Précis of Mathematical Logic. Translated by Bird, Otto. D. Reidel. doi:10.1007/978-94-017-0592-9. ISBN 978-94-017-0592-9.
• Enderton, H. (2001). A Mathematical Introduction to Logic (2nd ed.). Harcourt Academic Press. ISBN 0-12-238452-0.
• Quine, W.V. (1982). Methods of Logic (4th ed.). Harvard University Press. ISBN 978-0-674-57175-4.
External links
Wikimedia Commons has media related to Truth tables.
• "Truth table", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Truth Tables, Tautologies, and Logical Equivalence
• Anellis, Irving H. (2011). "Peirce's Truth-functional Analysis and the Origin of Truth Tables". arXiv:1108.2429 [math.HO].
• Converting truth tables into Boolean expressions
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Tsachik Gelander
Tsachik Gelander (צחיק גלנדר) is an Israeli mathematician working in the fields of Lie groups, topological groups, symmetric spaces, lattices and discrete subgroups (of Lie groups as well as general locally compact groups). He is a professor in Northwestern University.[1]
Tsachik Gelander
NationalityIsraeli
Scientific career
FieldsGeometric group theory, locally compact groups, Lie groups, symmetric spaces
InstitutionsNorthwestern University
Doctoral advisorShahar Mozes
Gelander earned his PhD from the Hebrew University of Jerusalem in 2003, under the supervision of Shahar Mozes.[2] His doctoral dissertation, Counting Manifolds and Tits Alternative, won the Haim Nessyahu Prize in Mathematics, awarded by the Israel Mathematical Union for the best annual doctoral dissertations in mathematics.[3] After holding a Gibbs Assistant Professorship at Yale University, and faculty positions at the Hebrew University of Jerusalem and the Weizmann Institute of Science, Gelander joined Northwestern as professor of mathematics in 2022.[4] He contributed to the theory of lattices, Fuchsian groups and local rigidity, and the work on Chern's conjecture and the Derivation Problem.[5]
He gave the distinguished Nachdiplom Lectures at ETH Zurich in 2011, and was an invited speaker at the 2018 International Congress of Mathematicians, giving a talk under the title of Asymptotic Invariants of Locally Symmetric Spaces.[6]
Selected publications
• Gelander, Tsachik (15 September 2004). "Homotopy type and volume of locally symmetric manifolds". Duke Mathematical Journal. Duke University Press. 124 (3). arXiv:math/0111165. doi:10.1215/s0012-7094-04-12432-7. ISSN 0012-7094. S2CID 14272953.
• Bader, Uri; Furman, Alex; Gelander, Tsachik; Monod, Nicolas (2007). "Property (T) and rigidity for actions on Banach spaces". Acta Mathematica. International Press of Boston. 198 (1): 57–105. arXiv:math/0506361. doi:10.1007/s11511-007-0013-0. ISSN 0001-5962. S2CID 5739931.
• Breuillard, Emmanuel; Gelander, Tsachik (1 September 2007). "A topological Tits alternative". Annals of Mathematics. 166 (2): 427–474. arXiv:math/0403043. doi:10.4007/annals.2007.166.427. ISSN 0003-486X. S2CID 14859975.
• Breuillard, E.; Gelander, T. (3 May 2008). "Uniform independence in linear groups". Inventiones Mathematicae. Springer Science and Business Media LLC. 173 (2): 225–263. arXiv:math/0611829. Bibcode:2008InMat.173..225B. doi:10.1007/s00222-007-0101-y. ISSN 0020-9910. S2CID 16029687.
• Belolipetsky, Mikhail; Gelander, Tsachik; Lubotzky, Alexander; Shalev, Aner (5 October 2010). "Counting arithmetic lattices and surfaces". Annals of Mathematics. 172 (3): 2197–2221. arXiv:0811.2482. doi:10.4007/annals.2010.172.2197. ISSN 0003-486X. S2CID 14172846.
• Bader, U.; Gelander, T.; Monod, N. (27 October 2011). "A fixed point theorem for L 1 spaces". Inventiones Mathematicae. Springer Science and Business Media LLC. 189 (1): 143–148. arXiv:1012.1488. doi:10.1007/s00222-011-0363-2. ISSN 0020-9910. S2CID 55594695.
• Abert, Miklos; Bergeron, Nicolas; Biringer, Ian; Gelander, Tsachik; Nikolav, Nikolay; Raimbault, Jean; Samet, Iddo (1 May 2017). "On the growth of $L^2$-invariants for sequences of lattices in Lie groups" (PDF). Annals of Mathematics. 185 (3). doi:10.4007/annals.2017.185.3.1. ISSN 0003-486X. S2CID 106398777.
• Gelander, Tsachik (2019). "A VIEW ON INVARIANT RANDOM SUBGROUPS AND LATTICES". Proceedings of the International Congress of Mathematicians (ICM 2018). WORLD SCIENTIFIC. pp. 1321–1344. arXiv:1807.06979. doi:10.1142/9789813272880_0099. ISBN 978-981-327-287-3.
• Fraczyk, Mikolaj; Gelander, Tsachik (January 2023). "Infinite volume and infinite injectivity radius". Annals of Mathematics. 197 (1): 389–421. arXiv:2101.00640.
References
1. https://www.math.northwestern.edu/people/faculty/tsachik-gelander.html
2. Tsachik Gelander at the Mathematics Genealogy Project
3. "The Haim Nessyahu Prize in Mathematics", MacTutor History of Mathematics Archive, retrieved 20 July 2020
4. https://www.math.northwestern.edu/people/faculty/tsachik-gelander.html
5. Guérin, Nicolas (22 November 2011), Mathematics: Mapping a fixed point EPFL
6. Asymptotic invariants of locally symmetric spaces – Tsachik Gelander – ICM2018
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Tsallis statistics
The term Tsallis statistics usually refers to the collection of mathematical functions and associated probability distributions that were originated by Constantino Tsallis. Using that collection, it is possible to derive Tsallis distributions from the optimization of the Tsallis entropic form. A continuous real parameter q can be used to adjust the distributions, so that distributions which have properties intermediate to that of Gaussian and Lévy distributions can be created. The parameter q represents the degree of non-extensivity of the distribution. Tsallis statistics are useful for characterising complex, anomalous diffusion.
Tsallis functions
The q-deformed exponential and logarithmic functions were first introduced in Tsallis statistics in 1994.[1] However, the q-deformation is the Box–Cox transformation for $q=1-\lambda $, proposed by George Box and David Cox in 1964.[2]
q-exponential
The q-exponential is a deformation of the exponential function using the real parameter q.[3]
$e_{q}(x)={\begin{cases}\exp(x)&{\text{if }}q=1,\\[6pt][1+(1-q)x]^{1/(1-q)}&{\text{if }}q\neq 1{\text{ and }}1+(1-q)x>0,\\[6pt]0^{1/(1-q)}&{\text{if }}q\neq 1{\text{ and }}1+(1-q)x\leq 0,\\[6pt]\end{cases}}$
Note that the q-exponential in Tsallis statistics is different from a version used elsewhere.
q-logarithm
The q-logarithm is the inverse of q-exponential and a deformation of the logarithm using the real parameter q.[3]
$\ln _{q}(x)={\begin{cases}\ln(x)&{\text{if }}x>0{\text{ and }}q=1\\[8pt]{\dfrac {x^{1-q}-1}{1-q}}&{\text{if }}x>0{\text{ and }}q\neq 1\\[8pt]{\text{Undefined }}&{\text{if }}x\leq 0\\[8pt]\end{cases}}$
Inverses
These functions have the property that
${\begin{cases}e_{q}(\ln _{q}(x))=x&(x>0)\\\ln _{q}(e_{q}(x))=x&(0<e_{q}(x)<\infty )\\\end{cases}}$
Analysis
The $q\to 1$ limits of the above expression can be understood by considering $\left(1+{\frac {x}{N}}\right)^{N}\approx {\rm {e}}^{x}$ for the exponential function and $N\left(x^{\frac {1}{N}}-1\right)\approx \log(x)$ for the logarithm.
See also
• Tsallis entropy
• Tsallis distribution
• q-Gaussian
• q-exponential distribution
• q-Weibull distribution
References
1. Tsallis, Constantino (1994). "What are the numbers that experiments provide?". Química Nova. 17: 468.
2. Box, George E. P.; Cox, D. R. (1964). "An analysis of transformations". Journal of the Royal Statistical Society, Series B. 26 (2): 211–252. JSTOR 2984418. MR 0192611.
3. Umarov, Sabir; Tsallis, Constantino; Steinberg, Stanly (2008). "On a q-Central Limit Theorem Consistent with Nonextensive Statistical Mechanics" (PDF). Milan J. Math. Birkhauser Verlag. 76: 307–328. doi:10.1007/s00032-008-0087-y. S2CID 55967725. Retrieved 2011-07-27.
• S. Abe, A.K. Rajagopal (2003). Letters, Science (11 April 2003), Vol. 300, issue 5617, 249–251. doi:10.1126/science.300.5617.249d
• S. Abe, Y. Okamoto, Eds. (2001) Nonextensive Statistical Mechanics and its Applications. Springer-Verlag. ISBN 978-3-540-41208-3
• G. Kaniadakis, M. Lissia, A. Rapisarda, Eds. (2002) "Special Issue on Nonextensive Thermodynamics and Physical Applications." Physica A 305, 1/2.
External links
• Tsallis statistics on arxiv.org
Tsallis statistics
• Constantino Tsallis
• Tsallis entropy
• Tsallis statistics
• Tsallis distribution
• q-Gaussian distribution
• q-exponential distribution
• q-Weibull distribution
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Tschirnhaus transformation
In mathematics, a Tschirnhaus transformation, also known as Tschirnhausen transformation, is a type of mapping on polynomials developed by Ehrenfried Walther von Tschirnhaus in 1683.[1]
Simply, it is a method for transforming a polynomial equation of degree $n\geq 2$ with some nonzero intermediate coefficients, $a_{1},...,a_{n-1}$, such that some or all of the transformed intermediate coefficients, $a'_{1},...,a'_{n-1}$, are exactly zero.
For example, finding a substitution
$y(x)=k_{1}x^{2}+k_{2}x+k_{3}$
for a cubic equation of degree $n=3$,
$f(x)=x^{3}+a_{2}x^{2}+a_{1}x+a_{0}$
such that substituting $x=x(y)$ yields a new equation
$f'(y)=y^{3}+a'_{2}y^{2}+a'_{1}y+a'_{0}$
such that $a'_{1}=0$, $a'_{2}=0$, or both.
More generally, it may be defined conveniently by means of field theory, as the transformation on minimal polynomials implied by a different choice of primitive element. This is the most general transformation of an irreducible polynomial that takes a root to some rational function applied to that root.
Definition
For a generic $n^{th}$ degree reducible monic polynomial equation $f(x)=0$ of the form $f(x)=g(x)/h(x)$, where $g(x)$ and $h(x)$ are polynomials and $h(x)$ does not vanish at $f(x)=0$,
$f(x)=x^{n}+a_{1}x^{n-1}+a_{2}x^{n-2}+...+a_{n-1}x+a_{n}=0$
the Tschirnhaus transformation is the function:
$y=k_{1}x^{n-1}+k_{2}x^{n-2}+...+k_{n-1}x+k_{n}$
Such that the new equation in $y$, $f'(y)$, has certain special properties, most commonly such that some coefficients, $a'_{1},...,a'_{n-1}$, are identically zero.[2][3]
Example: Tschirnhaus' method for cubic equations
In Tschirnhaus' 1683 paper,[1] he solved the equation
$f(x)=x^{3}-px^{2}+qx-r=0$
using the Tschirnhaus transformation
$y(x;a)=x-a\longleftrightarrow x(y;a)=x=y+a.$
Substituting yields the transformed equation
$f'(y;a)=y^{3}+(3a-p)y^{2}+(3a^{2}-2pa+q)y+(a^{3}-pa^{2}+qa-r)=0$
or
${\begin{cases}a'_{1}=3a-p\\a'_{2}=3a^{2}-2pa+q\\a'_{3}=a^{3}-pa^{2}+qa-r\end{cases}}.$
Setting $a'_{1}=0$ yields,
$3a-p=0\rightarrow a={\frac {p}{3}}$
and finally the Tschirnhaus transformation
$y=x+{\frac {p}{3}},$
Which may be substituted into $f'(y;a)$ to yield an equation of the form:
$f'(y)=y^{3}-q'y-r'.$
Tschirnhaus went on to describe how a Tschirnhaus transformation of the form:
$x^{2}(y;a,b)=x^{2}=bx+y+a$
may be used to eliminate two coefficients in a similar way.
Generalization
In detail, let $K$ be a field, and $P(t)$ a polynomial over $K$. If $P$ is irreducible, then the quotient ring of the polynomial ring $K[t]$ by the principal ideal generated by $P$,
$K[t]/(P(t))=L$,
is a field extension of $K$. We have
$L=K(\alpha )$
where $\alpha $ is $t$ modulo $(P)$. That is, any element of $L$ is a polynomial in $\alpha $, which is thus a primitive element of $L$. There will be other choices $\beta $ of primitive element in $L$: for any such choice of $\beta $ we will have by definition:
$\beta =F(\alpha ),\alpha =G(\beta )$,
with polynomials $F$ and $G$ over $K$. Now if $Q$ is the minimal polynomial for $\beta $ over $K$, we can call $Q$ a Tschirnhaus transformation of $P$.
Therefore the set of all Tschirnhaus transformations of an irreducible polynomial is to be described as running over all ways of changing $P$, but leaving $L$ the same. This concept is used in reducing quintics to Bring–Jerrard form, for example. There is a connection with Galois theory, when $L$ is a Galois extension of $K$. The Galois group may then be considered as all the Tschirnhaus transformations of $P$ to itself.
History
In 1683, Ehrenfried Walther von Tschirnhaus published a method for rewriting a polynomial of degree $n>2$ such that the $x^{n-1}$ and $x^{n-2}$ terms have zero coefficients. In his paper, Tschirnhaus referenced a method by Descartes to reduce a quadratic polynomial $(n=2)$ such that the $x$ term has zero coefficient.
In 1786, this work was expanded by E. S. Bring who showed that any generic quintic polynomial could be similarly reduced.
In 1834, G. B. Jerrard further expanded Tschirnhaus' work by showing a Tschirnhaus transformation may be used to eliminate the $x^{n-1}$, $x^{n-2}$, and $x^{n-3}$ for a general polynomial of degree $n>3$.[3]
See also
• Polynomial transformations
• Monic polynomial
• Reducible polynomial
• Quintic function
• Galois theory
• Abel-Ruffini theorem
References
1. von Tschirnhaus, Ehrenfried Walter; Green, R. F. (2003-03-01). "A method for removing all intermediate terms from a given equation". ACM SIGSAM Bulletin. 37 (1): 1–3. doi:10.1145/844076.844078. ISSN 0163-5824. S2CID 18911887.
2. Garver, Raymond (1927). "The Tschirnhaus Transformation". Annals of Mathematics. 29 (1/4): 319–333. doi:10.2307/1968002. ISSN 0003-486X. JSTOR 1968002.
3. Weisstein, Eric W. "Tschirnhausen Transformation". mathworld.wolfram.com. Retrieved 2022-02-02.
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Tsen's theorem
In mathematics, Tsen's theorem states that a function field K of an algebraic curve over an algebraically closed field is quasi-algebraically closed (i.e., C1). This implies that the Brauer group of any such field vanishes,[1] and more generally that all the Galois cohomology groups H i(K, K*) vanish for i ≥ 1. This result is used to calculate the étale cohomology groups of an algebraic curve.
The theorem was published by Chiungtze C. Tsen in 1933.
See also
• Tsen rank
References
1. Lorenz, Falko (2008). Algebra. Volume II: Fields with Structure, Algebras and Advanced Topics. Springer. p. 181. ISBN 978-0-387-72487-4. Zbl 1130.12001.
• Ding, Shisun; Kang, Ming-Chang; Tan, Eng-Tjioe (1999), "Chiungtze C. Tsen (1898–1940) and Tsen's theorems", Rocky Mountain Journal of Mathematics, 29 (4): 1237–1269, doi:10.1216/rmjm/1181070405, ISSN 0035-7596, MR 1743370, Zbl 0955.01031
• Lang, Serge (1952), "On quasi algebraic closure", Annals of Mathematics, Second Series, 55: 373–390, doi:10.2307/1969785, ISSN 0003-486X, JSTOR 1969785, Zbl 0046.26202
• Serre, J. P. (2002), Galois Cohomology, Springer Monographs in Mathematics, Translated from the French by Patrick Ion, Berlin: Springer-Verlag, ISBN 3-540-42192-0, Zbl 1004.12003
• Tsen, Chiungtze C. (1933), "Divisionsalgebren über Funktionenkörpern", Nachr. Ges. Wiss. Göttingen, Math.-Phys. Kl. (in German): 335–339, JFM 59.0160.01, Zbl 0007.29401
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Wikipedia
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Tsen rank
In mathematics, the Tsen rank of a field describes conditions under which a system of polynomial equations must have a solution in the field. The concept is named for C. C. Tsen, who introduced their study in 1936.
We consider a system of m polynomial equations in n variables over a field F. Assume that the equations all have constant term zero, so that (0, 0, ... ,0) is a common solution. We say that F is a Ti-field if every such system, of degrees d1, ..., dm has a common non-zero solution whenever
$n>d_{1}^{i}+\cdots +d_{m}^{i}.\,$
The Tsen rank of F is the smallest i such that F is a Ti-field. We say that the Tsen rank of F is infinite if it is not a Ti-field for any i (for example, if it is formally real).
Properties
• A field has Tsen rank zero if and only if it is algebraically closed.
• A finite field has Tsen rank 1: this is the Chevalley–Warning theorem.
• If F is algebraically closed then rational function field F(X) has Tsen rank 1.
• If F has Tsen rank i, then the rational function field F(X) has Tsen rank at most i + 1.
• If F has Tsen rank i, then an algebraic extension of F has Tsen rank at most i.
• If F has Tsen rank i, then an extension of F of transcendence degree k has Tsen rank at most i + k.
• There exist fields of Tsen rank i for every integer i ≥ 0.
Norm form
We define a norm form of level i on a field F to be a homogeneous polynomial of degree d in n=di variables with only the trivial zero over F (we exclude the case n=d=1). The existence of a norm form on level i on F implies that F is of Tsen rank at least i − 1. If E is an extension of F of finite degree n > 1, then the field norm form for E/F is a norm form of level 1. If F admits a norm form of level i then the rational function field F(X) admits a norm form of level i + 1. This allows us to demonstrate the existence of fields of any given Tsen rank.
Diophantine dimension
The Diophantine dimension of a field is the smallest natural number k, if it exists, such that the field of is class Ck: that is, such that any homogeneous polynomial of degree d in N variables has a non-trivial zero whenever N > dk. Algebraically closed fields are of Diophantine dimension 0; quasi-algebraically closed fields of dimension 1.[1]
Clearly if a field is Ti then it is Ci, and T0 and C0 are equivalent, each being equivalent to being algebraically closed. It is not known whether Tsen rank and Diophantine dimension are equal in general.
See also
• Tsen's theorem
References
1. Neukirch, Jürgen; Schmidt, Alexander; Wingberg, Kay (2008). Cohomology of Number Fields. Grundlehren der Mathematischen Wissenschaften. Vol. 323 (2nd ed.). Springer-Verlag. p. 361. ISBN 978-3-540-37888-4.
• Tsen, C. (1936). "Zur Stufentheorie der Quasi-algebraisch-Abgeschlossenheit kommutativer Körper". J. Chinese Math. Soc. 171: 81–92. Zbl 0015.38803.
• Lorenz, Falko (2008). Algebra. Volume II: Fields with Structure, Algebras and Advanced Topics. Springer. ISBN 978-0-387-72487-4.
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Wikipedia
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Tsetlin machine
A Tsetlin machine is an Artificial Intelligence algorithm based on propositional logic.
Part of a series on
Machine learning
and data mining
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• Supervised learning
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• Classification
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• density estimation
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(classification • regression)
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Related articles
• Glossary of artificial intelligence
• List of datasets for machine-learning research
• Outline of machine learning
Background
A Tsetlin machine is a form of learning automaton based upon algorithms from reinforcement learning to learn expressions from propositional logic. Ole-Christoffer Granmo gave the method its name after Michael Lvovitch Tsetlin and his Tsetlin automata. The method uses computationally simpler and more efficient primitives compared to more ordinary artificial neural networks.[1]
As of April 2018 it has shown promising results on a number of test sets.[2][3]
Types
• Original Tsetlin machine[1]
• Convolutional Tsetlin machine[4]
• Regression Tsetlin machine[5]
• Relational Tsetlin machine[6]
• Weighted Tsetlin machine[7][8]
• Arbitrarily deterministic Tsetlin machine[9]
• Parallel asynchronous Tsetlin machine[10]
• Coalesced multi-output Tsetlin machine[11]
• Tsetlin machine for contextual bandit problems[12]
• Tsetlin machine autoencoder[13]
Applications
• Keyword spotting[14]
• Aspect-based sentiment analysis[15]
• Word-sense disambiguation[16]
• Novelty detection[17]
• Intrusion detection[18]
• Semantic relation analysis[19]
• Image analysis[4]
• Text categorization[20]
• Fake news detection[21]
• Game playing[22]
• Batteryless sensing[23]
• Recommendation systems[24]
• Word embedding[13]
• ECG analysis[25]>
• Edge computing[26]
• Bayesian network learning[27]
Original Tsetlin machine
List of hyperparameters[28]
Description Symbol
Number of binary inputs $N_{\text{Inputs}}$
Number of classes $N_{\text{Classes}}$
Number of clauses per class $N_{\text{Clauses}}$
Number of automaton states $2n$
Automaton decision boundary n
Automaton initialization state $\varnothing _{\text{Init}}$
Feedback threshold T
Learning sensitivity s
Tsetlin automaton
The Tsetlin automaton is the fundamental learning unit of the Tsetlin machine. It tackles the multi-armed bandit problem, learning the optimal action in an environment from penalties and rewards. Computationally, it can be seen as a finite-state machine (FSM) that changes its states based on the inputs. The FSM will generate its outputs based on the current states.
• A quintuple describes a two-action Tsetlin automaton:
$\{{\underline {\Phi }},{\underline {\alpha }},{\underline {\beta }},F(\cdot ,\cdot ),G(\cdot )\}.$
• A Tsetlin automaton has $2n$ states, here 6:
${\underline {\Phi }}=\{\phi _{1},\phi _{2},\phi _{3},\phi _{4},\phi _{5},\phi _{6}\}$
• The FSM can be triggered by two input events
${\underline {\beta }}=\{\beta _{\mathrm {Penalty} },\beta _{\mathrm {Reward} }\}$
• The rules of state migration of the FSM are stated as
$F(\phi _{u},\beta _{v})={\begin{cases}\phi _{u+1},&{\text{if}}~1\leq u\leq 3~{\text{and}}~v={\text{Penalty}}\\\phi _{u-1},&{\text{if}}~4\leq u\leq 6~{\text{and}}~v={\text{Penalty}}\\\phi _{u-1},&{\text{if}}~1<u\leq 3~{\text{and}}~v={\text{Reward}}\\\phi _{u+1},&{\text{if}}~4\leq u<6~{\text{and}}~v={\text{Reward}}\\\phi _{u},&{\text{otherwise}}.\end{cases}}$
• It includes two output actions
${\underline {\alpha }}=\{\alpha _{1},\alpha _{2}\}$
• Which can be generated by the algorithm
$G(\phi _{u})={\begin{cases}\alpha _{1},&{\text{if}}~1\leq u\leq 3\\\alpha _{2},&{\text{if}}~4\leq u\leq 6.\end{cases}}$
Boolean input
A basic Tsetlin machine takes a vector $X=[x_{1},\ldots ,x_{o}]$ of o Boolean features as input, to be classified into one of two classes, $y=0$ or $y=1$. Together with their negated counterparts, ${\bar {x}}_{k}={\lnot }{x}_{k}=1-x_{k}$, the features form a literal set $L=\{x_{1},\ldots ,x_{o},{\bar {x}}_{1},\ldots ,{\bar {x}}_{o}\}$.
Clause computing module
A Tsetlin machine pattern is formulated as a conjunctive clause $C_{j}$, formed by ANDing a subset $L_{j}{\subseteq }L$ of the literal set:
$C_{j}(X)=\bigwedge _{{l}{\in }L_{j}}l=\prod _{{l}{\in }L_{j}}l$.
For example, the clause $C_{j}(X)=x_{1}\land {\lnot }x_{2}=x_{1}{\bar {x}}_{2}$ consists of the literals $L_{j}=\{x_{1},{\bar {x}}_{2}\}$ and outputs 1 iff $x_{1}=1$ and $x_{2}=0$.
Summation and thresholding module
The number of clauses employed is a user-configurable parameter n. Half of the clauses are assigned positive polarity. The other half is assigned negative polarity. The clause outputs, in turn, are combined into a classification decision through summation and thresholding using the unit step function $u(v)=1~{\text{if}}~v\geq 0~{\text{else}}~0$:
${\hat {y}}=u\left(\sum _{j=1}^{n/2}C_{j}^{+}(X)-\sum _{j=1}^{n/2}C_{j}^{-}(X)\right).$
In other words, classification is based on a majority vote, with the positive clauses voting for $y=1$ and the negative for $y=0$. The classifier
${\hat {y}}=u\left(x_{1}{\bar {x}}_{2}+{\bar {x}}_{1}x_{2}-x_{1}x_{2}-{\bar {x}}_{1}{\bar {x}}_{2}\right)$,
for instance, captures the XOR-relation.
Type I feedback
Type I Feedback
Action Clause 1 0
Literal 1 0 1 0
Include literal P(reward) ${\frac {s-1}{s}}$ — 0 0
P(inaction) ${\frac {1}{s}}$ — ${\frac {s-1}{s}}$ ${\frac {s-1}{s}}$
P(penalty) 0 — ${\frac {1}{s}}$ ${\frac {1}{s}}$
Exclude literal P(reward) 0 ${\frac {1}{s}}$ ${\frac {1}{s}}$ ${\frac {1}{s}}$
P(inaction) ${\frac {1}{s}}$ ${\frac {s-1}{s}}$ ${\frac {s-1}{s}}$ ${\frac {s-1}{s}}$
P(penalty) ${\frac {s-1}{s}}$ 0 0 0
Type II feedback
Type II Feedback
Action Clause 1 0
Literal 1 0 1 0
Include literal P(reward) 0 — 0 0
P(inaction) 1.0 — 1.0 1.0
P(penalty) 0 — 0 0
Exclude literal P(reward) 0 0 0 0
P(inaction) 1.0 0 1.0 1.0
P(penalty) 0 1.0 0 0
Resource allocation
Resource allocation dynamics ensure that clauses distribute themselves across the frequent patterns, rather than missing some and overconcentrating on others. That is, for any input X, the probability of reinforcing a clause gradually drops to zero as the clause output sum
$v=\sum _{j=1}^{n/2}C_{j}^{+}(X)-\sum _{j=1}^{n/2}C_{j}^{-}(X)$
approaches a user-set target T for $y=1$ ($-T$ for $y=0$).
If a clause is not reinforced, it does not give feedback to its Tsetlin automata, and these are thus left unchanged. In the extreme, when the voting sum v equals or exceeds the target T (the Tsetlin Machine has successfully recognized the input X), no clauses are reinforced. Accordingly, they are free to learn new patterns, naturally balancing the pattern representation resources.
Implementations
Software
• Tsetlin Machine in C,[29][30] Python,[31][32] multithreaded Python,[33] CUDA[34]
• Convolutional Tsetlin Machine [35][4]
• Weighted Tsetlin Machine in C++[36]
Hardware
• One of the first FPGA-based hardware implementation[37][38] of the Tsetlin Machine on the Iris dataset was developed by the µSystems (microSystems) Research Group at Newcastle University.
• They also presented the first ASIC[39][40] implementation of the Tsetlin Machine focusing on energy frugality, claiming it could deliver 10 trillion operation per Joule.[41] The ASIC design had demoed on DATA2020.[42]
Additional Read
Books
• An Introduction to Tsetlin Machines [43]
Conferences
• International Symposium on the Tsetlin Machine (ISTM) [44][45]
Videos
• Tsetlin Machine—A new paradigm for pervasive AI[42]
• Keyword Spotting Using Tsetlin Machines [46]
• IOLTS Presentation: Explainability and Dependability Analysis of Learning Automata based AI hardware [47]
• FPGA and uC co-design: Tsetlin Machine on Iris demo [37][38]
• The-Ruler-of-Tsetlin-Automaton [48]
• Interpretable clustering and dimension reduction with Tsetlin automata machine learning.[49]
• Predicting and explaining economic growth using real-time interpretable learning [50]
• Early detection of breast cancer from a simple blood test[51]
• Recent advances in Tsetlin Machines [52]
Papers
• On the Convergence of Tsetlin Machines for the XOR Operator [53]
• Learning Automata based Energy-efficient AI Hardware Design for IoT Applications [28]
• On the Convergence of Tsetlin Machines for the IDENTITY- and NOT Operators [54]
• The Tsetlin Machine - A Game Theoretic Bandit Driven Approach to Optimal Pattern Recognition with Propositional Logic [1]
Publications/news/articles
• A low-power AI alternative to neural networks [41]
References
1. Granmo, Ole-Christoffer (2018-04-04). "The Tsetlin Machine - A Game Theoretic Bandit Driven Approach to Optimal Pattern Recognition with Propositional Logic". arXiv:1804.01508 [cs.AI].
2. Christiansen, Atle. "The Tsetlin Machine outperforms neural networks - Center for Artificial Intelligence Research". cair.uia.no. Retrieved 2018-05-03.
3. Øyvann, Stig (23 March 2018). "AI-gjennombrudd i Agder | Computerworld". Computerworld (in Norwegian). Retrieved 2018-05-04.
4. Granmo, Ole-Christoffer; Glimsdal, Sondre; Jiao, Lei; Goodwin, Morten; Omlin, Christian W.; Berge, Geir Thore (2019-12-27). "The Convolutional Tsetlin Machine". arXiv:1905.09688 [cs.LG].
5. Abeyrathna, K. Darshana; Granmo, Ole-Christoffer; Zhang, Xuan; Jiao, Lei; Goodwin, Morten (2020). "The regression Tsetlin machine: a novel approach to interpretable nonlinear regression". Philosophical Transactions of the Royal Society A. 378 (2164). Bibcode:2020RSPTA.37890165D. doi:10.1098/rsta.2019.0165. hdl:11250/2651754. PMID 31865880. S2CID 209439954."
6. Saha, Rupsa; Granmo, Ole-Christoffer; Zadorozhny, Vladimir; Goodwin, Morten (2022). "A relational Tsetlin machine with applications to natural language understanding". Journal of Intelligent Information Systems. Springer. 59: 121–148. doi:10.1007/s10844-021-00682-5. S2CID 231986401.
7. Phoulady, Adrian; Granmo, Ole-Christoffer; Gorji, Saeed Rahimi; Phoulady, Hady Ahmady (2019-11-28). "The Weighted Tsetlin Machine: Compressed Representations with Weighted Clauses". arXiv:1911.12607 [cs.LG].
8. Abeyrathna, K. Darshana; Granmo, Ole-Christoffer; Goodwin, Morten (2021). "Extending the Tsetlin Machine With Integer-Weighted Clauses for Increased Interpretability". IEEE Access. 9: 8233–8248. doi:10.1109/ACCESS.2021.3049569. S2CID 218581474."
9. Abeyrathna, K. Darshana; Granmo, Ole-Christoffer; Shafik, Rishad; Yakovlev, Alex; Wheeldon, Adrian; Lei, Jie; Goodwin, Morten (2021). "A multi-step finite-state automaton for arbitrarily deterministic Tsetlin Machine learning". Expert Systems. Wiley: exsy.12836. doi:10.1111/exsy.12836. S2CID 242770808.
10. Abeyrathna, K. Darshana; Bhattarai, Bimal; Goodwin, Morten; Gorji, Saeed; Granmo, Ole-Christoffer; Jiao, Lei; Saha, Rupsa; Yadav, Rohan K. (2021). Massively Parallel and Asynchronous Tsetlin Machine Architecture Supporting Almost Constant-Time Scaling (PDF). Thirty-eighth International Conference on Machine Learning (ICML 2021).
11. Glimsdal, Sondre; Granmo, Ole-Christoffer (2021-08-17). "Coalesced Multi-Output Tsetlin Machines with Clause Sharing". arXiv:2108.07594 [cs.AI].
12. Seraj, Raihan; Sharma, Jivitesh; Granmo, Ole-Christoffer (2022). Tsetlin Machine for Solving Contextual Bandit Problems. Thirty-sixth Conference on Neural Information Processing Systems (NeurIPS 2022).
13. Bhattarai, Bimal; Granmo, Ole-Christoffer; Jiao, Lei; Yadav, Rohan; Sharma, Jivitesh (2023-01-03). "Tsetlin Machine Embedding: Representing Words Using Logical Expressions". arXiv:2301.00709 [cs.CL].
14. Lei, Jie; Shafik, Rishad; Wheeldon, Adrian; Yakovlev, Alex; Granmo, Ole-Christoffer; Kawsar, Fahim; Akhil, Mathur (2021-04-09). "Low-Power Audio Keyword Spotting using Tsetlin Machines". Journal of Low Power Electronics and Applications. 11 (2): 18. doi:10.3390/jlpea11020018.
15. Yadav, Rohan Kumar; Jiao, Lei; Granmo, Ole-Christoffer; Goodwin, Morten (2021). Human-Level Interpretable Learning for Aspect-Based Sentiment Analysis. The Thirty-Fifth AAAI Conference on Artificial Intelligence (AAAI-21). AAAI.
16. Yadav, Rohan Kumar; Jiao, Lei; Granmo, Ole-Christoffer; Goodwin, Morten (2021). Interpretability in Word Sense Disambiguation using Tsetlin Machine. 13th International Conference on Agents and Artificial Intelligence (ICAART 2021). INSTICC.
17. Bhattarai, Bimal; Granmo, Ole-Christoffer; Jiao, Lei (2022). "Word-level human interpretable scoring mechanism for novel text detection using Tsetlin Machines". Applied Intelligence. Springer. 52 (15): 17465–17489. doi:10.1007/s10489-022-03281-1.
18. Abeyrathna, K. Darshana; Pussewalage, Harsha S. Gardiyawasam; Ranasinghea, Sasanka N.; Oleshchuk, Vladimir A.; Granmo, Ole-Christoffer (2020). Intrusion Detection with Interpretable Rules Generated Using the Tsetlin Machine. 2020 IEEE Symposium Series on Computational Intelligence (SSCI). IEEE.
19. Saha, Rupsa; Granmo, Ole-Christoffer; Goodwin, Morten (2021). "Using Tsetlin Machine to discover interpretable rules in natural language processing applications". Expert Systems. Wiley. doi:10.1111/exsy.12873. S2CID 244096520.
20. Berge, Geir Thore; Granmo, Ole-Christoffer; Tveit, Tor O.; Goodwin, Morten; Jiao, Lei; Matheussen, Bernt Viggo (2019). "Using the Tsetlin Machine to Learn Human-Interpretable Rules for High-Accuracy Text Categorization with Medical Applications". IEEE Access. 7: 115134–115146. doi:10.1109/ACCESS.2019.2935416. S2CID 52195410."
21. Bhattarai, Bimal; Granmo, Ole-Christoffer; Jiao, Lei (2022). Explainable Tsetlin Machine framework for fake news detection with credibility score assessment (PDF). 13th Conference on Language Resources and Evaluation (LREC 2022).
22. Giri, Charul; Granmo, Ole-Christoffer; Hoof, Herke van; Blakely, Christian D. (2022-03-10). Logic-based AI for Interpretable Board Game Winner Prediction with Tsetlin Machine. The 2022 International Joint Conference on Neural Networks (IJCNN 2022). arXiv:2203.04378.
23. Bakar, Abu; Rahman, Tousif; Shafik, Rishad; Kawsar, Fahim; Montanari, Alessandro (2023-01-24). Adaptive Intelligence for Batteryless Sensors Using Software-Accelerated Tsetlin Machines. ACM SenSys 2022. pp. 236–249. doi:10.1145/3560905.3568512.
24. Borgersen, Karl Audun; Goodwin, Morten; Sharma, Jivitesh (2023). "A comparison between Tsetlin machines and deep neural networks in the context of recommendation systems". Proceedings of the Northern Lights Deep Learning Workshop. 4. arXiv:2212.10136. doi:10.7557/18.6807. S2CID 254877078.
25. Zhang, Jinbao; Zhang, Xuan; Jiao, Lei; Granmo, Ole-Christoffer; Qian, Yongjun; Pan, Fan (2023-01-25). "Interpretable Tsetlin Machine-based Premature Ventricular Contraction Identification". arXiv:2301.10181 [eess.SP)].
26. Maheshwari, Sidharth; Rahman, Tousif; Shafik, Rishad; Yakovlev, Alex; Rafiev, Ashur; Jiao, Lei; Granmo, Ole-Christoffer (2023). "REDRESS: Generating Compressed Models for Edge Inference Using Tsetlin Machines". IEEE Transactions on Pattern Analysis and Machine Intelligence. doi:10.1109/TPAMI.2023.3268415.
27. Blakely, Christian D. (2023-05-17). "Generating Bayesian Network Models from Data Using Tsetlin Machines". arXiv:2305.10538 [cs.AI)].
28. Wheeldon, A.; Shafik, R.; Rahman, T.; Lei, J.; Yakovlev, A.; Granmo, O. C. (2020). "Learning Automata based Energy-efficient AI Hardware Design for IoT Applications". Philosophical Transactions of the Royal Society A. 378 (2182). Bibcode:2020RSPTA.37890593W. doi:10.1098/rsta.2019.0593. PMC 7536019. PMID 32921236.
29. cair/TsetlinMachineC, Centre for Artificial Intelligence Research (CAIR), 2019-04-18, retrieved 2020-07-27
30. cair/FastTsetlinMachineC, Centre for Artificial Intelligence Research (CAIR), 2019-02-15, retrieved 2021-02-15
31. cair/pyTsetlinMachine, Centre for Artificial Intelligence Research (CAIR), 2020-07-07, retrieved 2020-07-27
32. cair/TsetlinMachine, Centre for Artificial Intelligence Research (CAIR), 2020-07-27, retrieved 2020-07-27
33. cair/pyTsetlinMachineParallel, Centre for Artificial Intelligence Research (CAIR), 2020-07-07, retrieved 2020-07-27
34. cair/PyTsetlinMachineCUDA, Centre for Artificial Intelligence Research (CAIR), 2020-07-27, retrieved 2020-07-27
35. "cair/convolutional-tsetlin-machine-tutorial". GitHub. Retrieved 2020-07-27.
36. Phoulady, Adrian (2020-04-13), adrianphoulady/weighted-tsetlin-machine-cpp, retrieved 2020-07-27
37. JieGH (2020-03-22), JieGH/Hardware_TM_Demo, retrieved 2020-07-22
38. JieGH. "Tsetlin Machine on Iris Data Set Demo, Handheld #MignonAI". Youtube.
39. "Logic-based AI Everywhere: Tsetlin Machines in Hardware". Twitter. Retrieved 2020-07-27.
40. "mignon". www.mignon.ai. Retrieved 2020-07-27.
41. Bush, Steve (2020-07-27). "A low-power AI alternative to neural networks". Electronics Weekly. Retrieved 2020-07-27.
42. "Tsetlin Machine -- A new paradigm for pervasive AI". YouTube.
43. Granmo, Ole-Christoffer (2021). An Introduction to Tsetlin Machines.
44. "International Symposium on the Tsetlin Machine (ISTM)".
45. "Proceedings of the 2022 International Symposium on the Tsetlin Machine (ISTM)".
46. "Keyword Spotting Using Tsetlin Machines". YouTube.
47. "IOLTS Presentation: Explainability and Dependability Analysis of Learning Automata based AI hardware". YouTube.
48. "The-Ruler-of-Tsetlin-Automaton". YouTube.
49. "Interpretable Clustering & Dimension Reduction with Tsetlin Automata machine learning". YouTube.
50. "Predicting and explaining economic growth using real-time interpretable learning". YouTube.
51. "Early detection of breast cancer from a simple blood test". YouTube.
52. "Recent advances in Tsetlin Machines". YouTube.
53. Jiao, Lei; Zhang, Xuan; Granmo, Ole-Christoffer; Abeyrathna, K. Darshana (2022). "On the Convergence of Tsetlin Machines for the XOR Operator". IEEE Transactions on Pattern Analysis and Machine Intelligence. PP: 1. arXiv:2101.02547. doi:10.1109/TPAMI.2022.3203150. PMID 36070276. S2CID 230799244.
54. Zhang, Xuan; Jiao, Lei; Granmo, Ole-Christoffer; Goodwin, Morten (2021). "On the Convergence of Tsetlin Machines for the IDENTITY- and NOT Operators". IEEE Transactions on Pattern Analysis and Machine Intelligence. PP (10): 6345–6359. arXiv:2007.14268. doi:10.1109/TPAMI.2021.3085591. PMID 34077353. S2CID 220831619.
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Wikipedia
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Qin Jiushao
Qin Jiushao (Chinese: 秦九韶; pinyin: Qín Jiǔsháo; Wade–Giles: Ch'in Chiu-shao, ca. 1202–1261), courtesy name Daogu (道古), was a Chinese mathematician, meteorologist, inventor, politician, and writer. He is credited for discovering Horner's method as well as inventing Tianchi basins, a type of rain gauge instrument used to gather meteorological data.[1]
Biography
Although Qin Jiushao was born in Ziyang, Sichuan, his family came from Shandong province. He is regarded as one of the greatest mathematicians in Chinese history. This is especially remarkable because Qin did not devote his life to mathematics. He was accomplished in many other fields and held a series of bureaucratic positions in several Chinese provinces.
Qin wrote Shùshū Jiǔzhāng ("Mathematical Treatise in Nine Sections") in 1247 CE. This treatise covered a variety of topics including indeterminate equations and the numerical solution of certain polynomial equations up to 10th order, as well as discussions on military matters and surveying. In the treatise Qin included a general form of the Chinese remainder theorem that used Da yan shu (大衍术) or algorithms to solve it. In geometry, he discovered "Qin Jiushao's formula" for finding the area of a triangle from the given lengths of three sides. This formula is the same as Heron's formula, proved by Heron of Alexandria about 60 BCE, though knowledge of the formula may go back to Archimedes.
As precipitation was important agriculture and food production, Qin developed precipitation gauges that was widely used in 1247 during the Mongol Empire/Southern Song dynasty to gather meteorological data. Qin Jiushao later records the application of rainfall measurements in the mathematical treatise. The book also discusses the use of large snow gauges made from bamboo situated in mountain passes and uplands which are speculated to be first referenced to snow measurement.[2][3]
Qin recorded the earliest explanation of how Chinese calendar experts calculated astronomical data according to the timing of the winter solstice. Among his accomplishments are the introduction techniques for solving certain types of algebraic equations using a numerical algorithm (equivalent to the 19th century Horner's method) and for finding sums of arithmetic series. He also introduced the use of the zero symbol into written Chinese mathematics.
After he completed his work on mathematics, he ventured into politics. As a government official he was boastful, corrupt, and was accused of bribery and of poisoning his enemies. As a result, he was relieved of his duties multiple times. Yet in spite of these problems he managed to become very wealthy (Katz, 1993).
Main work
• Shushu Jiuzhang (Mathematical Treatise in Nine Sections) (1248)
References
1. Selin, Helaine (2008). Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures (2nd ed.). Springer (published April 16, 2008). p. 736. ISBN 978-1402045592.
2. Strangeways, Ian (2011). Precipitation: Theory, Measurement and Distribution. Cambridge University Press (published April 14, 2011). p. 140. ISBN 978-0521172929.
3. Selin, Helaine (2008). Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures (2nd ed.). Springer (published April 16, 2008). p. 736. ISBN 978-1402045592.
Bibliography
• Guo, Shuchun. Encyclopedia of China (Mathematics Volume), 1st ed.
• Qin Jiushao, . (Chinese History Timeline), 2007.
• Ulrich Libbrecht: Chinese Mathematics in the Thirteenth Century (The Shu-Shu-Chiu-Chang of Chin Chiu shao) Dover Publication ISBN 0-486-44619-0
• Victor J. Katz "A history of mathematics: an introduction." New York (1993).
External links
• O'Connor, John J.; Robertson, Edmund F., "Qin Jiushao", MacTutor History of Mathematics Archive, University of St Andrews
• Simon Fraser University biography for "Qin Jiushao"
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Tsirelson space
In mathematics, especially in functional analysis, the Tsirelson space is the first example of a Banach space in which neither an ℓ p space nor a c0 space can be embedded. The Tsirelson space is reflexive.
It was introduced by B. S. Tsirelson in 1974. The same year, Figiel and Johnson published a related article (Figiel & Johnson (1974)) where they used the notation T for the dual of Tsirelson's example. Today, the letter T is the standard notation[1] for the dual of the original example, while the original Tsirelson example is denoted by T*. In T* or in T, no subspace is isomorphic, as Banach space, to an ℓ p space, 1 ≤ p < ∞, or to c0.
All classical Banach spaces known to Banach (1932), spaces of continuous functions, of differentiable functions or of integrable functions, and all the Banach spaces used in functional analysis for the next forty years, contain some ℓ p or c0. Also, new attempts in the early '70s[2] to promote a geometric theory of Banach spaces led to ask [3] whether or not every infinite-dimensional Banach space has a subspace isomorphic to some ℓ p or to c0. Moreover, it was shown by Baudier, Lancien, and Schlumprecht that ℓ p and c0 do not even coarsely embed into T*.
The radically new Tsirelson construction is at the root of several further developments in Banach space theory: the arbitrarily distortable space of Schlumprecht (Schlumprecht (1991)), on which depend Gowers' solution to Banach's hyperplane problem[4] and the Odell–Schlumprecht solution to the distortion problem. Also, several results of Argyros et al.[5] are based on ordinal refinements of the Tsirelson construction, culminating with the solution by Argyros–Haydon of the scalar plus compact problem.[6]
Tsirelson's construction
On the vector space ℓ∞ of bounded scalar sequences x = {xj } j∈N, let Pn denote the linear operator which sets to zero all coordinates xj of x for which j ≤ n.
A finite sequence $\{x_{n}\}_{n=1}^{N}$ of vectors in ℓ∞ is called block-disjoint if there are natural numbers $\textstyle \{a_{n},b_{n}\}_{n=1}^{N}$ so that $a_{1}\leq b_{1}<a_{2}\leq b_{2}<\cdots \leq b_{N}$, and so that $(x_{n})_{i}=0$ when $i<a_{n}$ or $i>b_{n}$, for each n from 1 to N.
The unit ball B∞ of ℓ∞ is compact and metrizable for the topology of pointwise convergence (the product topology). The crucial step in the Tsirelson construction is to let K be the smallest pointwise closed subset of B∞ satisfying the following two properties:[7]
a. For every integer j in N, the unit vector ej and all multiples $\lambda e_{j}$, for |λ| ≤ 1, belong to K.
b. For any integer N ≥ 1, if $\textstyle (x_{1},\ldots ,x_{N})$ is a block-disjoint sequence in K, then $\textstyle {{1 \over 2}P_{N}(x_{1}+\cdots +x_{N})}$ belongs to K.
This set K satisfies the following stability property:
c. Together with every element x of K, the set K contains all vectors y in ℓ∞ such that |y| ≤ |x| (for the pointwise comparison).
It is then shown that K is actually a subset of c0, the Banach subspace of ℓ∞ consisting of scalar sequences tending to zero at infinity. This is done by proving that
d: for every element x in K, there exists an integer n such that 2 Pn(x) belongs to K,
and iterating this fact. Since K is pointwise compact and contained in c0, it is weakly compact in c0. Let V be the closed convex hull of K in c0. It is also a weakly compact set in c0. It is shown that V satisfies b, c and d.
The Tsirelson space T* is the Banach space whose unit ball is V. The unit vector basis is an unconditional basis for T* and T* is reflexive. Therefore, T* does not contain an isomorphic copy of c0. The other ℓ p spaces, 1 ≤ p < ∞, are ruled out by condition b.
Properties
The Tsirelson space T* is reflexive (Tsirel'son (1974)) and finitely universal, which means that for some constant C ≥ 1, the space T* contains C-isomorphic copies of every finite-dimensional normed space, namely, for every finite-dimensional normed space X, there exists a subspace Y of the Tsirelson space with multiplicative Banach–Mazur distance to X less than C. Actually, every finitely universal Banach space contains almost-isometric copies of every finite-dimensional normed space,[8] meaning that C can be replaced by 1 + ε for every ε > 0. Also, every infinite-dimensional subspace of T* is finitely universal. On the other hand, every infinite-dimensional subspace in the dual T of T* contains almost isometric copies of $\scriptstyle {\ell _{n}^{1}}$, the n-dimensional ℓ1-space, for all n.
The Tsirelson space T is distortable, but it is not known whether it is arbitrarily distortable.
The space T* is a minimal Banach space.[9] This means that every infinite-dimensional Banach subspace of T* contains a further subspace isomorphic to T*. Prior to the construction of T*, the only known examples of minimal spaces were ℓ p and c0. The dual space T is not minimal.[10]
The space T* is polynomially reflexive.
Derived spaces
The symmetric Tsirelson space S(T) is polynomially reflexive and it has the approximation property. As with T, it is reflexive and no ℓ p space can be embedded into it.
Since it is symmetric, it can be defined even on an uncountable supporting set, giving an example of non-separable polynomially reflexive Banach space.
See also
• Distortion problem
• Sequence space, Schauder basis
• James' space
Notes
1. see for example Casazza & Shura (1989), p. 8; Lindenstrauss & Tzafriri (1977), p. 95; The Handbook of the Geometry of Banach Spaces, vol. 1, p. 276; vol. 2, p. 1060, 1649.
2. see Lindenstrauss (1970), Milman (1970).
3. The question is formulated explicitly in Lindenstrauss (1970), Milman (1970), Lindenstrauss (1971) on last page. Lindenstrauss & Tzafriri (1977), p. 95, say that this question was "a long standing open problem going back to Banach's book" (Banach (1932)), but the question does not appear in Banach's book. However, Banach compares the linear dimension of ℓ p to that of other classical spaces, a somewhat similar question.
4. The question is whether every infinite-dimensional Banach space is isomorphic to its hyperplanes. The negative solution is in Gowers, "A solution to Banach's hyperplane problem". Bull. London Math. Soc. 26 (1994), 523-530.
5. for example, S. Argyros and V. Felouzis, "Interpolating Hereditarily Indecomposable Banach spaces", Journal Amer. Math. Soc., 13 (2000), 243–294; S. Argyros and A. Tolias, "Methods in the theory of hereditarily indecomposable Banach spaces", Mem. Amer. Math. Soc. 170 (2004), no. 806.
6. S. Argyros and R. Haydon constructed a Banach space on which every bounded operator is a compact perturbation of a scalar multiple of the identity, in "A hereditarily indecomposable L∞-space that solves the scalar-plus-compact problem", Acta Mathematica (2011) 206: 1-54.
7. conditions b, c, d here are conditions (3), (2) and (4) respectively in Tsirel'son (1974), and a is a modified form of condition (1) from the same article.
8. this is because for every n, C and ε, there exists N such that every C-isomorph of ℓ∞N contains a (1 + ε)-isomorph of ℓ∞n, by James' blocking technique (see Lemma 2.2 in Robert C. James "Uniformly Non-Square Banach Spaces", Annals of Mathematics, Vol. 80, 1964, pp. 542-550), and because every finite-dimensional normed space (1 + ε)-embeds in ℓ∞n when n is large enough.
9. see Casazza & Shura (1989), p. 54.
10. see Casazza & Shura (1989), p. 56.
References
• Tsirel'son, B. S. (1974), "'Not every Banach space contains an imbedding of ℓ p or c0", Functional Analysis and Its Applications, 8: 138–141, doi:10.1007/BF01078599, MR 0350378.
• Banach, Stefan (1932). Théorie des Opérations Linéaires [Theory of Linear Operations] (PDF). Monografie Matematyczne (in French). Vol. 1. Warszawa: Subwencji Funduszu Kultury Narodowej. Zbl 0005.20901. Archived from the original (PDF) on 2014-01-11. Retrieved 2020-07-11.
• Figiel, T.; Johnson, W. B. (1974), "A uniformly convex Banach space which contains no ℓ p", Compositio Mathematica, 29: 179–190, MR 0355537.
• Casazza, Peter G.; Shura, Thaddeus J. (1989), Tsirelson's Space, Lecture Notes in Mathematics, vol. 1363, Berlin: Springer-Verlag, ISBN 3-540-50678-0, MR 0981801.
• Johnson, William B.; J. Lindenstrauss, Joram, eds. (2001), Handbook of the Geometry of Banach Spaces, vol. 1, Elsevier.
• Johnson, William B.; J. Lindenstrauss, Joram, eds. (2003), Handbook of the Geometry of Banach Spaces, vol. 2, Elsevier.
• Lindenstrauss, Joram (1970), "Some aspects of the theory of Banach spaces", Advances in Mathematics, 5: 159–180, doi:10.1016/0001-8708(70)90032-0.
• Lindenstrauss, Joram (1971), "The geometric theory of the classical Banach spaces", Actes du Congrès Intern. Math., Nice 1970: 365–372.
• Lindenstrauss, Joram; Tzafriri, Lior (1977), Classical Banach Spaces I, Sequence Spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 92, Berlin: Springer-Verlag, ISBN 3-540-08072-4.
• Milman, V. D. (1970), "Geometric theory of Banach spaces. I. Theory of basic and minimal systems", Uspekhi Mat. Nauk (in Russian), 25 no. 3: 113–174. English translation in Russian Math. Surveys 25 (1970), 111-170.
• Schlumprecht, Thomas (1991), "An arbitrary distortable Banach space", Israel Journal of Mathematics, 76: 81–95, arXiv:math/9201225, doi:10.1007/bf02782845, MR 1177333.
• Baudier, Florent; Lancien, Gilles; Schlumprecht, Thomas (2018), "The coarse geometry of Tsirelson's space and applications", Journal of the American Mathematical Society, 31: 699--717, arXiv:1705.06797, doi:10.1090/jams/899, MR 3787406.
External links
• Boris Tsirelson's reminiscences on his web page
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Chrysoula Tsogka
Chrysoula Tsogka is a Greek applied mathematician whose research involves remote sensing, wave propagation, and imaging through complex media. She is a professor of applied mathematics at the University of California, Merced.[1]
Education
Tsogka studied chemical engineering at the National Technical University of Athens, graduating with a bachelor's degree in 1995. She went to Paris Dauphine University for graduate study in applied mathematics, earning a master's degree in 1996 and completing her Ph.D. in 1999.[2] Her dissertation, Mathematical and Numerical Modeling of 3D Elastic Wave Propagation in Complex Media with Cracks, was supervised by Patrick Joly.[3]
Career
After working as a researcher for CNRS in the Laboratoire de Mecanique et d’Acoustique, and as a visiting researcher at Stanford University, she became an assistant professor at the University of Chicago in 2004. She moved to the University of Crete as an associate professor in 2007, and was promoted to full professor in 2014. She took her present position at the University of California, Merced in 2019.[2]
References
1. "Chrysoula Tsogka", Applied mathematics, University of California, Merced, retrieved 2022-08-28
2. Curriculum vitae (PDF), University of California, Merced, retrieved 2022-08-28
3. Chrysoula Tsogka at the Mathematics Genealogy Project
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Zu Chongzhi
Zu Chongzhi (Chinese: 祖沖之; 429–500 AD), courtesy name Wenyuan (Chinese: 文遠), was a Chinese astronomer, mathematician, politician, inventor, and writer during the Liu Song and Southern Qi dynasties. He was most notable for calculating pi as between 3.1415926 and 3.1415927, a record in accuracy which would not be surpassed for over 800 years.
Zu Chongzhi
Traditional Chinese祖沖之
Simplified Chinese祖冲之
Transcriptions
Standard Mandarin
Hanyu PinyinZǔ Chōngzhī
Wade–GilesTsu Ch'ung-chih
Wenyuan (courtesy name)
Traditional Chinese文遠
Simplified Chinese文远
Transcriptions
Standard Mandarin
Hanyu PinyinWényuǎn
Life and works
Chongzhi's ancestry was from modern Baoding, Hebei. To flee from the ravages of war, Zu's grandfather Zu Chang moved to the Yangtze, as part of the massive population movement during the Eastern Jin. Zu Chang (祖昌) at one point held the position of Chief Minister for the Palace Buildings (大匠卿) within the Liu Song and was in charge of government construction projects. Zu's father, Zu Shuozhi (祖朔之), also served the court and was greatly respected for his erudition.
Zu was born in Jiankang. His family had historically been involved in astronomical research, and from childhood Zu was exposed to both astronomy and mathematics. When he was only a youth his talent earned him much repute. When Emperor Xiaowu of Liu Song heard of him, he was sent to the Hualin Xuesheng (華林學省) academy, and later the Imperial Nanjing University (Zongmingguan) to perform research. In 461 in Nanxu (today Zhenjiang, Jiangsu), he was engaged in work at the office of the local governor.
Zu Chongzhi, along with his son Zu Gengzhi, wrote a mathematical text entitled Zhui Shu (綴述; "Methods for Interpolation"). It is said that the treatise contained formulas for the volume of a sphere, cubic equations and an accurate value of pi.[1] This book has been lost since the Song Dynasty.
His mathematical achievements included
• the Daming calendar (大明曆) introduced by him in 465.
• distinguishing the sidereal year and the tropical year. He measured 45 years and 11 months per degree between those two; today we know the difference is 70.7 years per degree.
• calculating one year as 365.24281481 days, which is very close to 365.24219878 days as we know today.
• calculating the number of overlaps between sun and moon as 27.21223, which is very close to 27.21222 as we know today; using this number he successfully predicted an eclipse four times during 23 years (from 436 to 459).
• calculating the Jupiter year as about 11.858 Earth years, which is very close to 11.862 as we know of today.
• deriving two approximations of pi, (3.1415926535897932...) which held as the most accurate approximation for π for over nine hundred years. His best approximation was between 3.1415926 and 3.1415927, with 355/113 (密率, milü, close ratio) and 22/7 (約率, yuelü, approximate ratio) being the other notable approximations. He obtained the result by approximating a circle with a 24,576 (= 213 × 3) sided polygon. This was an impressive feat for the time, especially considering that the counting rods he used for recording intermediate results were merely a pile of wooden sticks laid out in certain patterns. Japanese mathematician Yoshio Mikami pointed out, "22/7 was nothing more than the π value obtained several hundred years earlier by the Greek mathematician Archimedes, however milü π = 355/113 could not be found in any Greek, Indian or Arabian manuscripts, not until 1585 Dutch mathematician Adriaan Anthoniszoon obtained this fraction; the Chinese possessed this most extraordinary fraction over a whole millennium earlier than Europe". Hence Mikami strongly urged that the fraction 355/113 be named after Zu Chongzhi as Zu's fraction.[2] In Chinese literature, this fraction is known as "Zu's ratio". Zu's ratio is a best rational approximation to π, and is the closest rational approximation to π from all fractions with denominator less than 16600.[3]
• finding the volume of a sphere as πD3/6 where D is diameter (equivalent to 4πr3/3).
Astronomy
Zu was an accomplished astronomer who calculated the time values with unprecedented precision. His methods of interpolation and the use of integration were far ahead of his time. Even the results of the astronomer Yi Xing (who was beginning to utilize foreign knowledge) were not comparable. The Sung dynasty calendar was backwards to the "Northern barbarians" because they were implementing their daily lives with the Da Ming Li. It is said that his methods of calculation were so advanced, the scholars of the Sung dynasty and Indo influence astronomers of the Tang dynasty found it confusing.
Mathematics
Further information: Milü
Part of a series of articles on the
mathematical constant π
3.1415926535897932384626433...
Uses
• Area of a circle
• Circumference
• Use in other formulae
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• Irrationality
• Transcendence
Value
• Less than 22/7
• Approximations
• Madhava's correction term
• Memorization
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• Liu Hui
• Zu Chongzhi
• Aryabhata
• Madhava
• Jamshīd al-Kāshī
• Ludolph van Ceulen
• François Viète
• Seki Takakazu
• Takebe Kenko
• William Jones
• John Machin
• William Shanks
• Srinivasa Ramanujan
• John Wrench
• Chudnovsky brothers
• Yasumasa Kanada
History
• Chronology
• A History of Pi
In culture
• Indiana Pi Bill
• Pi Day
Related topics
• Squaring the circle
• Basel problem
• Six nines in π
• Other topics related to π
The majority of Zu's great mathematical works are recorded in his lost text the Zhui Shu. Most schools argue about his complexity since traditionally the Chinese had developed mathematics as algebraic and equational. Logically, scholars assume that the Zhui Shu yields methods of cubic equations. His works on the accurate value of pi describe the lengthy calculations involved. Zu used the Liu Hui's π algorithm described earlier by Liu Hui to inscribe a 12,288-gon. Zu's value of pi is precise to six decimal places and for a thousand years thereafter no subsequent mathematician computed a value this precise. Zu also worked on deducing the formula for the volume of a sphere.
Inventions and innovations
Hammer mills
In 488, Zu Chongzhi was responsible for erecting water powered trip hammer mills which was inspected by Emperor Wu of Southern Qi during the early 490s.[4][5][6]
Paddle boats
Zu is also credited with inventing Chinese paddle boats or Qianli chuan in the late 5th century AD during the Southern Qi Dynasty.[7][8][9][6] The boats made sailing a more reliable form of transportation and based on the shipbuilding technology of its day, numerous paddle wheel ships were constructed during the Tang era as the boats were able to cruise at faster speeds than the existing vessels at the time as well as being able to cover hundreds of kilometers of distance without the aid of wind.[7]
South pointing chariot
The south-pointing chariot device was first invented by the Chinese mechanical engineer Ma Jun (c. 200–265 AD). It was a wheeled vehicle that incorporated an early use of differential gears to operate a fixed figurine that would constantly point south, hence enabling one to accurately measure their directional bearings. This effect was achieved not by magnetics (like in a compass), but through intricate mechanics, the same design that allows equal amounts of torque applied to wheels rotating at different speeds for the modern automobile. After the Three Kingdoms period, the device fell out of use temporarily. However, it was Zu Chongzhi who successfully re-invented it in 478, as described in the texts of the Book of Song and the Book of Qi, with a passage from the latter below:
When Emperor Wu of Liu Song subdued Guanzhong he obtained the south-pointing carriage of Yao Xing, but it was only the shell with no machinery inside. Whenever it moved it had to have a man inside to turn (the figure). In the Sheng-Ming reign period, Gao Di commissioned Zi Zu Chongzhi to reconstruct it according to the ancient rules. He accordingly made new machinery of bronze, which would turn round about without a hitch and indicate the direction with uniformity. Since Ma Jun's time such a thing had not been.[10][11]
Literature
Zu's paradoxographical work Accounts of Strange Things [述異記] survives.[12][13]
Named after him
• π ≈ 355/113 as Zu Chongzhi's π ratio.
• The lunar crater Tsu Chung-Chi
• 1888 Zu Chong-Zhi is the name of asteroid 1964 VO1.
• ZUC stream cipher is a new encryption algorithm.
Notes
1. Ho, Peng Yoke, LI, QI and SHU, Hong Kong University Press, 1985. University of Washington Press edition, 1987. ISBN 0-295- 96362-X, p.76
2. Yoshio Mikami (1913). Development of Mathematics in China and Japan. B. G. Teubner. p. 50.
3. The next "best rational approximation" to π is 52163/16604 = 3.1415923874.
4. Liu, Heping (2002). ""The Water Mill" and Northern Song Imperial Patronage of Art, Commerce, and Science". The Art Bulletin. CAA. 84 (4): 574. doi:10.2307/3177285. JSTOR 3177285.
5. Needham, Joseph (1965). Science and Civilization in China, Vol. IV: Physics and Physical Technology, p.400. ISBN 978-0-521-05802-5.
6. Yongxiang Lu, ed. (2014). A History of Chinese Science and Technology, Volume 3. Springer. p. 280. ISBN 9783662441664.
7. Needham, 416
8. Selin, Helaine (2008). Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures (2nd ed.). Springer (published April 16, 2008). p. 1061. Bibcode:2008ehst.book.....S. ISBN 978-1402045592.
9. Wang, Hsien-Chun (January 1, 2019). "Discovering Steam Power in China, 1840s–1860s". Technology and Culture. Johns Hopkins University Press. 51: 38.
10. Needham, Volume 4, Part 2, 289.
11. Book of Qi, 52.905
12. 中国大百科全书（第二版） [Encyclopedia of China (2nd Edition)] (in Chinese). Vol. 30. Encyclopedia of China Publishing House. 2009. p. 205. ISBN 978-7-500-07958-3.
13. Owen, Stephen (2010). The Cambridge History of Chinese Literature. Vol. 1. Cambridge University Press. p. 242. ISBN 978-0-521-11677-0.
References
• Needham, Joseph (1986). Science and Civilization in China: Volume 4, Part 2. Cambridge University Press
• Du Shiran and He Shaogeng, "Zu Chongzhi". Encyclopedia of China (Mathematics Edition), 1st ed.
Further reading
• Needham, Joseph (1986). Science and Civilization in China: Volume 3, Mathematics and the Sciences of the Heavens and the Earth. Cambridge University Press.
• Xiao Zixian, ed. (1974) [537]. 南齊書 [Book of Qi]. Vol. 52. Beijing: Zhonghua Publishing. pp. 903–906.
• Li Dashi; Li Yanshou (李延壽) [in Chinese], eds. (1975) [659]. 南史 [History of the Southern Dynasties]. Vol. 72. Beijing: Zhonghua Publishing. pp. 1773–1774.
External links
• Encyclopædia Britannica's description of Zu Chongzhi
• Zu Chongzhi at Chinaculture.org
• Zu Chongzhi at the University of Maine Archived 2010-06-14 at the Wayback Machine
• O'Connor, John J.; Robertson, Edmund F., "Zu Chongzhi", MacTutor History of Mathematics Archive, University of St Andrews
Authority control
International
• ISNI
• VIAF
National
• Korea
• Netherlands
Academics
• MathSciNet
• zbMATH
Other
• IdRef
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Zu Gengzhi
Zu Geng or Zu Gengzhi (Chinese: 祖暅之; Wade–Giles: Tsu Keng-chih; ca. 480 – ca. 525) was a Chinese mathematician, politician, and writer. His courtesy name was Jingshuo (景爍). He was the son of the famous mathematician Zu Chongzhi.[1] He is known principally for deriving and proving the formula for the volume of a sphere. He additionally measured the angular distance between Polaris and the celestial north pole.
Zu Gengzhi
Traditional Chinese祖暅之
Simplified Chinese祖暅之
Transcriptions
Standard Mandarin
Hanyu PinyinZǔ Gèngzhī
Wade–GilesTsu Keng-chih
Jingshuo (courtesy name)
Traditional Chinese景爍
Simplified Chinese景烁
Transcriptions
Standard Mandarin
Hanyu PinyinJǐngshuò
See also
• List of Chinese mathematicians
References
1. "Zu Gengzhi". Encyclopædia Britannica. Retrieved 2014-09-22.
External links
• O'Connor, John J.; Robertson, Edmund F., "Zu Gengzhi", MacTutor History of Mathematics Archive, University of St Andrews
Authority control: Academics
• MathSciNet
• zbMATH
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Tube domain
In mathematics, a tube domain is a generalization of the notion of a vertical strip (or half-plane) in the complex plane to several complex variables. A strip can be thought of as the collection of complex numbers whose real part lie in a given subset of the real line and whose imaginary part is unconstrained; likewise, a tube is the set of complex vectors whose real part is in some given collection of real vectors, and whose imaginary part is unconstrained.
Tube domains are domains of the Laplace transform of a function of several real variables (see multidimensional Laplace transform). Hardy spaces on tubes can be defined in a manner in which a version of the Paley–Wiener theorem from one variable continues to hold, and characterizes the elements of Hardy spaces as the Laplace transforms of functions with appropriate integrability properties. Tubes over convex sets are domains of holomorphy. The Hardy spaces on tubes over convex cones have an especially rich structure, so that precise results are known concerning the boundary values of Hp functions. In mathematical physics, the future tube is the tube domain associated to the interior of the past null cone in Minkowski space, and has applications in relativity theory and quantum gravity.[1] Certain tubes over cones support a Bergman metric in terms of which they become bounded symmetric domains. One of these is the Siegel half-space which is fundamental in arithmetic.
Definition
Let Rn denote real coordinate space of dimension n and Cn denote complex coordinate space. Then any element of Cn can be decomposed into real and imaginary parts:
$a=(z_{1},\dots ,z_{n})=(x_{1}+iy_{1},\dots ,x_{n}+iy_{n})=(x_{1},\dots ,x_{n})+i(y_{1},\dots ,y_{n})=x+iy.$
Let A be an open subset of Rn. The tube over A, denoted TA, is the subset of Cn consisting of all elements whose real parts lie in A:[2][lower-alpha 1]
$T_{A}=\{z=x+iy\in \mathbb {C} ^{n}\mid x\in A\}.$
Tubes as domains of holomorphy
Main article: Bochner's tube theorem
Suppose that A is a connected open set. Then any complex-valued function that is holomorphic in a tube TA can be extended uniquely to a holomorphic function on the convex hull of the tube ch TA,[2] which is also a tube, and in fact
$\operatorname {ch} \,T_{A}=T_{\operatorname {ch} \,A}.$
Since any convex open set is a domain of holomorphy (holomorphically convex), a convex tube is also a domain of holomorphy. So the holomorphic envelope of any tube is equal to its convex hull.[3][4]
Hardy spaces
Let A be an open set in Rn. The Hardy space H p(TA) is the set of all holomorphic functions F in TA such that
$\int _{\mathbb {R} ^{n}}|F(x+iy)|^{p}\,dy<\infty $
for all x in A.
In the special case of p = 2, functions in H2(TA) can be characterized as follows.[5] Let ƒ be a complex-valued function on Rn satisfying
$\sup _{x\in A}\int _{\mathbb {R} ^{n}}|f(t)|^{2}e^{-4\pi x\cdot t}\,dt<\infty .$
The Fourier–Laplace transform of ƒ is defined by
$F(x+iy)=\int _{\mathbb {R} ^{n}}e^{2\pi z\cdot t}f(t)\,dt.$
Then F is well-defined and belongs to H2(TA). Conversely, every element of H2(TA) has this form.
A corollary of this characterization is that H2(TA) contains a nonzero function if and only if A contains no straight line.
Tubes over cones
Let A be an open convex cone in Rn. This means that A is an open convex set such that, whenever x lies in A, so does the entire ray from the origin to x. Symbolically,
$x\in A\implies tx\in A\ \ \ {\text{for all}}\ t>0.$
If A is a cone, then the elements of H2(TA) have L2 boundary limits in the sense that[5]
$\lim _{y\to 0}F(x+iy)$
exists in L2(B). There is an analogous result for Hp(TA), but it requires additional regularity of the cone (specifically, the dual cone A* needs to have nonempty interior).
See also
• Reinhardt domain
• Siegel domain
Notes
1. Some conventions instead define a tube to be a domain such that the imaginary part lies in A (Stein & Weiss 1971).
Citations
1. Gibbons 2000.
2. Hörmander 1990.
3. Chirka 2001.
4. Carmignani 1973.
5. Stein & Weiss 1971.
Sources
• Chirka, E.M. (2001) [First published 1994], "Tube domain", Encyclopedia of Mathematics, EMS Press.
• Gibbons, G.W. (2000), "Holography and the future tube", Classical and Quantum Gravity, 17 (5): 1071–1079, arXiv:hep-th/9911027, Bibcode:2000CQGra..17.1071G, doi:10.1088/0264-9381/17/5/316, S2CID 14045117.
• Hörmander, Lars (1990), Introduction to complex analysis in several variables, New York: North-Holland, ISBN 0-444-88446-7.
• Stein, Elias; Weiss, Guido (1971), Introduction to Fourier Analysis on Euclidean Spaces, Princeton, N.J.: Princeton University Press, ISBN 978-0-691-08078-9 – via Internet Archive.
• Carmignani, Robert (1973). "Envelopes of Holomorphy and Holomorphic Convexity". Transactions of the American Mathematical Society. 179: 415–431. doi:10.1090/S0002-9947-1973-0316748-1. hdl:1911/14576. JSTOR 1996512..
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Tube lemma
In mathematics, particularly topology, the tube lemma, also called Wallace's theorem, is a useful tool in order to prove that the finite product of compact spaces is compact.
Statement
The lemma uses the following terminology:
• If $X$ and $Y$ are topological spaces and $X\times Y$ is the product space, endowed with the product topology, a slice in $X\times Y$ is a set of the form $\{x\}\times Y$ for $x\in X$.
• A tube in $X\times Y$ is a subset of the form $U\times Y$ where $U$ is an open subset of $X$. It contains all the slices $\{x\}\times Y$ for $x\in U$.
Tube Lemma — Let $X$ and $Y$ be topological spaces with $Y$ compact, and consider the product space $X\times Y.$ If $N$ is an open set containing a slice in $X\times Y,$ then there exists a tube in $X\times Y$ containing this slice and contained in $N.$
Using the concept of closed maps, this can be rephrased concisely as follows: if $X$ is any topological space and $Y$ a compact space, then the projection map $X\times Y\to X$ is closed.
Generalized Tube Lemma 1 — Let $X$ and $Y$ be topological spaces and consider the product space $X\times Y.$ Let $A$ be a compact subset of $X$ and $B$ be a compact subset of $Y.$ If $N$ is an open set containing $A\times B,$ then there exists $U$ open in $X$ and $V$ open in $Y$ such that $A\times B\subseteq U\times V\subseteq N.$
Generalized Tube Lemma 2 — Let $X_{i},i\in I$ be topological spaces and consider the product space $\prod _{i\in I}X_{i}.$ For each $i\in I$, let $A_{i}$ be a compact subset of $X_{i}.$ If $N$ is an open set containing $\prod _{i\in I}A_{i},$ then there exists $U_{i}$ open in $X_{i}$ with $U_{i}=X_{i}$ for all but finite amount of $i\in I$, such that $\prod _{i\in I}A_{i}\subseteq \prod _{i\in I}U_{i}\subseteq N.$
Examples and properties
1. Consider $\mathbb {R} \times \mathbb {R} $ in the product topology, that is the Euclidean plane, and the open set $N=\{(x,y)\in \mathbb {R} \times \mathbb {R} ~:~|xy|<1\}.$ The open set $N$ contains $\{0\}\times \mathbb {R} ,$ but contains no tube, so in this case the tube lemma fails. Indeed, if $W\times \mathbb {R} $ is a tube containing $\{0\}\times \mathbb {R} $ and contained in $N,$ $W$ must be a subset of $\left(-1/x,1/x\right)$ for all $x>0$ which means $W=\{0\}$ contradicting the fact that $W$ is open in $\mathbb {R} $ (because $W\times \mathbb {R} $ is a tube). This shows that the compactness assumption is essential.
2. The tube lemma can be used to prove that if $X$ and $Y$ are compact spaces, then $X\times Y$ is compact as follows:
Let $\{G_{a}\}$ be an open cover of $X\times Y$. For each $x\in X$, cover the slice $\{x\}\times Y$ by finitely many elements of $\{G_{a}\}$ (this is possible since $\{x\}\times Y$ is compact, being homeomorphic to $Y$). Call the union of these finitely many elements $N_{x}.$ By the tube lemma, there is an open set of the form $W_{x}\times Y$ containing $\{x\}\times Y$ and contained in $N_{x}.$ The collection of all $W_{x}$ for $x\in X$ is an open cover of $X$ and hence has a finite subcover $\{W_{x_{1}},\dots ,W_{x_{n}}\}$. Thus the finite collection $\{W_{x_{1}}\times Y,\dots ,W_{x_{n}}\times Y\}$ covers $X\times Y$. Using the fact that each $W_{x_{i}}\times Y$ is contained in $N_{x_{i}}$ and each $N_{x_{i}}$ is the finite union of elements of $\{G_{a}\}$, one gets a finite subcollection of $\{G_{a}\}$ that covers $X\times Y$.
3. By part 2 and induction, one can show that the finite product of compact spaces is compact.
4. The tube lemma cannot be used to prove the Tychonoff theorem, which generalizes the above to infinite products.
Proof
The tube lemma follows from the generalized tube lemma by taking $A=\{x\}$ and $B=Y.$ It therefore suffices to prove the generalized tube lemma. By the definition of the product topology, for each $(a,b)\in A\times B$ there are open sets $U_{a,b}\subseteq X$ and $V_{a,b}\subseteq Y$ such that $(a,b)\in U_{a,b}\times V_{a,b}\subseteq N.$ For any $a\in A,$ $\left\{V_{a,b}~:~b\in B\right\}$ is an open cover of the compact set $B$ so this cover has a finite subcover; namely, there is a finite set $B_{0}(a)\subseteq B$ such that $V_{a}:=\bigcup _{b\in B_{0}(a)}V_{a,b}$ contains $B,$ where observe that $V_{a}$ is open in $Y.$ For every $a\in A,$ let $U_{a}:=\bigcap _{b\in B_{0}(a)}U_{a,b},$ which is an open in $X$ set since $B_{0}(a)$ is finite. Moreover, the construction of $U_{a}$ and $V_{a}$ implies that $\{a\}\times B\subseteq U_{a}\times V_{a}\subseteq N.$ We now essentially repeat the argument to drop the dependence on $a.$ Let $A_{0}\subseteq A$ be a finite subset such that $U:=\bigcup _{a\in A_{0}}U_{a}$ contains $A$ and set $V:=\bigcap _{a\in A_{0}}V_{a}.$ It then follows by the above reasoning that $A\times B\subseteq U\times V\subseteq N$ and $U\subseteq X$ and $V\subseteq Y$ are open, which completes the proof.
See also
• Alexander's sub-base theorem – Collection of subsets that generate a topologyPages displaying short descriptions of redirect targets
• Tubular neighborhood – neighborhood of a submanifold homeomorphic to that submanifold’s normal bundlePages displaying wikidata descriptions as a fallback
• Tychonoff theorem – Product of any collection of compact topological spaces is compactPages displaying short descriptions of redirect targets
References
• James Munkres (1999). Topology (2nd ed.). Prentice Hall. ISBN 0-13-181629-2.
• Joseph J. Rotman (1988). An Introduction to Algebraic Topology. Springer. ISBN 0-387-96678-1. (See Chapter 8, Lemma 8.9)
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Tubular neighborhood
In mathematics, a tubular neighborhood of a submanifold of a smooth manifold is an open set around it resembling the normal bundle.
The idea behind a tubular neighborhood can be explained in a simple example. Consider a smooth curve in the plane without self-intersections. On each point on the curve draw a line perpendicular to the curve. Unless the curve is straight, these lines will intersect among themselves in a rather complicated fashion. However, if one looks only in a narrow band around the curve, the portions of the lines in that band will not intersect, and will cover the entire band without gaps. This band is a tubular neighborhood.
In general, let S be a submanifold of a manifold M, and let N be the normal bundle of S in M. Here S plays the role of the curve and M the role of the plane containing the curve. Consider the natural map
$i:N_{0}\to S$
which establishes a bijective correspondence between the zero section $N_{0}$ of N and the submanifold S of M. An extension j of this map to the entire normal bundle N with values in M such that $j(N)$ is an open set in M and j is a homeomorphism between N and $j(N)$ is called a tubular neighbourhood.
Often one calls the open set $T=j(N),$ rather than j itself, a tubular neighbourhood of S, it is assumed implicitly that the homeomorphism j mapping N to T exists.
Normal tube
A normal tube to a smooth curve is a manifold defined as the union of all discs such that
• all the discs have the same fixed radius;
• the center of each disc lies on the curve; and
• each disc lies in a plane normal to the curve where the curve passes through that disc's center.
Formal definition
Let $S\subseteq M$ be smooth manifolds. A tubular neighborhood of $S$ in $M$ is a vector bundle $\pi :E\to S$ together with a smooth map $J:E\to M$ such that
• $J\circ 0_{E}=i$ where $i$ is the embedding $S\hookrightarrow M$ and $0_{E}$ the zero section
• there exists some $U\subseteq E$ and some $V\subseteq M$ with $0_{E}[S]\subseteq U$ and $S\subseteq V$ such that $J\vert _{U}:U\to V$ is a diffeomorphism.
The normal bundle is a tubular neighborhood and because of the diffeomorphism condition in the second point, all tubular neighborhood have the same dimension, namely (the dimension of the vector bundle considered as a manifold is) that of $M.$
Generalizations
Generalizations of smooth manifolds yield generalizations of tubular neighborhoods, such as regular neighborhoods, or spherical fibrations for Poincaré spaces.
These generalizations are used to produce analogs to the normal bundle, or rather to the stable normal bundle, which are replacements for the tangent bundle (which does not admit a direct description for these spaces).
See also
• Parallel curve (aka offset curve)
• Tube lemma – proof in topologyPages displaying wikidata descriptions as a fallback
References
• Raoul Bott, Loring W. Tu (1982). Differential forms in algebraic topology. Berlin: Springer-Verlag. ISBN 0-387-90613-4.
• Morris W. Hirsch (1976). Differential Topology. Berlin: Springer-Verlag. ISBN 0-387-90148-5.
• Waldyr Muniz Oliva (2002). Geometric Mechanics. Berlin: Springer-Verlag. ISBN 3-540-44242-1.
Wikimedia Commons has media related to Tubular neighborhood.
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Tucker decomposition
In mathematics, Tucker decomposition decomposes a tensor into a set of matrices and one small core tensor. It is named after Ledyard R. Tucker[1] although it goes back to Hitchcock in 1927.[2] Initially described as a three-mode extension of factor analysis and principal component analysis it may actually be generalized to higher mode analysis, which is also called higher-order singular value decomposition (HOSVD).
It may be regarded as a more flexible PARAFAC (parallel factor analysis) model. In PARAFAC the core tensor is restricted to be "diagonal".
In practice, Tucker decomposition is used as a modelling tool. For instance, it is used to model three-way (or higher way) data by means of relatively small numbers of components for each of the three or more modes, and the components are linked to each other by a three- (or higher-) way core array. The model parameters are estimated in such a way that, given fixed numbers of components, the modelled data optimally resemble the actual data in the least squares sense. The model gives a summary of the information in the data, in the same way as principal components analysis does for two-way data.
For a 3rd-order tensor $T\in F^{n_{1}\times n_{2}\times n_{3}}$, where $F$ is either $\mathbb {R} $ or $\mathbb {C} $, Tucker Decomposition can be denoted as follows,
$T={\mathcal {T}}\times _{1}U^{(1)}\times _{2}U^{(2)}\times _{3}U^{(3)}$
where ${\mathcal {T}}\in F^{d_{1}\times d_{2}\times d_{3}}$ is the core tensor, a 3rd-order tensor that contains the 1-mode, 2-mode and 3-mode singular values of $T$, which are defined as the Frobenius norm of the 1-mode, 2-mode and 3-mode slices of tensor ${\mathcal {T}}$ respectively. $U^{(1)},U^{(2)},U^{(3)}$ are unitary matrices in $F^{d_{1}\times n_{1}},F^{d_{2}\times n_{2}},F^{d_{3}\times n_{3}}$ respectively. The j-mode product (j = 1, 2, 3) of ${\mathcal {T}}$ by $U^{(j)}$ is denoted as ${\mathcal {T}}\times U^{(j)}$ with entries as
${\begin{aligned}({\mathcal {T}}\times _{1}U^{(1)})(n_{1},d_{2},d_{3})&=\sum _{i_{1}=1}^{d_{1}}{\mathcal {T}}(i_{1},d_{2},d_{3})U^{(1)}(i_{1},n_{1})\\({\mathcal {T}}\times _{2}U^{(2)})(d_{1},n_{2},d_{3})&=\sum _{i_{2}=1}^{d_{2}}{\mathcal {T}}(d_{1},i_{2},d_{3})U^{(2)}(i_{2},n_{2})\\({\mathcal {T}}\times _{3}U^{(3)})(d_{1},d_{2},n_{3})&=\sum _{i_{3}=1}^{d_{3}}{\mathcal {T}}(d_{1},d_{2},i_{3})U^{(3)}(i_{3},n_{3})\end{aligned}}$
Taking $d_{i}=n_{i}$ for all $i$ is always sufficient to represent $T$ exactly, but often $T$ can be compressed or efficiently approximately by choosing $d_{i}<n_{i}$. A common choice is $d_{1}=d_{2}=d_{3}=\min(n_{1},n_{2},n_{3})$, which can be effective when the difference in dimension sizes is large.
There are two special cases of Tucker decomposition:
Tucker1: if $U^{(2)}$ and $U^{(3)}$ are identity, then $T={\mathcal {T}}\times _{1}U^{(1)}$
Tucker2: if $U^{(3)}$ is identity, then $T={\mathcal {T}}\times _{1}U^{(1)}\times _{2}U^{(2)}$ .
RESCAL decomposition [3] can be seen as a special case of Tucker where $U^{(3)}$ is identity and $U^{(1)}$ is equal to $U^{(2)}$ .
See also
• Higher-order singular value decomposition
• Multilinear principal component analysis
References
1. Ledyard R. Tucker (September 1966). "Some mathematical notes on three-mode factor analysis". Psychometrika. 31 (3): 279–311. doi:10.1007/BF02289464. PMID 5221127.
2. F. L. Hitchcock (1927). "The expression of a tensor or a polyadic as a sum of products". Journal of Mathematics and Physics. 6: 164–189.
3. Nickel, Maximilian; Tresp, Volker; Kriegel, Hans-Peter (28 June 2011). A Three-Way Model for Collective Learning on Multi-Relational Data. ICML. Vol. 11. pp. 809–816.
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Tucker's lemma
In mathematics, Tucker's lemma is a combinatorial analog of the Borsuk–Ulam theorem, named after Albert W. Tucker.
Let T be a triangulation of the closed n-dimensional ball $B_{n}$. Assume T is antipodally symmetric on the boundary sphere $S_{n-1}$. That means that the subset of simplices of T which are in $S_{n-1}$ provides a triangulation of $S_{n-1}$ where if σ is a simplex then so is −σ. Let $L:V(T)\to \{+1,-1,+2,-2,...,+n,-n\}$ be a labeling of the vertices of T which is an odd function on $S_{n-1}$, i.e, $L(-v)=-L(v)$ for every vertex $v\in S_{n-1}$. Then Tucker's lemma states that T contains a complementary edge - an edge (a 1-simplex) whose vertices are labelled by the same number but with opposite signs.[1]
Proofs
The first proofs were non-constructive, by way of contradiction.[2]
Later, constructive proofs were found, which also supplied algorithms for finding the complementary edge.[3][4] Basically, the algorithms are path-based: they start at a certain point or edge of the triangulation, then go from simplex to simplex according to prescribed rules, until it is not possible to proceed any more. It can be proved that the path must end in a simplex which contains a complementary edge.
An easier proof of Tucker's lemma uses the more general Ky Fan lemma, which has a simple algorithmic proof.
The following description illustrates the algorithm for $n=2$.[5] Note that in this case $B_{n}$ is a disc and there are 4 possible labels: $-2,-1,1,2$, like the figure at the top-right.
Start outside the ball and consider the labels of the boundary vertices. Because the labeling is an odd function on the boundary, the boundary must have both positive and negative labels:
• If the boundary contains only $\pm 1$ or only $\pm 2$, there must be a complementary edge on the boundary. Done.
• Otherwise, the boundary must contain $(+1,-2)$ edges. Moreover, the number of $(+1,-2)$ edges on the boundary must be odd.
Select an $(+1,-2)$ edge and go through it. There are three cases:
• You are now in a $(+1,-2,+2)$ simplex. Done.
• You are now in a $(+1,-2,-1)$ simplex. Done.
• You are in a simplex with another $(+1,-2)$ edge. Go through it and continue.
The last case can take you outside the ball. However, since the number of $(+1,-2)$ edges on the boundary must be odd, there must be a new, unvisited $(+1,-2)$ edge on the boundary. Go through it and continue.
This walk must end inside the ball, either in a $(+1,-2,+2)$ or in a $(+1,-2,-1)$ simplex. Done.
Run-time
The run-time of the algorithm described above is polynomial in the triangulation size. This is considered bad, since the triangulations might be very large. It would be desirable to find an algorithm which is logarithmic in the triangulation size. However, the problem of finding a complementary edge is PPA-complete even for $n=2$ dimensions. This implies that there is not too much hope for finding a fast algorithm.[6]
Equivalent results
There are several fixed-point theorems which come in three equivalent variants: an algebraic topology variant, a combinatorial variant and a set-covering variant. Each variant can be proved separately using totally different arguments, but each variant can also be reduced to the other variants in its row. Additionally, each result in the top row can be deduced from the one below it in the same column.[7]
Algebraic topologyCombinatoricsSet covering
Brouwer fixed-point theoremSperner's lemmaKnaster–Kuratowski–Mazurkiewicz lemma
Borsuk–Ulam theoremTucker's lemmaLusternik–Schnirelmann theorem
See also
• Topological combinatorics
References
1. Matoušek, Jiří (2003), Using the Borsuk–Ulam Theorem, Springer-Verlag, pp. 34–45, ISBN 3-540-00362-2
2. Tucker, Albert W. (1946), "Some topological properties of disk and sphere", Proc. First Canadian Math. Congress, Montreal, 1945, Toronto: University of Toronto Press, pp. 285–309, MR 0020254
3. Freund, Robert M.; Todd, Michael J. (1981), "A constructive proof of Tucker's combinatorial lemma", Journal of Combinatorial Theory, Series A, 30 (3): 321–325, doi:10.1016/0097-3165(81)90027-3, MR 0618536
4. Freund, Robert M.; Todd, Michael J. (1980), A constructive proof of Tucker's combinatorial lemma, archived from the original on June 22, 2015
5. Meunier, Frédéric (2010), Sperner and Tucker lemmas (PDF), Algorithms and Pretty Theorems Blog, pp. 46–64, retrieved 25 May 2015
6. Aisenberg, James; Bonet, Maria Luisa; Buss, Sam (2015), 2-D Tucker is PPA complete, ECCC TR15-163
7. Nyman, Kathryn L.; Su, Francis Edward (2013), "A Borsuk–Ulam equivalent that directly implies Sperner's lemma", The American Mathematical Monthly, 120 (4): 346–354, doi:10.4169/amer.math.monthly.120.04.346, JSTOR 10.4169/amer.math.monthly.120.04.346, MR 3035127
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Tukey lambda distribution
Formalized by John Tukey, the Tukey lambda distribution is a continuous, symmetric probability distribution defined in terms of its quantile function. It is typically used to identify an appropriate distribution (see the comments below) and not used in statistical models directly.
Tukey lambda distribution
Probability density function
Notation Tukey(λ)
Parameters λ ∈ R — shape parameter
Support x ∈ [−1/λ, 1/λ] for λ > 0,
x ∈ R for λ ≤ 0
PDF $(Q(p;\lambda ),q(p;\lambda )^{-1}),\,0\leq \,p\,\leq \,1$
CDF $(e^{-x}+1)^{-1},\,\,\lambda \,=\,0\,$(special case)
$(Q(p;\lambda ),p),\,0\leq \,p\,\leq \,1\,$(general case)
Mean $0,\,\,\lambda >-1$
Median 0
Mode 0
Variance ${\frac {2}{\lambda ^{2}}}{\bigg (}{\frac {1}{1+2\lambda }}-{\frac {\Gamma (\lambda +1)^{2}}{\Gamma (2\lambda +2)}}{\bigg )},\,\,\lambda >-1/2$
${\frac {\pi ^{2}}{3}},\,\,\lambda \,=\,0$
Skewness $0,\,\,\lambda >-1/3$
Ex. kurtosis ${\frac {(2\lambda +1)^{2}}{2(4\lambda +1)}}\,{\frac {g_{2}^{2}{\big (}3g_{2}^{2}-4g_{1}g_{3}+g_{4}{\big )}}{g_{4}{\big (}g_{1}^{2}-g_{2}{\big )}^{2}}}\,-\,3,$
$1.2,\,\,\lambda \,=\,0,$
${\text{where}}\,g_{k}\,=\,\Gamma (k\lambda +1)\,{\text{and}}\,\lambda \,>\,-1/4.$
Entropy $h(\lambda )=\int _{0}^{1}\log(q(p;\lambda ))\,dp$[1]
CF $\phi (t;\lambda )=\int _{0}^{1}\exp(\,it\,Q(p;\lambda ))\,dp$[2]
The Tukey lambda distribution has a single shape parameter, λ, and as with other probability distributions, it can be transformed with a location parameter, μ, and a scale parameter, σ. Since the general form of probability distribution can be expressed in terms of the standard distribution, the subsequent formulas are given for the standard form of the function.
Quantile function
For the standard form of the Tukey lambda distribution, the quantile function, $~Q(p)~,$ (i.e. the inverse function to the cumulative distribution function) and the quantile density function ($~q=\operatorname {d} Q/\operatorname {d} p~$ are
$Q\left(p;\lambda \right)~=~{\begin{cases}{\frac {1}{\lambda }}\left[p^{\lambda }-(1-p)^{\lambda }\right]~,&{\mbox{if }}\lambda \neq 0~,\\\log({\frac {p}{1-p}})~,&{\mbox{if }}\lambda =0~.\end{cases}}$
$q\left(p;\lambda \right)~=~\operatorname {d} Q/\operatorname {d} p~=~p^{(\lambda -1)}+\left(1-p\right)^{(\lambda -1)}~.$
For most values of the shape parameter, λ, the probability density function (PDF) and cumulative distribution function (CDF) must be computed numerically. The Tukey lambda distribution has a simple, closed form for the CDF and / or PDF only for a few exceptional values of the shape parameter, for example: λ ∈ { 2, 1, 1/2, 0 } (see uniform distribution [case λ = 1] and the logistic distribution [case λ = 0]).
However, for any value of λ both the CDF and PDF can be tabulated for any number of cumulative probabilities, p, using the quantile function Q to calculate the value x, for each cumulative probability p, with the probability density given by 1/q, the reciprocal of the quantile density function. As is the usual case with statistical distributions, the Tukey lambda distribution can readily be used by looking up values in a prepared table.
Moments
The Tukey lambda distribution is symmetric around zero, therefore the expected value of this distribution is equal to zero. The variance exists for λ > −1/2 and is given by the formula (except when λ = 0)
$\operatorname {Var} [X]={\frac {2}{\lambda ^{2}}}{\bigg (}{\frac {1}{1+2\lambda }}-{\frac {\Gamma (\lambda +1)^{2}}{\Gamma (2\lambda +2)}}{\bigg )}.$
More generally, the n-th order moment is finite when λ > −1/n and is expressed in terms of the beta function Β(x,y) (except when λ = 0) :
$\mu _{n}=\operatorname {E} [X^{n}]={\frac {1}{\lambda ^{n}}}\sum _{k=0}^{n}(-1)^{k}{n \choose k}\,\mathrm {B} (\lambda k+1,\,\lambda (n-k)+1).$
Note that due to symmetry of the density function, all moments of odd orders are equal to zero.
L-moments
Differently from the central moments, L-moments can be expressed in a closed form. The L-moment of order r>1 is given by[3]
$L_{r}={\frac {\left[1+(-1)^{r}\right]}{\lambda }}\sum _{k=0}^{r-1}(-1)^{r-1-k}{\binom {r-1}{k}}{\binom {r+k-1}{k}}\left({\frac {1}{k+1+\lambda }}\right).$
The first six L-moments can be presented as follows:[3]
$L_{1}=0$
$L_{2}={\frac {2}{\lambda }}\left[-{\frac {1}{1+\lambda }}+{\frac {2}{2+\lambda }}\right]$
$L_{3}=0$
$L_{4}={\frac {2}{\lambda }}\left[-{\frac {1}{1+\lambda }}+{\frac {12}{2+\lambda }}-{\frac {30}{3+\lambda }}+{\frac {30}{4+\lambda }}\right]$
$L_{5}=0$
$L_{6}={\frac {2}{\lambda }}\left[-{\frac {1}{1+\lambda }}+{\frac {30}{2+\lambda }}-{\frac {210}{3+\lambda }}+{\frac {560}{4+\lambda }}-{\frac {630}{5+\lambda }}+{\frac {252}{6+\lambda }}\right]\,.$
Comments
The Tukey lambda distribution is actually a family of distributions that can approximate a number of common distributions. For example,
λ = −1 approx. Cauchy C(0,π)
λ = 0 exactly logistic
λ = 0.14 approx. normal N(0, 2.142)
λ = 0.5 strictly concave ($\cap $-shaped)
λ = 1 exactly uniform U(−1, 1)
λ = 2 exactly uniform U(−1/2, 1/2)
The most common use of this distribution is to generate a Tukey lambda PPCC plot of a data set. Based on the PPCC plot, an appropriate model for the data is suggested. For example, if the best-fit of the curve to the data occurs for a value of λ at or near 0.14, then the data could be well-modeled with a normal distribution. Values of λ less than 0.14 suggests a heavier-tailed distribution; a milepost at λ = 0 (logistic) would indicate quite fat tails, with the extreme limit at λ = −1, approximating Cauchy. That is, as the best-fit value of λ varies from 0.14 towards −1, a bell-shaped PDF with increasingly heavy tails is suggested. Similarly, for an optimal value of λ becomes greater than 0.14 suggests a distribution with exceptionally thin tails (based on the point of view that the normal distribution itself is thin-tailed to begin with).
Except for values of λ very close to 0, all the suggested PDF functions have finite support, between −1 /|λ| and +1 /|λ| .
Since the Tukey lambda distribution is a symmetric distribution, the use of the Tukey lambda PPCC plot to determine a reasonable distribution to model the data only applies to symmetric distributions. A histogram of the data should provide evidence as to whether the data can be reasonably modeled with a symmetric distribution.[4]
References
1. Vasicek, Oldrich (1976), "A Test for Normality Based on Sample Entropy", Journal of the Royal Statistical Society, Series B, 38 (1): 54–59.
2. Shaw, W. T.; McCabe, J. (2009), "Monte Carlo sampling given a Characteristic Function: Quantile Mechanics in Momentum Space", arXiv:0903.1592 [q-fin.CP]
3. Karvanen, Juha; Nuutinen, Arto (2008). "Characterizing the generalized lambda distribution by L-moments". Computational Statistics & Data Analysis. 52 (4): 1971–1983. arXiv:math/0701405. doi:10.1016/j.csda.2007.06.021. S2CID 939977.
4. Joiner, Brian L.; Rosenblatt, Joan R. (1971). "Some properties of the range in samples from Tukey's symmetric lambda distributions". Journal of the American Statistical Association. 66 (334): 394–399. doi:10.2307/2283943. JSTOR 2283943.
External links
• "1.3.6.6.15 – Tukey-Lambda distribution". Gallery of Distributions. Engineering Statistics Handbook. US NIST – Information Technology Laboratory. EDA 366F.
This article incorporates public domain material from the National Institute of Standards and Technology.
Probability distributions (list)
Discrete
univariate
with finite
support
• Benford
• Bernoulli
• beta-binomial
• binomial
• categorical
• hypergeometric
• negative
• Poisson binomial
• Rademacher
• soliton
• discrete uniform
• Zipf
• Zipf–Mandelbrot
with infinite
support
• beta negative binomial
• Borel
• Conway–Maxwell–Poisson
• discrete phase-type
• Delaporte
• extended negative binomial
• Flory–Schulz
• Gauss–Kuzmin
• geometric
• logarithmic
• mixed Poisson
• negative binomial
• Panjer
• parabolic fractal
• Poisson
• Skellam
• Yule–Simon
• zeta
Continuous
univariate
supported on a
bounded interval
• arcsine
• ARGUS
• Balding–Nichols
• Bates
• beta
• beta rectangular
• continuous Bernoulli
• Irwin–Hall
• Kumaraswamy
• logit-normal
• noncentral beta
• PERT
• raised cosine
• reciprocal
• triangular
• U-quadratic
• uniform
• Wigner semicircle
supported on a
semi-infinite
interval
• Benini
• Benktander 1st kind
• Benktander 2nd kind
• beta prime
• Burr
• chi
• chi-squared
• noncentral
• inverse
• scaled
• Dagum
• Davis
• Erlang
• hyper
• exponential
• hyperexponential
• hypoexponential
• logarithmic
• F
• noncentral
• folded normal
• Fréchet
• gamma
• generalized
• inverse
• gamma/Gompertz
• Gompertz
• shifted
• half-logistic
• half-normal
• Hotelling's T-squared
• inverse Gaussian
• generalized
• Kolmogorov
• Lévy
• log-Cauchy
• log-Laplace
• log-logistic
• log-normal
• log-t
• Lomax
• matrix-exponential
• Maxwell–Boltzmann
• Maxwell–Jüttner
• Mittag-Leffler
• Nakagami
• Pareto
• phase-type
• Poly-Weibull
• Rayleigh
• relativistic Breit–Wigner
• Rice
• truncated normal
• type-2 Gumbel
• Weibull
• discrete
• Wilks's lambda
supported
on the whole
real line
• Cauchy
• exponential power
• Fisher's z
• Kaniadakis κ-Gaussian
• Gaussian q
• generalized normal
• generalized hyperbolic
• geometric stable
• Gumbel
• Holtsmark
• hyperbolic secant
• Johnson's SU
• Landau
• Laplace
• asymmetric
• logistic
• noncentral t
• normal (Gaussian)
• normal-inverse Gaussian
• skew normal
• slash
• stable
• Student's t
• Tracy–Widom
• variance-gamma
• Voigt
with support
whose type varies
• generalized chi-squared
• generalized extreme value
• generalized Pareto
• Marchenko–Pastur
• Kaniadakis κ-exponential
• Kaniadakis κ-Gamma
• Kaniadakis κ-Weibull
• Kaniadakis κ-Logistic
• Kaniadakis κ-Erlang
• q-exponential
• q-Gaussian
• q-Weibull
• shifted log-logistic
• Tukey lambda
Mixed
univariate
continuous-
discrete
• Rectified Gaussian
Multivariate
(joint)
• Discrete:
• Ewens
• multinomial
• Dirichlet
• negative
• Continuous:
• Dirichlet
• generalized
• multivariate Laplace
• multivariate normal
• multivariate stable
• multivariate t
• normal-gamma
• inverse
• Matrix-valued:
• LKJ
• matrix normal
• matrix t
• matrix gamma
• inverse
• Wishart
• normal
• inverse
• normal-inverse
• complex
Directional
Univariate (circular) directional
Circular uniform
univariate von Mises
wrapped normal
wrapped Cauchy
wrapped exponential
wrapped asymmetric Laplace
wrapped Lévy
Bivariate (spherical)
Kent
Bivariate (toroidal)
bivariate von Mises
Multivariate
von Mises–Fisher
Bingham
Degenerate
and singular
Degenerate
Dirac delta function
Singular
Cantor
Families
• Circular
• compound Poisson
• elliptical
• exponential
• natural exponential
• location–scale
• maximum entropy
• mixture
• Pearson
• Tweedie
• wrapped
• Category
• Commons
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Wikipedia
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Tukey depth
In statistics and computational geometry, the Tukey depth [1] is a measure of the depth of a point in a fixed set of points. The concept is named after its inventor, John Tukey. Given a set of n points ${\mathcal {X}}_{n}=\{X_{1},\dots ,X_{n}\}$ in d-dimensional space, Tukey's depth of a point x is the smallest fraction (or number) of points in any closed halfspace that contains x.
Tukey's depth measures how extreme a point is with respect to a point cloud. It is used to define the bagplot, a bivariate generalization of the boxplot.
For example, for any extreme point of the convex hull there is always a (closed) halfspace that contains only that point, and hence its Tukey depth as a fraction is 1/n.
Definitions
Sample Tukey's depth of point x, or Tukey's depth of x with respect to the point cloud ${\mathcal {X}}_{n}$, is defined as
$D(x;{\mathcal {X}}_{n})=\inf _{v\in \mathbb {R} ^{d},\|v\|=1}{\frac {1}{n}}\sum _{i=1}^{n}\mathbf {1} \{v^{T}(X_{i}-x)\geq 0\},$
where $\mathbf {1} \{\cdot \}$ is the indicator function that equals 1 if its argument holds true or 0 otherwise.
Population Tukey's depth of x wrt to a distribution $P_{X}$ is
$D(x;P_{X})=\inf _{v\in \mathbb {R} ^{d},\|v\|=1}P(v^{T}(X-x)\geq 0),$
where X is a random variable following distribution $P_{X}$.
Tukey mean and relation to centerpoint
A centerpoint c of a point set of size n is nothing else but a point of Tukey depth of at least n/(d + 1).
See also
• Centerpoint (geometry)
References
1. Tukey, John W (1975). Mathematics and the Picturing of Data. Proceedings of the International Congress of Mathematicians. p. 523-531.
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Wikipedia
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Centerpoint (geometry)
In statistics and computational geometry, the notion of centerpoint is a generalization of the median to data in higher-dimensional Euclidean space. Given a set of points in d-dimensional space, a centerpoint of the set is a point such that any hyperplane that goes through that point divides the set of points in two roughly equal subsets: the smaller part should have at least a 1/(d + 1) fraction of the points. Like the median, a centerpoint need not be one of the data points. Every non-empty set of points (with no duplicates) has at least one centerpoint.
Related concepts
Closely related concepts are the Tukey depth of a point (the minimum number of sample points on one side of a hyperplane through the point) and a Tukey median of a point set (a point maximizing the Tukey depth). A centerpoint is a point of depth at least n/(d + 1), and a Tukey median must be a centerpoint, but not every centerpoint is a Tukey median. Both terms are named after John Tukey.
For a different generalization of the median to higher dimensions, see geometric median.
Existence
A simple proof of the existence of a centerpoint may be obtained using Helly's theorem. Suppose there are n points, and consider the family of closed half-spaces that contain more than dn/(d + 1) of the points. Fewer than n/(d + 1) points are excluded from any one of these halfspaces, so the intersection of any subset of d + 1 of these halfspaces must be nonempty. By Helly's theorem, it follows that the intersection of all of these halfspaces must also be nonempty. Any point in this intersection is necessarily a centerpoint.
Algorithms
For points in the Euclidean plane, a centerpoint may be constructed in linear time.[1] In any dimension d, a Tukey median (and therefore also a centerpoint) may be constructed in time O(nd − 1 + n log n).[2]
A randomized algorithm that repeatedly replaces sets of d + 2 points by their Radon point can be used to compute an approximation to a centerpoint of any point set, in the sense that its Tukey depth is linear in the sample set size, in an amount of time that is polynomial in both the number of points and the dimension.[3]
References
Citations
1. Jadhav & Mukhopadhyay (1994).
2. Chan (2004).
3. Clarkson et al. (1996).
Sources
• Chan, Timothy M. (2004), "An optimal randomized algorithm for maximum Tukey depth", Proc. 15th ACM–SIAM Symp. on Discrete Algorithms (SODA 2004), pp. 430–436.
• Clarkson, Kenneth L.; Eppstein, David; Miller, Gary L.; Sturtivant, Carl; Teng, Shang-Hua (September 1996), "Approximating center points with iterated Radon points" (PDF), International Journal of Computational Geometry & Applications, 6 (3): 357–377, MR 1409651.
• Edelsbrunner, Herbert (1987), Algorithms in Combinatorial Geometry, Berlin: Springer-Verlag, ISBN 0-387-13722-X.
• Jadhav, S.; Mukhopadhyay, A. (1994), "Computing a centerpoint of a finite planar set of points in linear time", Discrete and Computational Geometry, 12 (1): 291–312, doi:10.1007/BF02574382.
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Wikipedia
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Tullio Levi-Civita
Tullio Levi-Civita, ForMemRS[1] (English: /ˈtʊlioʊ ˈlɛvi ˈtʃɪvɪtə/, Italian: [ˈtulljo ˈlɛːvi ˈtʃiːvita]; 29 March 1873 – 29 December 1941) was an Italian mathematician, most famous for his work on absolute differential calculus (tensor calculus) and its applications to the theory of relativity, but who also made significant contributions in other areas. He was a pupil of Gregorio Ricci-Curbastro, the inventor of tensor calculus. His work included foundational papers in both pure and applied mathematics, celestial mechanics (notably on the three-body problem), analytic mechanics (the Levi-Civita separability conditions in the Hamilton–Jacobi equation)[2] and hydrodynamics.[3][4]
This article is about the mathematician. For the mathematical symbol, see Levi-Civita symbol.
Tullio Levi-Civita
Tullio Levi-Civita
Born(1873-03-29)29 March 1873
Padua, Italy
Died29 December 1941(1941-12-29) (aged 68)
Rome, Italy
Alma materUniversity of Padua
Known for
• Tensor calculus
• Levi-Civita symbol
• Levi-Civita connection
• Levi-Civita field
• Levi-Civita parallelogramoid
AwardsSylvester Medal (1922)
FRS (1930)
Scientific career
FieldsMathematics
InstitutionsUniversity of Rome
Doctoral advisorGregorio Ricci-Curbastro
Doctoral students
• Evan Tom Davies
• Albert Joseph McConnell
• Octav Onicescu
• Attilio Palatini
• Antonio Signorini
• Libera Trevisani
• Giovanni Lampariello
• Gheorghe Vrânceanu
Biography
Born into an Italian Jewish family in Padua, Levi-Civita was the son of Giacomo Levi-Civita, a lawyer and former senator. He graduated in 1892 from the University of Padua Faculty of Mathematics. In 1894 he earned a teaching diploma after which he was appointed to the Faculty of Science teacher's college in Pavia. In 1898 he was appointed to the Padua Chair of Rational Mechanics (left uncovered by death of Ernesto Padova) where he met and, in 1914, married Libera Trevisani, one of his pupils.[5] He remained in his position at Padua until 1918, when he was appointed to the Chair of Higher Analysis at the University of Rome; in another two years he was appointed to the Chair of Mechanics there.
In 1900 he and Ricci-Curbastro published the theory of tensors in Méthodes de calcul différentiel absolu et leurs applications,[6] which Albert Einstein used as a resource to master the tensor calculus, a critical tool in the development of the theory of general relativity. In 1917 he introduced the notion of parallel transport[7][8] in Riemannian geometry, motivated by the will to simplify the computation of the curvature of a Riemannian manifold.[9] Levi-Civita's series of papers on the problem of a static gravitational field were also discussed in his 1915–1917 correspondence with Einstein. The correspondence was initiated by Levi-Civita, as he found mathematical errors in Einstein's use of tensor calculus to explain the theory of relativity. Levi-Civita methodically kept all of Einstein's replies to him; and even though Einstein had not kept Levi-Civita's, the entire correspondence could be re-constructed from Levi-Civita's archive. It is evident from this that, after numerous letters, the two men had grown to respect each other. In one of the letters, regarding Levi-Civita's new work, Einstein wrote "I admire the elegance of your method of computation; it must be nice to ride through these fields upon the horse of true mathematics while the like of us have to make our way laboriously on foot". In 1933 Levi-Civita contributed to Paul Dirac's equations in quantum mechanics as well.[10]
His textbook on tensor calculus, The Absolute Differential Calculus (originally a set of lecture notes in Italian co-authored with Ricci-Curbastro), remains one of the standard texts almost a century after its first publication, with several translations available.
In 1936, receiving an invitation from Einstein, Levi-Civita traveled to Princeton, United States and lived there with him for a year. But when the risk of war in Europe again rose, he returned to Italy. The 1938 race laws enacted by the Italian Fascist government deprived Levi-Civita of his professorship and of his membership of all scientific societies. Isolated from the scientific world, he died in his apartment in Rome in 1941.
Among his PhD students were Octav Onicescu, Attilio Palatini, Giovanni Lampariello and Gheorghe Vrânceanu.
Later on, when asked what he liked best about Italy, Einstein said "spaghetti and Levi-Civita".[11]
Other studies and honors
Analytical dynamics was another aspect of Levi-Civita's studies: many of his articles examine the three-body problem. He wrote articles on hydrodynamics and on systems of differential equations. He is credited with improvements to the Cauchy–Kowalevski theorem, on which he wrote a book in 1931. In 1933, he contributed to work on the Dirac equation. He developed the Levi-Civita field, a system of numbers that includes infinitesimal quantities.
Levi-Civita was elected an international honorary member of the American Academy of Arts and Sciences in 1917.[12] The Royal Society awarded him the Sylvester Medal in 1922 and elected him as a foreign member in 1930. He became an honorary member of the London Mathematical Society, of the Royal Society of Edinburgh, and of the Edinburgh Mathematical Society, following his participation in their colloquium in 1930 at the University of St Andrews. He was also a member of the Accademia dei Lincei, the Pontifical Academy of Sciences, and the American Philosophical Society.[13]
Works
All his mathematical works, except for the monographs, treatises and textbooks, were posthumously gathered in the six volumes of his "Collected works", in a revised typographical form amending both typographical errors and author's oversights.
Articles
• Ricci, Gregorio; Levi-Civita, Tullio (1900), "Méthodes de calcul différentiel absolu et leurs applications" [Methods of the absolute differential calculus and their applications], Mathematische Annalen (in French), 54 (1–2): 125–201, doi:10.1007/BF01454201, JFM 31.0297.01, S2CID 120009332.
• Levi-Civita, Tullio (1904), "Sulla integrazione della equazione di Hamilton-Jacobi per separazione di variabili" [On the integration of the Hamilton-Jacobi equation by separation of variables], Mathematische Annalen (in Italian), 59 (3): 383–397, doi:10.1007/bf01445149, JFM 35.0362.02, S2CID 123144759.
• Levi-Civita, Tullio (1917), "Nozione di parallelismo in una varietà qualunque e conseguente specificazione geometrica della curvatura riemanniana" [Notion of parallelism in any variety and consequent geometric specification of the Riemannian curvature], Rendiconti del Circolo Matematico di Palermo (in Italian), 42: 173–205, doi:10.1007/BF03014898, JFM 46.1125.02, S2CID 122088291.
Books
• Tullio Levi-Civita and Ugo Amaldi Lezioni di meccanica razionale (Bologna: N. Zanichelli, 1923)
• Tullio Levi-Civita Questioni di meccanica classica e relativistica (Bologna, N. Zanichelli, 1924)
• Tullio Levi-Civita Lezioni di calcolo differenziale assoluto (Roma: Alberto Stock Editore 1925)
• The Absolute Differential Calculus (London & Glasgow, Blackie & Son 1927) (edited by Enrico Persico, trans. by Marjorie Long)[14]
• Tullio Levi-Civita and Enrico Persico Fondamenti di meccanica relativistica (Bologna : N. Zanichelli, 1928)
• Tullio Levi-Civita Caratteristiche dei sistemi differenziali e propagazione ondosa (Bologna, N. Zanichelli 1931)
• Tullio Levi-Civita and Ugo Amaldi Nozioni di balistica esterna (Bologna: N. Zanichelli, 1935)
• Tullio Levi Problème des N Corps en relativité générale (Gauthier-Villars, Paris, 1950, Mémorial des sciences mathématiques ISSN 0025-9187)
• Levi-Civita, Tullio (1954), Opere Matematiche. Memorie e Note [Collected mathematical works. Memoirs and notes] (PDF) (in French and Italian), vol. primo (1893−1900), Pubblicate a cura dell'Accademia Nazionale dei Lincei, Roma: Zanichelli Editore, pp. XXX, 564.
• Levi-Civita, Tullio (1956), Opere Matematiche. Memorie e Note [Collected mathematical works. Memoirs and notes] (PDF) (in French and Italian), vol. secondo (1901−1907), Pubblicate a cura dell'Accademia Nazionale dei Lincei, Roma: Zanichelli Editore, pp. VI, 636.
• Levi-Civita, Tullio (1957), Opere Matematiche. Memorie e Note [Collected mathematical works. Memoirs and notes] (PDF) (in French and Italian), vol. terzo (1908−1916), Pubblicate a cura dell'Accademia Nazionale dei Lincei, Roma: Zanichelli Editore, pp. VI, 600.
• Levi-Civita, Tullio (1960), Opere Matematiche. Memorie e Note [Collected mathematical works. Memoirs and notes] (PDF) (in French and Italian), vol. quarto (1917−1928), Pubblicate a cura dell'Accademia Nazionale dei Lincei, Roma: Zanichelli Editore, pp. VI, 608.
• Levi-Civita, Tullio (1970), Opere Matematiche. Memorie e Note [Collected mathematical works. Memoirs and notes] (in French and Italian), vol. quinto (1929−1937), Pubblicate a cura dell'Accademia Nazionale dei Lincei, Roma: Zanichelli Editore, pp. VI, 670.
• Levi-Civita, Tullio (1970), Opere Matematiche. Memorie e Note [Collected mathematical works. Memoirs and notes] (in French and Italian), vol. sesto (1938−1941), Pubblicate a cura dell'Accademia Nazionale dei Lincei, Roma: Zanichelli Editore, pp. VI, 502.
• Levi-Civita, Tullio (2007) [1895], Pamphlets, mathematics, University of Michigan, retrieved 14 January 2017. A collection of some of his published papers (in their original typographical form), probably an unordered uncorrected collection of offprints.
See also
• Levi-Civita connection
• Levi-Civita crater
• Levi-Civita field
• Levi-Civita parallelogramoid
• Levi-Civita symbol
Notes
1. Tullio Levi-Civita. Nndb.com. Retrieved on 2011-08-14.
2. (Levi-Civita 1904)
3. O'Connor, John J.; Robertson, Edmund F., "Tullio Levi-Civita", MacTutor History of Mathematics Archive, University of St Andrews
4. Tullio Levi-Civita at the Mathematics Genealogy Project
5. Goodstein, Judith R. (2018). Einstein's Italian mathematicians : Ricci, Levi-Civita, and the birth of general relativity. American Mathematical Society. pp. 115–117. ISBN 978-1470428464.
6. (Ricci & Levi-Civita 1900).
7. (Levi-Civita 1917)
8. Levi-Civita, Tullio (2022). "Notion of Parallelism on a Generic Manifold and Consequent Geometrical Specification of the Riemannian Curvature". arXiv:2210.13239 [gr-qc].
9. Iurato, Giuseppe (2016). "On the history of Levi-Civita's parallel transport". arXiv:1608.04986. Bibcode:2016arXiv160804986I. {{cite journal}}: Cite journal requires |journal= (help)
10. C Cattani and M De Maria, Geniality and rigor: the Einstein – Levi-Civita correspondence (1915–1917), Riv. Stor. Sci. (2) 4 (1) (1996), 1–22; as cited in MacTutor archive.
11. Jackson, Allyn (1996). "Celebrating the 100th Annual Meeting of the AMS". In Case, Bettye Anne (ed.). A Century of Mathematical Meetings. Providence, RI: American Mathematical Society. pp. 10–18. ISBN 0-8218-0465-0.
12. "Tullio Levi-Civita". American Academy of Arts & Sciences. Retrieved 2023-05-04.
13. "APS Member History". search.amphilsoc.org. Retrieved 2023-05-04.
14. Rainich, G. Y. (1928). "Levi-Civita on Tensor Calculus" (PDF). Bull. Amer. Math. Soc. 34: 775–777. doi:10.1090/s0002-9904-1928-04644-x.
References
Biographical references
• "Professor T. Levi-Civita, Member of Vatican Academy," The Jewish Chronicle (UK), February 6, 1942.
General references
• Segre, Beniamino (1975), "Parole introduttive al Convegno", in Segre, Beniamino; Cattaneo, Carlo; Bompiani, Enrico; Colombo, Giuseppe; Finzi, Bruno; Graffi, Dario; Radicati di Brozolo, Luigi; Tricomi, Francesco Giacomo (eds.), Tullio Levi-Civita. Convegno internazionale celebrativo del centenario della nascita (Roma, 17–19 dicembre 1973) [Tullio Levi-Civita. International congress for the celebration of the centenary of his birth (Rome, 17–19 December 1973)], Atti dei Convegni Lincei (in Italian), vol. 8, Roma: Accademia Nazionale dei Lincei, pp. 171–177, ISSN 0391-805X. The "Inaugural address" (English translation of the contribution title) of Beniamino Segre, a commemoration describing briefly many aspects of the life and the work of Levi-Civita.
Scientific references
• Aczel, Amir D. (1999), God's Equation, New York: MJF Books, pp. 236, ISBN 1-56858-139-4.
• Graffi, Dario (1975), "L'Elettromagnetismo in Levi-Civita", in Segre, Beniamino; Cattaneo, Carlo; Bompiani, Enrico; Colombo, Giuseppe; Finzi, Bruno; Graffi, Dario; Radicati di Brozolo, Luigi; Tricomi, Francesco Giacomo (eds.), Tullio Levi-Civita. Convegno internazionale celebrativo del centenario della nascita (Roma, 17–19 dicembre 1973) [Tullio Levi-Civita. International congress for the celebration of the centenary of his birth (Rome, 17–19 December 1973)], Atti dei Convegni Lincei (in Italian), vol. 8, Roma: Accademia Nazionale dei Lincei, pp. 171–177, ISSN 0391-805X. "Electromagnetism in the work of Levi-Civita" (English translation of the contribution title) is a survey of some of the works of Levi-Civita on the theory of electromagnetism.
• Loinger, Angelo (2007). "Einstein, Levi-Civita, and Bianchi relations". arXiv:physics/0702244..
Publications dedicated to his memory
• Segre, Beniamino; Cattaneo, Carlo; Bompiani, Enrico; Colombo, Giuseppe; Finzi, Bruno; Graffi, Dario; Radicati di Brozolo, Luigi; Tricomi, Francesco Giacomo, eds. (1975), Tullio Levi-Civita. Convegno internazionale celebrativo del centenario della nascita (Roma, 17–19 dicembre 1973) [Tullio Levi-Civita. International congress for the celebration of the centenary of his birth (Rome, 17–19 December 1973)], Atti dei Convegni Lincei (in English, French, Italian, and Russian), vol. 8, Roma: Accademia Nazionale dei Lincei, p. 316, ISSN 0391-805X.
External links
• Media related to Tullio Levi-Civita at Wikimedia Commons
• Tullio Levi-Civita at the Mathematics Genealogy Project
• Scienceworld biography
• Another short biography
• An Italian short biography of Tullio Levi-Civita in Edizione Nazionale Mathematica Italiana online.
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Tuna Altınel
Tuna Altınel is a Turkish mathematician, born February 12, 1966 in Istanbul, who has worked at the University Lyon 1 in France since 1996.[1] He is a specialist in group theory and mathematical logic. With Alexandre Borovik and Gregory Cherlin, he proved a major case of the Cherlin–Zilber conjecture.[2]
In the political sphere, Altınel is active in the Academics for Peace movement, which supports a peaceful resolution of the conflict in south-eastern Turkey, and calls for the human rights of the civilian population to be respected.[3] Accused by the Turkish authorities of membership in a terrorist organization, Altınel has been imprisoned since May 11, 2019, at the Kepsut prison in Turkey.[4]
Education and career
After undergraduate studies in mathematics and computer science at Boğaziçi University, Istanbul, Altınel received his doctorate from Rutgers University (New Jersey, USA) under the direction of Gregory Cherlin.[5] In 1996 he joined the department of mathematics of the university Lyon-1, as maître de conférences, and completed his French habilitation in 2001.[1]
Altınel has written 26 mathematical articles, principally on the subject of groups in model theory, more particularly groups of finite Morley rank and the Cherlin–Zilber Algebraicity Conjecture, concerning the structure of the simple groups of finite Morley rank. He is joint author with Alexandre Borovik and Gregory Cherlin of a book in which this conjecture is proved in the case of infinite 2-rank, after the development of a body of machinery analogous to certain chapters of finite simple group theory.[2]
Altınel's doctoral advisees include Éric Jaligot, winner of the 2000 Sacks Prize,[6] a prize given annually for an outstanding doctoral thesis in mathematical logic[7] (doctoral thesis supervised jointly by Tuna Altınel and Bruno Poizat[8]). He is active in the domain of scientific cooperation with Turkey; in particular, he was an organizer of an international mathematics conference held in Istanbul in 2016 in honor of Alexandre Borovik and Ali Nesin (Leelavati prize winner, 2018).[9]
Political activities
Overview
Altınel has been an active supporter of a peaceful resolution of the conflict in southeastern Turkey and of human rights and civil liberties in Turkey.[10]
With regard to the Kurdish conflict in southeastern Turkey, he was one of 116 academics who signed a 2003 letter in support of a peaceful resolution of that conflict,[11] among the first group of signatories of a similar peace petition in January 2016 that garnered 1128 signatures at the time of its promulgation under the title "We will not be parties to this crime," [12] among the 132 intellectuals calling for assistance to those wounded in the conflict at Cizre,[13] and one of 170 academics to sign a letter in 2018 opposing the Afrin operation.[14] On February 21, 2019, he acted as translator for a former member of parliament of the Peoples' Democratic Party (HDP) at a public meeting in Lyon, France, in which a documentary on the Cizre massacres was shown, followed by a discussion.[15]
With the resumption of active conflict in August 2015 following a period of relative calm, Altınel reached out to the affected community and began to visit the areas involved in September 2015.[10] His own account of these activities is quoted below, from subsequent court testimony.
With the trials of the signatories of the January 2016 petition and the broader wave of repression following the attempted coup of July 2016, described in more detail below, questions of academic freedom and freedom of speech become more prominent. Altınel's actions in this direction include
• a petition responding to the suicide of Mehmet Fatih Traş, an academic fired for his involvement with the peace petition (February 2017)[16]
• denunciation of the role of the Turkish research council TÜBİTAK in the state of emergency following the attempted coup d'état of 2016 (April 2017); the CNRS Scientific Council voted unanimously to recommend to the CNRS to reconsider its agreements concerning collaboration with TÜBİTAK (April 24–25, 2017).[17]
• publication of a review article on the trials of the Academics for Peace entitled "Les procès contre les Universitaires pour la paix : extraits d’une comédie politico-juridique (The trials of the Academics for Peace: scenes from a politico-juridical spectacle)".[18]
• petition in support of Academic for Peace Füsün Üstel[19]
These activities have led to two separate court cases against Altınel in Turkey and his social media postings have been used to justify the second of these cases.[20]
January 2016 petition and Academics for Peace
Altınel was one of the first signatories of the January 2016 peace petition entitled "We will not be parties to this crime!", which was promulgated by the Academics for Peace on January 11, 2016.[12]
The following day, President Erdoğan publicly criticized the signatories, and within a few days 27 had been arrested."[21] At the same time foreign reaction was strongly supportive of the signatories.[22]
The peace petition ultimately garnered 2212 signatures of academics, largely in Turkey.[3] Altınel is one of over 750 signatories from the first group of 1128 such who have been prosecuted or sentenced as individuals for that act under Turkish Anti-Terrorism legislation, through June 2019,[23] on a charge of "propaganda in support of a terrorist organization." Since 2016 Altınel has been an active and vocal supporter both of the content of this petition and of the civil rights of its signers.
In the second hearing in his case, February 28, 2019, at the 29th Central Criminal Court, Çağlayan Courthouse, Istanbul, Altınel testified that he had aided civilian victims of military operations that took place in the towns placed under military curfew:[24]
Since September 2015, I have traveled several times to a number of provinces, including some of those mentioned in the Peace Petition which I signed. ... I carried bag upon bag of provisions to help the victims of destruction and forced migration, I spoke with those who had lost their homes and relatives. I did all of this on my own initiative, and my principle was as follows: If every Turkish citizen will do what I do, we will come closer to peace. You can find the traces of my efforts where I sojourned in the towns of Sur, Nusaybin, Cizre, Hakkari, and Yüksekova. The Prosecutor may use this as evidence against me. ... I did not simply sign the Peace Petition. I thought about it, felt it, lived it. I wrote that text.[25] I stand behind every sentence.
The sentencing hearing for Altınel's trial for "propaganda on behalf of a terrorist organization" in the context of the Academics for Peace Trials is scheduled for July 16, 2019.[26]
2019 charge and imprisonment
On April 12, 2019, on arriving for a visit to Turkey, Altınel's passport was confiscated at the airport. On May 10 he requested a new passport at the Balıkesir prefecture and was taken into custody for interrogation and placed in pre-trial detention on the following day. It was learned later that a new charge had been filed against him on April 30, 2019 at the prosecutor general's office in Balıkesir.[27]
This new charge is "membership in a terrorist organization",[28] based on his participation on February 21, 2019, at a public meeting in Villeurbanne, near Lyon, France. This meeting was organized by the local Kurdish Society; a documentary was shown on the subject of the Cizre massacres and a discussion was held with a former member of the Turkish parliament, Faysal Sarıyıldız (HDP), now in exile.[32] At that public meeting, Altınel acted as translator for the former MP.[15]
On May 8 Füsun Üstel was incarcerated and began serving a 15-month sentence for signing the peace petition of January 2016. Altınel was arrested on May 11.[33] After his first hearing on the new charge was scheduled for July 30, 2019, he was released.[34]
Press reports
Altınel's May 11 arrest was widely reported in the press, notably in France and in Turkey.
Some early reports of the arrest in Turkey quoting variously from Altınel's lawyer or Academics for Peace put the case in the context of the Academics for Peace trials and the conference held in Lyon, France.[35] Other reports originating with the İhlas News Agency and reported on Habertürk and elsewhere described the case as the capture of a wanted terrorist; one of these reports stated that an anti-terrorist operation captured five members of Gülen Movement and the Kurdistan Workers' Party (PKK), listing Altınel's arrest as the fifth.[36]
The first article in France, in Mediapart,[37] appeared that same day and was followed rapidly by articles in Le Progrès, Le Monde, 20 minutes, Lyon Capitale, Lyon Mag, Le Figaro Étudiant, Le Figaro, Le Canard enchaîné, Libération, and L’Humanité.[38] Altınel was featured as L’Humanité's Man of the Day on May 16, 2019. Euronews TV reported on the case on May 30, 2019.[39]
Official reactions
Less than weeks after the confiscation of Altınel's passport, on April 23, 2019, the French Applied Mathematics Society and the French Mathematical Society wrote jointly to President Macron of France.[40]
On May 11, the day of Altınel's arrest, the Turkish Consul General in Lyon, Mehmet Özgür Çakar, stated "Tuna Altınel organized, and moderated, a meeting in Lyon consisting entirely of propaganda in favor of the PKK. ... It is possible that this had a negative effect on his situation."[41] The consul also noted that the PKK remains classified a terrorist group by Ankara, the United States, and the European Union.[42] The French Ministry of Europe and Foreign Affairs expressed its "disquiet" on May 13, 2019.[43] A support committee formed at Lyon created a website to document the evolution of the affair, and on May 23 the committee launched a petition in favor of the liberation of Altınel, with over 6000 signatories as of June, 2019, predominantly academics, along with approximately 60 members of the French National Assembly.[44]
Professional societies from a number of countries, including mathematics societies in the United States, France, Great Britain, Germany Austria, Italy, and Belgium, as well as the European Mathematical Society, the Association for Symbolic Logic, and the Committee of Concerned Scientists have issued statements in support of Altınel.[45]
National Assembly, France
On June 11, 2019, the French mathematician and politician Cédric Villani (LREM), Member of Parliament for Essonne's fifth district and Fields medalist, who is a colleague and an outspoken supporter of Altınel,[46] posed a question on the subject during a session of the National Assembly to the Minister for Europe and Foreign Affairs Jean-Yves Le Drian, who stated that the government was committed to doing "everything in its power" in favor of his liberation, notably on the occasion of his June 13 visit to Turkey to consult his counterpart there.[47]
See also
• Stable group,
• Presidency of Recep Tayyip Erdoğan, State of emergency and purges
• Censorship in Turkey: Article 301
• Kurdish–Turkish conflict (2015–present)
References
1. Tuna Altınel, CV
2. Altinel, Tuna; Borovik, Alexandre; Cherlin, Gregory (2008). Simple Groups of Finite Morley Rank. Mathematical Surveys and Monographs. Vol. 145. Providence, RI: American Mathematical Society.; Macpherson, Dugald (2010). "Simple Groups of Finite Morley Rank (Book Review)". Bulletin of the American Mathematical Society. doi:10.1090/S0273-0979-10-01287-5.
3. Peace Petition: "We will not be parties to this crime!", Academics for Peace
4. Article on Altınel's first hearing, on the Bianet website, June 10, 2019
5. Tuna Altınel, on the website Mathematics Genealogy Project.
6. List of Sacks Prize winners, on the Association for Symbolic Logic website.
7. Sacks Prize conditions, on the Association for Symbolic Logic website.
8. In Memoriam: ÉRIC JALIGOT, Adrien Deloro, Bull. Symbolic Logic 20, pp. 103–104, March 2014
9. 73 participants from 9 countries. List of participants, BN-Pair Conference in honor of Alexandre Borovik and Ali Nesin
10. "A man who has done his duty for this land, and for peace," Nurcan Baysal, Ahval News website, June 25, 2019; Gözlerini, aklını, kalbini kapatmayanlardan biri: Tuna Altınel (One who does not shut his eyes, mind, or heart: Tuna Altınel), Nurcan Baysal, Ahval News website, June 18, 2019 (Turkish) "August 2015;" "September 2017;" "never stopped showing his support";
11. "116 University professors call for peace", Bianet website, March 13, 2003] (letter signed also by Füsün Üstel)
12. Text of the petition "We will not be parties to this crime:, Website of the Academics for Peace; list of the first 1128 signatories of the petition (one missing: accessed June 20, 2019) (Turkish)
13. "Aydinlar_ve_sanatcilardan_devlete_Cizre_cagrisi.html "Intellectuals and artists call on the state on behalf of Cizre", Cumhuriyet, February 1, 2016 (Turkish); "We are ready to save those in Cizre if you won't," Bianet website, February 2, 2016; Cizre’de İnsanlık Ölüyor! "(People are dying in Cizre!)," Human Rights Association, Turkey, February 1, 2016 (Turkish)
14. "Afrin Operasyonuna hayır diyen 170 kişilik ihanet listesi! (The list of 170 traitors who say no to the Afrin Operation)," Akasyam News website, January 28, 2018, "Investigation launched against signatories of Afrin Letter"
15. Cédric Villani, June 11, French National Assembly, in French with English translation.
16. Among the first signatories: "Turquie : appel pour la liberté des universitaires (Turkey: a petition for academic freedom)", Liberation, December 5, 2017; "Universitaires pour la Paix • Une pétition à diffuser (Academics for Peace: a petition to be circulated," Kedistan website, March 1, 2017; "Mehmet Fatih Traş suicidé par le régime turc (Mehmet Fatih Traş suicided by the Turkish regime)," Kedistan website, February 26, 2017
17. "Tübitak: organisme scientifique ou outil politique? (Tübitak: scientific organization or political instrument?)", Mediapart, April 17, 2017 (French). "This appeal by the Turkish mathematician Tuna Altınel ..."; "Recommandation: Les relations avec le Conseil de la recherche scientifique et technique de Turquie (TÜBİTAK)" (Recommendation: Relations with the scientific and technical research council of Turkey (TÜBİTAK)), CNRS Scientific Council, April 24--25, 2017
18. Tuna Altınel, "Les procès contre les Universitaires pour la paix : extraits d’une comédie politico-juridique (The trials of the Academics for Peace: scenes from a politico-juridical spectacle)", chapter in Table of Contents, "Liberté(s)! En Turquie ? En Méditerranée ! (Freedom(s)! In Turkey? In the Mediterranean!"), Mathieu Touzeil-Divina, L'Epitoge, June 2018, Revue Méditerrqanéenne de Droit Public Vol. 9, ISBN 979-1-0926-8433-9, Part I: Freedom(s) of speech for our Turkish colleagues!
19. One of first 63 signatories; Entry by Igor Babou on seenthis.net, cached at
20. "PKK propagandası yapan akademisyen tutuklandı (Academic who made propaganda for the PKK arrested)" IHA, May 11, 2019 (Turkish): "sosyal medya araştırmasında" (investigation of social media)
21. President Erdoğan: State of Emergency: "terrorist propaganda", "darkest of people", "enemy of the state", "pit of treachery"; "Turkish Academe Under Attack," Elizabeth Redden, Inside Higher Ed, February 12, 2016: "treason" Turkish prosecutors to investigate academics over Erdoğan petition, Guardian news site, January 14, 2019: "treason", "fifth columns"
22. "Turkish Academe Under Attack," Elizabeth Redden, Inside Higher Ed, February 12, 2016: "chilling effect", "extremely worrying", "dismayed ... grave concern; Academics for peace: Foreign reactions (French): U.S. Ambassador, Germany, professional associations, international associations, PEN, etc.
23. Academics for Peace—Hearing Statistics, accessed June 19, 2019
24. Translation of statement by Altınel, Feb. 28, 2019
25. This sentence follows one that is obviously metaphorical; it may be intended in the same spirit.
26. Lyon: Tuna Altinel, the mathematician arrested en Turkey, judged in July, Justin Boche, Lyon Capitale, June 10, 2019 (French)
27. Article: Academic who acted as translator at PKK [sic] conference captured and arrested, Balıkesir Press Agency, May 11, 2019 (Turkish)
28. Either the [PKK] or the legally registered Lyon Society of Kurds; not specified in published legal documents
29. "Sirnak, a city leveled, symbol of the war between Turkey and the Kurds", on France 24, November 18, 2016.
30. Statement by President Erdoğan regarding Faysal Sarıyıldız, reported in the newspaper Sabah, April 19, 2016. (Turkish)
31. Interview with Faysal Sarıyıldız, in L'Humanité, July 12, 2016; Interview with Faysal Sarıyıldız, in Mediapart, August 26, 2016. (Both in French)
32. Parliamentary deputy from the province of Şırnak from 2011 to 2017, in exile since 2016. After the resumption of the Turkish-Kurd conflict in summer 2015,[29] and the December 2015 Cizre basement massacre, accused by Erdoğan of transporting firearms intended for terrorists in his car,[30] a charge he has characterized as "an odious fabrication.".[31] Honorary citizen of the town of Champigny-sur-Marne (94) and frequently called on to discuss the conflict with the Kurds in Turkey and the Cizre basement massacres.
33. Turkey, France to discuss Syria, strained bilateral affairs
34. 2nd Central Criminal Court in Balıkesir. "First hearing for an Academic for Peace", Bianet website, June 10, 2019. (Turkish)
35. "Barış Akademisyeni Tuna Altınel tutuklandı" (Academic for Peace Tuna Altinel Arrested), Evrensel, May 11, 2019, "Barış Akademisyeni Tuna Altınel tutuklandı" (Academic for Peace Tuna Altınel tutuklandı" (Academic for Peace Tuna Altinel Arrested), Cumhuriyet, May 11, 2019, "Akademisyen Tuna Altinel tutuklandi" (Academic Tuna Altınel arrested), Nüpelö May 11, 2019
36. Arrest of an academic making propaganda for the PKK", Habertürk, May 11, 2019, Source: İHA; Balıkesir merkezli FETÖ ve PKK/KCK operasyonu: 5 kişi yakalandı (FETÖ PKK Operation Based in Balıkesır: 5 People Captured, May 11, 2019. Habertürk, May 11, 2019, Source: İHA. (Turkish)
37. Arrest of a Mathematician from the university Lyon-1 in Turkey Mediapart, May 11, 2019. (French)
38. Survey of press with respect to the arrest of Tuna Altınel, support committee website, Lyon, accessed June 20, 2019
39. Video: Support for an imprisoned Turkish academic, Euronews, May 30, 2019; 2:20
40. Letter to President Macron, April 23, 2019, from the Presidents of the French Mathematical Society and the Society for Applied and Industrial Mathematics (France) (French)
41. "President Erdoğan's dirty war: Thirty years of conflict", in Le Monde diplomatique, July 2016.
42. Lyon: Movement in solidarity with an academic imprisoned in Turkey, Le Figaro, May 14, 2019.
43. "Turkey: Freedom of Speech", French Ministry of Europe and Foreign Affairs
44. "Free Tuna Altinel," on-line petition
45. List of statements of support by professional institutions in support of Altinel; website of the support committee for Tuna Altınel, Lyon, accessed June 20, 2019.
46. Tweet, Cédric Villani, May 23, 2019
47. Le Drian promises to make "every effort" to achieve the release of a professor from Lyon in Turkey, Le Figaro, June 11, 2019; France is making “every effort” to achieve the release of a Turkish mathematician from the University of Lyon, on Local French television, June 11, 2019.
External links
• Tuna Altınel: CV
• Altinel, Tuna; Borovik, Alexandre; Cherlin, Gregory (2008), Simple Groups of Finite Morley Rank, Mathematical Surveys and Monographs, vol. 145, Providence, RI: American Mathematical Society, pp. xx+556, doi:10.1090/surv/145, ISBN 978-0-8218-4305-5, MR 2400564
• Altinel Support Committee, Lyon
• Webpage, Academics for Peace
• Observations from 2ith February 2019, in the Caglayan Courts ("The Turkish State vs. Academics for Peace"), David Bradley-Williams, April/May 2019
• Translation of statement by Altınel, Feb. 28, 2019, Çağlayan Courthouse
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Tunnel number
In mathematics, the tunnel number of a knot, as first defined by Bradd Clark, is a knot invariant, given by the minimal number of arcs (called tunnels) that must be added to the knot so that the complement becomes a handlebody. The tunnel number can equally be defined for links. The boundary of a regular neighbourhood of the union of the link and its tunnels forms a Heegaard splitting of the link exterior.
Examples
• The unknot is the only knot with tunnel number 0.
• The trefoil knot has tunnel number 1. In general, any nontrivial torus knot has tunnel number 1.[1]
Every link L has a tunnel number. This can be seen, for example, by adding a 'vertical' tunnel at every crossing in a diagram of L. It follows from this construction that the tunnel number of a knot is always less than or equal to its crossing number.
References
• Clark, Bradd (1980), "The Heegaard Genus Of Manifolds Obtained By Surgery On Links And Knots", International Journal of Mathematics and Mathematical Sciences, 3 (3): 583–589, doi:10.1155/S0161171280000440
• Boileau, Michel; Lustig, Martin; Moriah, Yoav (1994), "Links with super-additive tunnel number", Mathematical Proceedings of the Cambridge Philosophical Society, 115 (1): 85–95, Bibcode:1994MPCPS.115...85B, doi:10.1017/S0305004100071930, MR 1253284.
• Kobayashi, Tsuyoshi; Rieck, Yo'av (2006), "On the growth rate of the tunnel number of knots", Journal für die Reine und Angewandte Mathematik, 2006 (592): 63–78, arXiv:math/0402025, doi:10.1515/CRELLE.2006.023, MR 2222730.
• Scharlemann, Martin (1984), "Tunnel number one knots satisfy the Poenaru conjecture", Topology and Its Applications, 18 (2–3): 235–258, doi:10.1016/0166-8641(84)90013-0, MR 0769294.
• Scharlemann, Martin (2004), "There are no unexpected tunnel number one knots of genus one", Transactions of the American Mathematical Society, 356 (4): 1385–1442, arXiv:math/0106017, doi:10.1090/S0002-9947-03-03182-9, MR 2034312.
Knot theory (knots and links)
Hyperbolic
• Figure-eight (41)
• Three-twist (52)
• Stevedore (61)
• 62
• 63
• Endless (74)
• Carrick mat (818)
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• (−2,3,7) pretzel (12n242)
• Whitehead (52
1
)
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2
)
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Satellite
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Torus
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1
)
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1
)
• Solomon's (42
1
)
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1. Boileau, Michel; Rost, Markus; Zieschang, Heiner (1 January 1988). "On Heegaard decompositions of torus knot exteriors and related Seifert fibre spaces". Mathematische Annalen. 279 (3): 553–581. doi:10.1007/BF01456287. ISSN 1432-1807.
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Wikipedia
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Tunnell's theorem
In number theory, Tunnell's theorem gives a partial resolution to the congruent number problem, and under the Birch and Swinnerton-Dyer conjecture, a full resolution.
Congruent number problem
Main article: Congruent number problem
The congruent number problem asks which positive integers can be the area of a right triangle with all three sides rational. Tunnell's theorem relates this to the number of integral solutions of a few fairly simple Diophantine equations.
Theorem
For a given square-free integer n, define
${\begin{aligned}A_{n}&=\#\{(x,y,z)\in \mathbb {Z} ^{3}\mid n=2x^{2}+y^{2}+32z^{2}\},\\B_{n}&=\#\{(x,y,z)\in \mathbb {Z} ^{3}\mid n=2x^{2}+y^{2}+8z^{2}\},\\C_{n}&=\#\{(x,y,z)\in \mathbb {Z} ^{3}\mid n=8x^{2}+2y^{2}+64z^{2}\},\\D_{n}&=\#\{(x,y,z)\in \mathbb {Z} ^{3}\mid n=8x^{2}+2y^{2}+16z^{2}\}.\end{aligned}}$
Tunnell's theorem states that supposing n is a congruent number, if n is odd then 2An = Bn and if n is even then 2Cn = Dn. Conversely, if the Birch and Swinnerton-Dyer conjecture holds true for elliptic curves of the form $y^{2}=x^{3}-n^{2}x$, these equalities are sufficient to conclude that n is a congruent number.
History
The theorem is named for Jerrold B. Tunnell, a number theorist at Rutgers University, who proved it in Tunnell (1983).
Importance
The importance of Tunnell's theorem is that the criterion it gives is testable by a finite calculation. For instance, for a given $n$, the numbers $A_{n},B_{n},C_{n},D_{n}$ can be calculated by exhaustively searching through $x,y,z$ in the range $-{\sqrt {n}},\ldots ,{\sqrt {n}}$.
See also
• Birch and Swinnerton-Dyer conjecture
• Congruent number
References
• Koblitz, Neal (2012), Introduction to Elliptic Curves and Modular Forms, Graduate Texts in Mathematics (Book 97) (2nd ed.), Springer-Verlag, ISBN 978-1-4612-6942-7
• Tunnell, Jerrold B. (1983), "A classical Diophantine problem and modular forms of weight 3/2", Inventiones Mathematicae, 72 (2): 323–334, doi:10.1007/BF01389327, hdl:10338.dmlcz/137483
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Wikipedia
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Forbidden subgraph problem
In extremal graph theory, the forbidden subgraph problem is the following problem: given a graph $G$, find the maximal number of edges $\operatorname {ex} (n,G)$ an $n$-vertex graph can have such that it does not have a subgraph isomorphic to $G$. In this context, $G$ is called a forbidden subgraph.[1]
An equivalent problem is how many edges in an $n$-vertex graph guarantee that it has a subgraph isomorphic to $G$?[2]
Definitions
The extremal number $\operatorname {ex} (n,G)$ is the maximum number of edges in an $n$-vertex graph containing no subgraph isomorphic to $G$. $K_{r}$ is the complete graph on $r$ vertices. $T(n,r)$ is the Turán graph: a complete $r$-partite graph on $n$ vertices, with vertices distributed between parts as equally as possible. The chromatic number $\chi (G)$ of $G$ is the minimum number of colors needed to color the vertices of $G$ such that no two adjacent vertices have the same color.
Upper bounds
Turán's theorem
See also: Turán's theorem
Turán's theorem states that for positive integers $n,r$ satisfying $n\geq r\geq 3$,[3] $ \operatorname {ex} (n,K_{r})=\left(1-{\frac {1}{r-1}}\right){\frac {n^{2}}{2}}.$
This solves the forbidden subgraph problem for $G=K_{r}$. Equality cases for Turán's theorem come from the Turán graph $T(n,r-1)$.
This result can be generalized to arbitrary graphs $G$ by considering the chromatic number $\chi (G)$ of $G$. Note that $T(n,r)$ can be colored with $r$ colors and thus has no subgraphs with chromatic number greater than $r$. In particular, $T(n,\chi (G)-1)$ has no subgraphs isomorphic to $G$. This suggests that the general equality cases for the forbidden subgraph problem may be related to the equality cases for $G=K_{r}$. This intuition turns out to be correct, up to $o(n^{2})$ error.
Erdős–Stone theorem
See also: Erdős–Stone theorem
Erdős–Stone theorem states that for all positive integers $n$ and all graphs $G$,[4] $ \operatorname {ex} (n,G)=\left(1-{\frac {1}{\chi (G)-1}}+o(1)\right){\binom {n}{2}}.$
When $G$ is not bipartite, this gives us a first-order approximation of $\operatorname {ex} (n,G)$.
Bipartite graphs
For bipartite graphs $G$, the Erdős–Stone theorem only tells us that $\operatorname {ex} (n,G)=o(n^{2})$. The forbidden subgraph problem for bipartite graphs is known as the Zarankiewicz problem, and it is unsolved in general.
Progress on the Zarankiewicz problem includes following theorem:
Kővári–Sós–Turán theorem. For every pair of positive integers $s,t$ with $t\geq s\geq 1$, there exists some constant $C$ (independent of $n$) such that $ \operatorname {ex} (n,K_{s,t})\leq Cn^{2-{\frac {1}{s}}}$ for every positive integer $n$.[5]
Another result for bipartite graphs is the case of even cycles, $G=C_{2k},k\geq 2$. Even cycles are handled by considering a root vertex and paths branching out from this vertex. If two paths of the same length $k$ have the same endpoint and do not overlap, then they create a cycle of length $2k$. This gives the following theorem.
Theorem (Bondy and Simonovits, 1974). There exists some constant $C$ such that $ \operatorname {ex} (n,C_{2k})\leq Cn^{1+{\frac {1}{k}}}$ for every positive integer $n$ and positive integer $k\geq 2$.[6]
A powerful lemma in extremal graph theory is dependent random choice. This lemma allows us to handle bipartite graphs with bounded degree in one part:
Theorem (Alon, Krivelevich, and Sudakov, 2003). Let $G$ be a bipartite graph with vertex parts $A$ and $B$ such that every vertex in $A$ has degree at most $r$. Then there exists a constant $C$ (dependent only on $G$) such that $ \operatorname {ex} (n,G)\leq Cn^{2-{\frac {1}{r}}}$for every positive integer $n$.[7]
In general, we have the following conjecture.:
Rational Exponents Conjecture (Erdős and Simonovits). For any finite family ${\mathcal {L}}$ of graphs, if there is a bipartite $L\in {\mathcal {L}}$, then there exists a rational $\alpha \in [0,1)$ such that $\operatorname {ex} (n,{\mathcal {L}})=\Theta (n^{1+\alpha })$.[8]
A survey by Füredi and Simonovits describes progress on the forbidden subgraph problem in more detail.[8]
Lower bounds
There are various techniques used for obtaining the lower bounds.
Probabilistic method
While this method mostly gives weak bounds, the theory of random graphs is a rapidly developing subject. It is based on the idea that if we take a graph randomly with a sufficiently small density, the graph would contain only a small number of subgraphs of $G$ inside it. These copies can be removed by removing one edge from every copy of $G$ in the graph, giving us a $G$ free graph.
The probabilistic method can be used to prove $\operatorname {ex} (n,G)\geq cn^{2-{\frac {v(G)-2}{e(G)-1}}}$where $c$ is a constant only depending on the graph $G$.[9] For the construction we can take the Erdős-Rényi random graph $G(n,p)$, that is the graph with $n$ vertices and the edge been any two vertices drawn with probability $p$, independently. After computing the expected number of copies of $G$ in $G(n,p)$ by linearity of expectation, we remove one edge from each such copy of $G$ and we are left with a $G$-free graph in the end. The expected number of edges remaining can be found to be $\operatorname {ex} (n,G)\geq cn^{2-{\frac {v(G)-2}{e(G)-1}}}$ for a constant $c$ depending on $G$. Therefore, at least one $n$-vertex graph exists with at least as many edges as the expected number.
This method can also be used to find the constructions of a graph for bounds on the girth of the graph. The girth, denoted by $g(G)$, is the length of the shortest cycle of the graph. Note that for $g(G)>2k$, the graph must forbid all the cycles with length less than equal to $2k$. By linearity of expectation,the expected number of such forbidden cycles is equal to the sum of the expected number of cycles $C_{i}$ (for $i=3,...,n-1,n$.). We again remove the edges from each copy of a forbidden graph and end up with a graph free of smaller cycles and $g(G)>2k$, giving us $c_{0}n^{1+{\frac {1}{2k-1}}}$ edges in the graph left.
Algebraic constructions
For specific cases, improvements have been made by finding algebraic constructions. A common feature for such constructions is that it involves the use of geometry to construct a graph, with vertices representing geometric objects and edges according to the algebraic relations between the vertices. We end up with no subgraph of $G$, purely due to purely geometric reasons, while the graph has a large number of edges to be a strong bound due to way the incidences were defined. The following proof by Erdős, Rényi, and Sős[10] establishing the lower bound on $\operatorname {ex} (n,K_{2,2})$ as$\left({\frac {1}{2}}-o(1)\right)n^{3/2}.$, demonstrates the power of this method.
First, suppose that $n=p^{2}-1$ for some prime $p$. Consider the polarity graph $G$ with vertices elements of $\mathbb {F} _{p}^{2}-\{0,0\}$ and edges between vertices $(x,y)$ and $(a,b)$ if and only if $ax+by=1$ in $\mathbb {F} _{p}$. This graph is $K_{2,2}$-free because a system of two linear equations in $\mathbb {F} _{p}$ cannot have more than one solution. A vertex $(a,b)$ (assume $b\neq 0$) is connected to $\left(x,{\frac {1-ax}{b}}\right)$ for any $x\in \mathbb {F} _{p}$, for a total of at least $p-1$ edges (subtracted 1 in case $(a,b)=\left(x,{\frac {1-ax}{b}}\right)$). So there are at least ${\frac {1}{2}}(p^{2}-1)(p-1)=\left({\frac {1}{2}}-o(1)\right)p^{3}=\left({\frac {1}{2}}-o(1)\right)n^{3/2}$ edges, as desired. For general $n$, we can take $p=(1-o(1)){\sqrt {n}}$ with $p\leq {\sqrt {n+1}}$ (which is possible because there exists a prime $p$ in the interval$[k-k^{0.525},k]$ for sufficiently large $k$[11]) and construct a polarity graph using such $p$, then adding $n-p^{2}+1$ isolated vertices, which do not affect the asymptotic value.
The following theorem is a similar result for $K_{3,3}$.
Theorem (Brown, 1966). $\operatorname {ex} (n,K_{3,3})\geq \left({\frac {1}{2}}-o(1)\right)n^{5/3}.$[12]
Proof outline.[13] Like in the previous theorem, we can take $n=p^{3}$ for prime $p$ and let the vertices of our graph be elements of $\mathbb {F} _{p}^{3}$. This time, vertices $(a,b,c)$ and $(x,y,z)$ are connected if and only if $(x-a)^{2}+(y-b)^{2}+(z-c)^{2}=u$ in $\mathbb {F} _{p}$, for some specifically chosen $u$. Then this is $K_{3,3}$-free since at most two points lie in the intersection of three spheres. Then since the value of $(x-a)^{2}+(y-b)^{2}+(z-c)^{2}$ is almost uniform across $\mathbb {F} _{p}$, each point should have around $p^{2}$ edges, so the total number of edges is $\left({\frac {1}{2}}-o(1)\right)p^{2}\cdot p^{3}=\left({\frac {1}{2}}-o(1)\right)n^{5/3}$.
However, it remains an open question to tighten the lower bound for $\operatorname {ex} (n,K_{t,t})$ for $t\geq 4$.
Theorem (Alon et al., 1999) For $t\geq (s-1)!+1$, $\operatorname {ex} (n,K_{s,t})=\Theta (n^{2-{\frac {1}{s}}}).$[14]
Randomized Algebraic constructions
This technique combines the above two ideas. It uses random polynomial type relations when defining the incidences between vertices, which are in some algebraic set. Using this technique to prove the following theorem.
Theorem: For every $s\geq 2$, there exists some $t$ such that $\operatorname {ex} (n,K_{s,t})\geq \left({\frac {1}{2}}-o(1)\right)n^{1-{\frac {1}{s}}}$.
Proof outline: We take the largest prime power $q$ with $q^{s}\leq n$. Due to the prime gaps, we have $q=(1-o(1))n^{\frac {1}{s}}$. Let $f\in \mathbb {F} _{q}[x_{1},x_{2},\cdots ,x_{s},y_{1},y_{2},\cdots ,y_{s}]_{\leq d}$ be a random polynomial in $\mathbb {F} _{q}$ with degree at most $d=s^{2}$ in $X=(X_{1},X_{2},...,X_{s})$ and $Y=(Y_{1},Y_{2},...,Y_{s})$ and satisfying $f(X,Y)=f(Y,X)$. Let the graph $G$ have the vertex set $\mathbb {F} _{q}^{s}$ such that two vertices $x,y$ are adjacent if $f(x,y)=0$.
We fix a set $U\subset \mathbb {F} _{q}^{s}$, and defining a set $Z_{U}$ as the elements of $\mathbb {F} _{q}^{s}$ not in $U$ satisfying $f(x,u)=0$ for all elements $u\in U$. By the Lang–Weil bound, we obtain that for $q$ sufficiently large enough, we have $|Z_{U}|\leq C$ or $|Z_{U}|>{\frac {q}{2}}$ for some constant $C$.Now, we compute the expected number of $U$ such that $Z_{U}$ has size greater than $C$, and remove a vertex from each such $U$. The resulting graph turns out to be $K_{s,C+1}$ free, and at least one graph exists with the expectation of the number of edges of this resulting graph.
Supersaturation
Supersaturation refers to a variant of the forbidden subgraph problem, where we consider when some $h$-uniform graph $G$ contains many copies of some forbidden subgraph $H$. Intuitively, one would expect this to once $G$ contains significantly more than $\operatorname {ex} (n,H)$ edges. We introduce Turán density to formalize this notion.
Turán density
The Turán density of a $h$-uniform graph $G$ is defined to be
$\pi (G)=\lim _{n\to \infty }{\frac {\operatorname {ex} (n,G)}{\binom {n}{h}}}.$
It is true that ${\frac {\operatorname {ex} (n,G)}{\binom {n}{h}}}$ is in fact positive and monotone decreasing, so the limit must therefore exist. [15]
As an example, Turán's Theorem gives that $\pi (K_{r+1})=1-{\frac {1}{r}}$, and the Erdős–Stone theorem gives that $\pi (G)=1-{\frac {1}{\chi (H)-1}}$. In particular, for bipartite $G$, $\pi (G)=0$. Determining the Turán density $\pi (H)$ is equivalent to determining $\operatorname {ex} (n,G)$ up to an $o(n^{2})$ error.[16]
Supersaturation Theorem
Consider an $h$-uniform hypergraph $H$ with $v$ vertices. The supersaturation theorem states that for every $\epsilon >0$, there exists a $\delta >0$ such that if $G$ is a graph on $n$ vertices and at least $(\pi (H)+\epsilon ){\binom {n}{2}}$ edges for $n$ sufficiently large, then there are at least $\delta n^{v(H)}$ copies of $H$. [17]
Equivalently, we can restate this theorem as the following: If a graph $G$ with $n$ vertices has $o(n^{v(H)})$ copies of $H$, then there are at most $\pi (H){\binom {n}{2}}+o(n^{2})$ edges in $G$.
Applications
We may solve various forbidden subgraph problems by considering supersaturation-type problems. We restate and give a proof sketch of the Kővári–Sós–Turán theorem below:
Kővári–Sós–Turán theorem. For every pair of positive integers $s,t$ with $t\geq s\geq 1$, there exists some constant $C$ (independent of $n$) such that $ \operatorname {ex} (n,K_{s,t})\leq Cn^{2-{\frac {1}{s}}}$ for every positive integer $n$.[18]
Proof. Let $G$ be a $2$-graph on $n$ vertices, and consider the number of copies of $K_{1,s}$ in $G$. Given a vertex of degree $d$, we get exactly ${\binom {d}{s}}$ copies of $K_{1,s}$ rooted at this vertex, for a total of $\sum _{v\in V(G)}{\binom {\operatorname {deg} (v)}{s}}$ copies. Here, ${\binom {k}{s}}=0$ when $0\leq k<s$. By convexity, there are at total of at least $n{\binom {2e(G)/n}{s}}$ copies of $K_{1,s}$. Moreover, there are clearly ${\binom {n}{s}}$ subsets of $s$ vertices, so if there are more than $(t-1){\binom {n}{s}}$ copies of $K_{1,s}$, then by the Pigeonhole Principle there must exist a subset of $s$ vertices which form the set of leaves of at least $t$ of these copies, forming a $K_{s,t}$. Therefore, there exists an occurrence of $K_{s,t}$ as long as we have $n{\binom {2e(G)/n}{s}}>(t-1){\binom {n}{s}}$. In other words, we have an occurrence if ${\frac {e(G)^{s}}{n^{s-1}}}\geq O(n^{s})$, which simplifies to $e(G)\geq O(n^{2-{\frac {1}{s}}})$, which is the statement of the theorem. [19]
In this proof, we are using the supersaturation method by considering the number of occurrences of a smaller subgraph. Typically, applications of the supersaturation method do not use the supersaturation theorem. Instead, the structure often involves finding a subgraph $H'$ of some forbidden subgraph $H$ and showing that if it appears too many times in $G$, then $H$ must appear in $G$ as well. Other theorems regarding the forbidden subgraph problem which can be solved with supersaturation include:
• $\operatorname {ex} (n,C_{2t})\leq O(n^{1+1/t})$. [20]
• For any $t$ and $k\geq 2$, $\operatorname {ex} (n,C_{2k},C_{2k-1})\leq O\left(\left({\frac {n}{2}}\right)^{1+1/t}\right)$. [20]
• If $Q$ denotes the graph determined by the vertices and edges of a cube, and $Q^{*}$ denotes the graph obtained by joining two opposite vertices of the cube, then $\operatorname {ex} (n,Q)\leq \operatorname {ex} (n,Q^{*})=O(n^{8/5})$. [19]
Generalizations
The problem may be generalized for a set of forbidden subgraphs $S$: find the maximal number of edges in an $n$-vertex graph which does not have a subgraph isomorphic to any graph from $S$.[21]
There are also hypergraph versions of forbidden subgraph problems that are much more difficult. For instance, Turán's problem may be generalized to asking for the largest number of edges in an $n$-vertex 3-uniform hypergraph that contains no tetrahedra. The analog of the Turán construction would be to partition the vertices into almost equal subsets $V_{1},V_{2},V_{3}$, and connect vertices $x,y,z$ by a 3-edge if they are all in different $V_{i}$s, or if two of them are in $V_{i}$ and the third is in $V_{i+1}$ (where $V_{4}=V_{1}$). This is tetrahedron-free, and the edge density is $5/9$. However, the best known upper bound is 0.562, using the technique of flag algebras.[22]
See also
• Biclique-free graph
• Erdős–Hajnal conjecture
• Turán number
• Subgraph isomorphism problem
• Forbidden graph characterization
References
1. Combinatorics: Set Systems, Hypergraphs, Families of Vectors and Probabilistic Combinatorics, Béla Bollobás, 1986, ISBN 0-521-33703-8, p. 53, 54
2. "Modern Graph Theory", by Béla Bollobás, 1998, ISBN 0-387-98488-7, p. 103
3. Turán, Pál (1941). "On an extremal problem in graph theory". Matematikai és Fizikai Lapok (in Hungarian). 48: 436–452.
4. Erdős, P.; Stone, A. H. (1946). "On the structure of linear graphs". Bulletin of the American Mathematical Society. 52 (12): 1087–1091. doi:10.1090/S0002-9904-1946-08715-7.
5. Kővári, T.; T. Sós, V.; Turán, P. (1954), "On a problem of K. Zarankiewicz" (PDF), Colloq. Math., 3: 50–57, doi:10.4064/cm-3-1-50-57, MR 0065617
6. Bondy, J. A.; Simonovits, M. (April 1974). "Cycles of even length in graphs". Journal of Combinatorial Theory. Series B. 16 (2): 97–105. doi:10.1016/0095-8956(74)90052-5. MR 0340095.
7. Alon, Noga; Krivelevich, Michael; Sudakov, Benny. "Turán numbers of bipartite graphs and related Ramsey-type questions". Combinatorics, Probability and Computing. MR 2037065.
8. Füredi, Zoltán; Simonovits, Miklós (2013-06-21). "The history of degenerate (bipartite) extremal graph problems". arXiv:1306.5167 [math.CO].
9. Zhao, Yufei. "Graph Theory and Additive Combinatorics" (PDF). pp. 32–37. Retrieved 2 December 2019.
10. Erdős, P.; Rényi, A.; Sós, V. T. (1966). "On a problem of graph theory". Studia Sci. Math. Hungar. 1: 215–235. MR 0223262.
11. Baker, R. C.; Harman, G.; Pintz, J. (2001), "The difference between consecutive primes. II.", Proc. London Math. Soc., Series 3, 83 (3): 532–562, doi:10.1112/plms/83.3.532, MR 1851081
12. Brown, W. G. (1966). "On graphs that do not contain a Thomsen graph". Canad. Math. Bull. 9 (3): 281–285. doi:10.4153/CMB-1966-036-2. MR 0200182.
13. Zhao, Yufei. "Graph Theory and Additive Combinatorics" (PDF). pp. 32–37. Retrieved 2 December 2019.
14. Alon, Noga; Rónyai, Lajos; Szabó, Tibor (1999). "Norm-graphs: variations and applications". Journal of Combinatorial Theory. Series B. 76 (2): 280–290. doi:10.1006/jctb.1999.1906. MR 1699238.
15. Erdős, Paul; Simonovits, Miklós. "Supersaturated Graphs and Hypergraphs" (PDF). p. 3. Retrieved 27 November 2021.
16. Zhao, Yufei. "Graph Theory and Additive Combinatorics" (PDF). pp. 16–17. Retrieved 2 December 2019.
17. Simonovits, Miklós. "Extremal Graph Problems, Degenerate Extremal Problems, and Supersaturated Graphs" (PDF). p. 17. Retrieved 25 November 2021.
18. Kővári, T.; T. Sós, V.; Turán, P. (1954), "On a problem of K. Zarankiewicz" (PDF), Colloq. Math., 3: 50–57, doi:10.4064/cm-3-1-50-57, MR 0065617
19. Simonovits, Miklós. "Extremal Graph Problems, Degenerate Extremal Problems, and Supersaturated Graphs" (PDF). Retrieved 27 November 2021.
20. Erdős, Paul; Simonovits, Miklós. "Compactness Results in Extremal Graph Theory" (PDF). Retrieved 27 November 2021.
21. Handbook of Discrete and Combinatorial Mathematics By Kenneth H. Rosen, John G. Michaels p. 590
22. Keevash, Peter. "Hypergraph Turán Problems" (PDF). Retrieved 2 December 2019.
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Wikipedia
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Turing's method
In mathematics, Turing's method is used to verify that for any given Gram point gm there lie m + 1 zeros of ζ(s), in the region 0 < Im(s) < Im(gm), where ζ(s) is the Riemann zeta function.[1] It was discovered by Alan Turing and published in 1953,[2] although that proof contained errors and a correction was published in 1970 by R. Sherman Lehman.[3]
For every integer i with i < n we find a list of Gram points $\{g_{i}\mid 0\leqslant i\leqslant m\}$ and a complementary list $\{h_{i}\mid 0\leqslant i\leqslant m\}$, where gi is the smallest number such that
$(-1)^{i}Z(g_{i}+h_{i})>0,$
where Z(t) is the Hardy Z function. Note that gi may be negative or zero. Assuming that $h_{m}=0$ and there exists some integer k such that $h_{k}=0$, then if
$1+{\frac {1.91+0.114\log(g_{m+k}/2\pi )+\sum _{j=m+1}^{m+k-1}h_{j}}{g_{m+k}-g_{m}}}<2,$
and
$-1-{\frac {1.91+0.114\log(g_{m}/2\pi )+\sum _{j=1}^{k-1}h_{m-j}}{g_{m}-g_{m-k}}}>-2,$
Then the bound is achieved and we have that there are exactly m + 1 zeros of ζ(s), in the region 0 < Im(s) < Im(gm).
References
1. Edwards, H. M. (1974). Riemann's zeta function. Pure and Applied Mathematics. Vol. 58. New York-London: Academic Press. ISBN 0-12-232750-0. Zbl 0315.10035.
2. Turing, A. M. (1953). "Some Calculations of the Riemann Zeta‐Function". Proceedings of the London Mathematical Society. s3-3 (1): 99–117. doi:10.1112/plms/s3-3.1.99.
3. Lehman, R. S. (1970). "On the Distribution of Zeros of the Riemann Zeta‐Function". Proceedings of the London Mathematical Society. s3-20 (2): 303–320. doi:10.1112/plms/s3-20.2.303.
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Wikipedia
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Turing's proof
Turing's proof is a proof by Alan Turing, first published in January 1937 with the title "On Computable Numbers, with an Application to the Entscheidungsproblem". It was the second proof (after Church's theorem) of the negation of Hilbert's Entscheidungsproblem; that is, the conjecture that some purely mathematical yes–no questions can never be answered by computation; more technically, that some decision problems are "undecidable" in the sense that there is no single algorithm that infallibly gives a correct "yes" or "no" answer to each instance of the problem. In Turing's own words: "what I shall prove is quite different from the well-known results of Gödel ... I shall now show that there is no general method which tells whether a given formula U is provable in K [Principia Mathematica]".[1]
Turing followed this proof with two others. The second and third both rely on the first. All rely on his development of typewriter-like "computing machines" that obey a simple set of rules and his subsequent development of a "universal computing machine".
Summary of the proofs
In his proof that the Entscheidungsproblem can have no solution, Turing proceeded from two proofs that were to lead to his final proof. His first theorem is most relevant to the halting problem, the second is more relevant to Rice's theorem.
First proof: that no "computing machine" exists that can decide whether or not an arbitrary "computing machine" (as represented by an integer 1, 2, 3, . . .) is "circle-free" (i.e. goes on printing its number in binary ad infinitum): "...we have no general process for doing this in a finite number of steps" (p. 132, ibid.). Turing's proof, although it seems to use the "diagonal process", in fact shows that his machine (called H) cannot calculate its own number, let alone the entire diagonal number (Cantor's diagonal argument): "The fallacy in the argument lies in the assumption that B [the diagonal number] is computable"[2] The proof does not require much mathematics.
Second proof: This one is perhaps more familiar to readers as Rice's theorem: "We can show further that there can be no machine E which, when supplied with the S.D ["program"] of an arbitrary machine M, will determine whether M ever prints a given symbol (0 say)"[lower-alpha 1]
Third proof: "Corresponding to each computing machine M we construct a formula Un(M) and we show that, if there is a general method for determining whether Un(M) is provable, then there is a general method for determining whether M ever prints 0".[1]
The third proof requires the use of formal logic to prove a first lemma, followed by a brief word-proof of the second:
Lemma 1: If S1 [symbol "0"] appears on the tape in some complete configuration of M, then Un(M) is provable.[3]
Lemma 2: [The converse] If Un(M) is provable then S1 [symbol "0"] appears on the tape in some complete configuration of M.[4]
Finally, in only 64 words and symbols Turing proves by reductio ad absurdum that "the Hilbert Entscheidungsproblem can have no solution".[1]
Summary of the first proof
Turing created a thicket of abbreviations. See the glossary at the end of the article for definitions.
Some key clarifications:
Turing's machine H is attempting to print a diagonal number of 0s and 1s.
This diagonal number is created when H actually "simulates" each "successful" machine under evaluation and prints the R-th "figure" (1 or 0) of the R-th "successful" machine.
Turing spent much of his paper actually "constructing" his machines to convince us of their truth. This was required by his use of the reductio ad absurdum form of proof. We must emphasize the "constructive" nature of this proof. Turing describes what could be a real machine, really buildable. The only questionable element is the existence of machine D, which this proof will eventually show to be impossible.
Turing begins the proof with the assertion of the existence of a “decision/determination” machine D. When fed any S.D (string of symbols A, C, D, L, R, N, semicolon “;”) it will determine if this S.D (symbol string) represents a "computing machine" that is either "circular" — and therefore "un-satisfactory u" — or "circle-free" — and therefore "satisfactory s".
Turing has previously demonstrated in his commentary that all "computing machines — machines that compute a number as 1s and 0s forever — can be written as an S.D on the tape of the “universal machine” U. Most of his work leading up to his first proof is spent demonstrating that a universal machine truly exists, i.e.
There truly exists a universal machine U
For each number N, there truly exists a unique S.D,
Every Turing machine has an S.D
Every S.D on U’s tape can be “run” by U and will produce the same “output” (figures 1, 0) as the original machine.
Turing makes no comment about how machine D goes about its work. For sake of argument, we suppose that D would first look to see if the string of symbols is "well-formed" (i.e. in the form of an algorithm and not just a scramble of symbols), and if not then discard it. Then it would go “circle-hunting”. To do this perhaps it would use “heuristics” (tricks: taught or learned). For purposes of the proof, these details are not important.
Turing then describes (rather loosely) the algorithm (method) to be followed by a machine he calls H. Machine H contains within it the decision-machine D (thus D is a “subroutine” of H). Machine H’s algorithm is expressed in H’s table of instructions, or perhaps in H’s Standard Description on tape and united with the universal machine U; Turing does not specify this.
In the course of describing universal machine U, Turing has demonstrated that a machine’s S.D (string of letters similar to a “program”) can be converted to an integer (base 8) and vice versa. Any number N (in base 8) can be converted to an S.D with the following replacements: 1 by A, 2 by C, 3 by D, 4 by L, 5 by R, 6 by N, 7 by semicolon ";".
As it turns out, machine H's unique number (D.N) is the number "K". We can infer that K is some hugely long number, maybe tens-of-thousands of digits long. But this is not important to what follows.
Machine H is responsible for converting any number N into an equivalent S.D symbol string for sub-machine D to test. (In programming parlance: H passes an arbitrary "S.D” to D, and D returns “satisfactory” or “unsatisfactory”.) Machine H is also responsible for keeping a tally R (“Record”?) of successful numbers (we suppose that the number of “successful” S.D's, i.e. R, is much less than the number of S.D's tested, i.e. N). Finally, H prints on a section of its tape a diagonal number “beta-primed” B’. H creates this B’ by “simulating” (in the computer-sense) the “motions” of each “satisfactory” machine/number; eventually this machine/number under test will arrive at its Rth “figure” (1 or 0), and H will print it. H then is responsible for “cleaning up the mess” left by the simulation, incrementing N and proceeding onward with its tests, ad infinitum.
Note: All these machines that H is hunting for are what Turing called "computing machines". These compute binary-decimal-numbers in an endless stream of what Turing called "figures": only the symbols 1 and 0.
An example to illustrate the first proof
An example: Suppose machine H has tested 13472 numbers and produced 5 satisfactory numbers, i.e. H has converted the numbers 1 through 13472 into S.D's (symbol strings) and passed them to D for test. As a consequence H has tallied 5 satisfactory numbers and run the first one to its 1st "figure", the second to its 2nd figure, the third to its 3rd figure, the fourth to its 4th figure, and the fifth to its 5th figure. The count now stands at N = 13472, R = 5, and B' = ".10011" (for example). H cleans up the mess on its tape, and proceeds:
H increments N = 13473 and converts "13473" to symbol string ADRLD. If sub-machine D deems ADLRD unsatisfactory, then H leaves the tally-record R at 5. H will increment the number N to 13474 and proceed onward. On the other hand, if D deems ADRLD satisfactory then H will increment R to 6. H will convert N (again) into ADLRD [this is just an example, ADLRD is probably useless] and “run” it using the universal machine U until this machine-under-test (U "running" ADRLD) prints its 6th “figure” i.e. 1 or 0. H will print this 6th number (e.g. “0”) in the “output” region of its tape (e.g. B’ = “.100110”).
H cleans up the mess, and then increments the number N to 13474.
The whole process unravels when H arrives at its own number K. We will proceed with our example. Suppose the successful-tally/record R stands at 12. H finally arrives at its own number minus 1, i.e. N = K-1 = 4335...3214, and this number is unsuccessful. Then H increments N to produce K = 4355...3215, i.e. its own number. H converts this to “LDDR...DCAR” and passes it to decision-machine D. Decision-machine D must return “satisfactory” (that is: H must by definition go on and on testing, ad infinitum, because it is "circle-free"). So H now increments tally R from 12 to 13 and then re-converts the number-under-test K into its S.D and uses U to simulate it. But this means that H will be simulating its own motions. What is the first thing the simulation will do? This simulation K-aka-H either creates a new N or “resets” the “old” N to 1. This "K-aka-H" either creates a new R or “resets” the “old” R to 0. Old-H “runs” new "K-aka-H" until it arrives at its 12th figure.
But it never makes it to the 13th figure; K-aka-H eventually arrives at 4355...3215, again, and K-aka-H must repeat the test. K-aka-H will never reach the 13th figure. The H-machine probably just prints copies of itself ad infinitum across blank tape. But this contradicts the premise that H is a satisfactory, non-circular computing machine that goes on printing the diagonal numbers's 1's and 0's forever. (We will see the same thing if N is reset to 1 and R is reset to 0.)
If the reader does not believe this, they can write a "stub" for decision-machine D (stub "D" will return "satisfactory") and then see for themselves what happens at the instant machine H encounters its own number.
Summary of the second proof
Less than one page long, the passage from premises to conclusion is obscure.
Turing proceeds by reductio ad absurdum. He asserts the existence of a machine E, which when given the S.D (Standard Description, i.e. "program") of an arbitrary machine M, will determine whether M ever prints a given symbol (0 say). He does not assert that this M is a "computing machine".
Given the existence of machine E, Turing proceeds as follows:
1. If machine E exists then a machine G exists that determines if M prints 0 infinitely often, AND
2. If E exists then another process exists [we can call the process/machine G' for reference] that determines if M prints 1 infinitely often, THEREFORE
3. When we combine G with G' we have a process that determines if M prints an infinity of figures, AND
4. IF the process "G with G'" determines M prints an infinity of figures, THEN "G with G'" has determined that M is circle-free, BUT
5. This process "G with G'" that determine if M is circle-free, by proof 1, cannot exist, THEREFORE
6. Machine E does not exist.
Details of second proof
The difficulty in the proof is step 1. The reader will be helped by realizing that Turing is not explaining his subtle handiwork. (In a nutshell: he is using certain equivalencies between the “existential-“ and “universal-operators” together with their equivalent expressions written with logical operators.)
Here's an example: Suppose we see before us a parking lot full of hundreds of cars. We decide to go around the entire lot looking for: “Cars with flat (bad) tires”. After an hour or so we have found two “cars with bad tires.” We can now say with certainty that “Some cars have bad tires”. Or we could say: “It’s not true that ‘All the cars have good tires’”. Or: “It is true that: ‘not all the cars have good tires”. Let us go to another lot. Here we discover that “All the cars have good tires.” We might say, “There’s not a single instance of a car having a bad tire.” Thus we see that, if we can say something about each car separately then we can say something about ALL of them collectively.
This is what Turing does: From M he creates a collection of machines {M1, M2, M3, M4, ..., Mn} and about each he writes a sentence: “X prints at least one 0” and allows only two “truth values”, True = blank or False = :0:. One by one he determines the truth value of the sentence for each machine and makes a string of blanks or :0:, or some combination of these. We might get something like this: “M1 prints a 0” = True AND “M2 prints a 0” = True AND “M3 prints a 0” = True AND “M4 prints a 0” = False, ... AND “Mn prints a 0” = False. He gets the string
BBB:0::0::0: ... :0: ... ad infinitum
if there are an infinite number of machines Mn. If on the other hand if every machine had produced a "True" then the expression on the tape would be
BBBBB....BBBB ... ad infinitum
Thus Turing has converted statements about each machine considered separately into a single "statement" (string) about all of them. Given the machine (he calls it G) that created this expression, he can test it with his machine E and determine if it ever produces a 0. In our first example above we see that indeed it does, so we know that not all the M's in our sequence print 0s. But the second example shows that, since the string is blanks then every Mn in our sequence has produced a 0.
All that remains for Turing to do is create a process to create the sequence of Mn's from a single M.
Suppose M prints this pattern:
M => ...AB01AB0010AB…
Turing creates another machine F that takes M and crunches out a sequence of Mn's that successively convert the first n 0's to “0-bar” (0):
He states, without showing details, that this machine F is truly build-able. We can see that one of a couple things could happen. F may run out of machines that have 0's, or it may have to go on ad infinitum creating machines to “cancel the zeros”.
Turing now combines machines E and F into a composite machine G. G starts with the original M, then uses F to create all the successor-machines M1, M2,. . ., Mn. Then G uses E to test each machine starting with M. If E detects that a machine never prints a zero, G prints :0: for that machine. If E detects that a machine does print a 0 (we assume, Turing doesn’t say) then G prints :: or just skips this entry, leaving the squares blank. We can see that a couple things can happen.
G will print no 0’s, ever, if all the Mn’s print 0’s, OR,
G will print ad infinitum 0’s if all the M’s print no 0’s, OR,
G will print 0’s for a while and then stop.
Now, what happens when we apply E to G itself?
If E(G) determines that G never prints a 0 then we know that all the Mn’s have printed 0’s. And this means that, because all the Mn came from M, that M itself prints 0’s ad infinitum, OR
If E(G) determines that G does print a 0 then we know that not all the Mn’s print 0’s; therefore M does not print 0’s ad infinitum.
As we can apply the same process for determining if M prints 1 infinitely often. When we combine these processes, we can determine that M does, or does not, go on printing 1's and 0's ad infinitum. Thus we have a method for determining if M is circle-free. By Proof 1 this is impossible. So the first assertion that E exists, is wrong: E does not exist.
Summary of the third proof
Here Turing proves "that the Hilbert Entscheidungsproblem can have no solution".[1] Here he
…show(s) that there can be no general process for determining whether a given formula U of the functional calculus K is provable. (ibid.)
Both Lemmas #1 and #2 are required to form the necessary "IF AND ONLY IF" (i.e. logical equivalence) required by the proof:
A set E is computably decidable if and only if both E and its complement are computably enumerable (Franzén, p. 67)
Turing demonstrates the existence of a formula Un(M) which says, in effect, that "in some complete configuration of M, 0 appears on the tape" (p. 146). This formula is TRUE, that is, it is "constructible", and he shows how to go about this.
Then Turing proves two Lemmas, the first requiring all the hard work. (The second is the converse of the first.) Then he uses reductio ad absurdum to prove his final result:
1. There exists a formula Un(M). This formula is TRUE, AND
2. If the Entscheidungsproblem can be solved THEN a mechanical process exists for determining whether Un(M) is provable (derivable), AND
3. By Lemmas 1 and 2: Un(M) is provable IF AND ONLY IF 0 appears in some "complete configuration" of M, AND
4. IF 0 appears in some "complete configuration" of M THEN a mechanical process exists that will determine whether arbitrary M ever prints 0, AND
5. By Proof 2 no mechanical process exists that will determine whether arbitrary M ever prints 0, THEREFORE
6. Un(M) is not provable (it is TRUE, but not provable) which means that the Entscheidungsproblem is unsolvable.
Details of the third proof
[If readers intend to study the proof in detail they should correct their copies of the pages of the third proof with the corrections that Turing supplied. Readers should also come equipped with a solid background in (i) logic (ii) the paper of Kurt Gödel: "On Formally Undecidable Propositions of Principia Mathematica and Related Systems".[lower-alpha 2] For assistance with Gödel's paper they may consult e.g. Ernest Nagel and James R. Newman, Gödel's Proof, New York University Press, 1958.]
To follow the technical details, the reader will need to understand the definition of "provable" and be aware of important "clues".
"Provable" means, in the sense of Gödel, that (i) the axiom system itself is powerful enough to produce (express) the sentence "This sentence is provable", and (ii) that in any arbitrary "well-formed" proof the symbols lead by axioms, definitions, and substitution to the symbols of the conclusion.
First clue: "Let us put the description of M into the first standard form of §6". Section 6 describes the very specific "encoding" of machine M on the tape of a "universal machine" U. This requires the reader to know some idiosyncrasies of Turing's universal machine U and the encoding scheme.
(i) The universal machine is a set of "universal" instructions that reside in an "instruction table". Separate from this, on U's tape, a "computing machine" M will reside as "M-code". The universal table of instructions can print on the tape the symbols A, C, D, 0, 1, u, v, w, x, y, z, : . The various machines M can print these symbols only indirectly by commanding U to print them.
(ii) The "machine code" of M consists of only a few letters and the semicolon, i.e. D, C, A, R, L, N, ; . Nowhere within the "code" of M will the numerical "figures" (symbols) 1 and 0 ever appear. If M wants U to print a symbol from the collection blank, 0, 1 then it uses one of the following codes to tell U to print them. To make things more confusing, Turing calls these symbols S0, S1, and S2, i.e.
blank = S0 = D
0 = S1 = DC
1 = S2 = DCC
(iii) A "computing machine", whether it is built directly into a table (as his first examples show), or as machine-code M on universal-machine U's tape, prints its number on blank tape (to the right of M-code, if there is one) as 1s and 0s forever proceeding to the right.
(iv) If a "computing machine" is U+"M-code", then "M-code" appears first on the tape; the tape has a left end and the "M-code" starts there and proceeds to the right on alternate squares. When the M-code comes to an end (and it must, because of the assumption that these M-codes are finite algorithms), the "figures" will begin as 1s and 0s on alternate squares, proceeding to the right forever. Turing uses the (blank) alternate squares (called "E"- "eraseable"- squares) to help U+"M-code" keep track of where the calculations are, both in the M-code and in the "figures" that the machine is printing.
(v) A "complete configuration" is a printing of all symbols on the tape, including M-code and "figures" up to that point, together with the figure currently being scanned (with a pointer-character printed to the left of the scanned symbol?). If we have interpreted Turing's meaning correctly, this will be a hugely long set of symbols. But whether the entire M-code must be repeated is unclear; only a printing of the current M-code instruction is necessary plus the printing of all figures with a figure-marker).
(vi) Turing reduced the vast possible number of instructions in "M-code" (again: the code of M to appear on the tape) to a small canonical set, one of three similar to this: {qi Sj Sk R ql} e.g. If machine is executing instruction #qi and symbol Sj is on the square being scanned, then Print symbol Sk and go Right and then go to instruction ql: The other instructions are similar, encoding for "Left" L and "No motion" N. It is this set that is encoded by the string of symbols qi = DA...A, Sj = DC...C, Sk = DC...C, R, ql = DA....A. Each instruction is separated from another one by the semicolon. For example, {q5, S1 S0 L q3} means: Instruction #5: If scanned symbol is 0 then print blank, go Left, then go to instruction #3. It is encoded as follows
; D A A A A A D C D L D A A A
Second clue: Turing is using ideas introduced in Gödel's paper, that is, the "Gödelization" of (at least part of) the formula for Un(M). This clue appears only as a footnote on page 138 (Davis (1965), p. 138): "A sequence of r primes is denoted by ^(r)" (ibid.) [Here, r inside parentheses is "raised".] This "sequence of primes" appears in a formula called F^(n).
Third clue: This reinforces the second clue. Turing's original attempt at the proof uses the expression:
(Eu)N(u) & (x)(... etc. ...)[5]
Earlier in the paper Turing had previously used this expression (p. 138) and defined N(u) to mean "u is a non-negative integer" (ibid.) (i.e. a Gödel number). But, with the Bernays corrections, Turing abandoned this approach (i.e. the use of N(u)) and the only place where "the Gödel number" appears explicitly is where he uses F^(n).
What does this mean for the proof? The first clue means that a simple examination of the M-code on the tape will not reveal if a symbol 0 is ever printed by U+"M-code". A testing-machine might look for the appearance of DC in one of the strings of symbols that represent an instruction. But will this instruction ever be "executed?" Something has to "run the code" to find out. This something can be a machine, or it can be lines in a formal proof, i.e. Lemma #1.
The second and third clues mean that, as its foundation is Gödel's paper, the proof is difficult.
In the example below we will actually construct a simple "theorem"—a little Post–Turing machine program "run it". We will see just how mechanical a properly designed theorem can be. A proof, we will see, is just that, a "test" of the theorem that we do by inserting a "proof example" into the beginning and see what pops out at the end.
Both Lemmas #1 and #2 are required to form the necessary "IF AND ONLY IF" (i.e. logical equivalence) required by the proof:
A set E is computably decidable if and only if both E and its complement are computably enumerable. (Franzén, p. 67)
To quote Franzén:
A sentence A is said to be decidable in a formal system S if either A or its negation is provable in S. (Franzén, p. 65)
Franzén has defined "provable" earlier in his book:
A formal system is a system of axioms (expressed in some formally defined language) and rules of reasoning (also called inference rules), used to derive the theorems of the system. A theorem is any statement in the language of the system obtainable by a series of applications of the rules of reasoning, starting from the axioms. A proof is a finite sequence of such applications, leading to a theorem as its conclusion. (ibid. p. 17)
Thus a "sentence" is a string of symbols, and a theorem is a string of strings of symbols.
Turing is confronted with the following task:
to convert a Universal Turing machine "program", and the numerical symbols on the tape (Turing's "figures", symbols "1" and "0"), into a "theorem"—that is, a (monstrously long) string of sentences that define the successive actions of the machine, (all) the figures of the tape, and the location of the "tape head".
Thus the "string of sentences" will be strings of strings of symbols. The only allowed individual symbols will come from Gödel's symbols defined in his paper.(In the following example we use the "<" and ">" around a "figure" to indicate that the "figure" is the symbol being scanned by the machine).
An example to illustrate the third proof
In the following, we have to remind ourselves that every one of Turing's “computing machines” is a binary-number generator/creator that begins work on “blank tape”. Properly constructed, it always cranks away ad infinitum, but its instructions are always finite. In Turing's proofs, Turing's tape had a “left end” but extended right ad infinitum. For sake of example below we will assume that the “machine” is not a Universal machine, but rather the simpler “dedicated machine” with the instructions in the Table.
Our example is based on a modified Post–Turing machine model of a Turing Machine. This model prints only the symbols 0 and 1. The blank tape is considered to be all b's. Our modified model requires us to add two more instructions to the 7 Post–Turing instructions. The abbreviations that we will use are:
R, RIGHT: look to the right and move tape to left, or move tape head right
L, LEFT : look to the left and to move tape right, or move tape head left
E, ERASE scanned square (e.g. make square blank)
P0,: PRINT 0 in scanned square
P1,: PRINT 1 in scanned square
Jb_n, JUMP-IF-blank-to-instruction_#n,
J0_n, JUMP-IF-0-to-instruction_#n,
J1_n, JUMP-IF-1-to-instrucntion_#n,
HALT.
In the cases of R, L, E, P0, and P1 after doing its task the machine continues on to the next instruction in numerical sequence; ditto for the jumps if their tests fail.
But, for brevity, our examples will only use three squares. And these will always start as there blanks with the scanned square on the left: i.e. bbb. With two symbols 1, 0 and blank we can have 27 distinct configurations:
bbb, bb0, bb1, b0b, b00, b01, b1b, b10, b11, 0bb, 0b0, 0b1, 00b, 000, 001, 01b, 010, 011, 1bb, 1b0, 1b1, 10b, 100, 101, 11b, 110, 111
We must be careful here, because it is quite possible that an algorithm will (temporarily) leave blanks in between figures, then come back and fill something in. More likely, an algorithm may do this intentionally. In fact, Turing's machine does this—it prints on alternate squares, leaving blanks between figures so it can print locator symbols.
Turing always left alternate squares blank so his machine could place a symbol to the left of a figure (or a letter if the machine is the universal machine and the scanned square is actually in the “program”). In our little example we will forego this and just put symbols ( ) around the scanned symbol, as follows:
b(b)0 this means, "Tape is blanks-to-the-left of left blank but left blank is 'in play', the scanned-square-is-blank, '0', blanks-to-right"
1(0)1 this means, "Tape is blanks-to-the-left, then 1, scanned square is '0'"
Let us write a simple program:
start: P1, R, P1, R, P1, H
Remember that we always start with blank tape. The complete configuration prints the symbols on the tape followed by the next instruction:
start config: (b) P1,
config #1: (1) R,
config #2: 1(b) P1,
config #3: 1(1) R,
config #4: 11(b) P1,
config #5: 11(1) H
Let us add “jump” into the formula. When we do this we discover why the complete configuration must include the tape symbols. (Actually, we see this better, below.) This little program prints three “1”s to the right, reverses direction and moves left printing 0’s until it hits a blank. We will print all the symbols that our machine uses:
start: P1, R, P1, R, P1, P0, L, J1_7, H
(b)bb P1,
(1)bb R,
1(b)b P1,
1(1)b R,
11(b) P1,
11(1) P0,
11(0) L,
1(1)0 J1_7
1(1)0 L
(1)10 J0_7
(1)10 L
(b)110 J0_7
(b)110 H
Here at the end we find that a blank on the left has “come into play” so we leave it as part of the total configuration.
Given that we have done our job correctly, we add the starting conditions and see “where the theorem goes”. The resulting configuration—the number 110—is the PROOF.
• Turing's first task had to write a generalized expression using logic symbols to express exactly what his Un(M) would do.
• Turing's second task is to "Gödelize" this hugely long string-of-string-of-symbols using Gödel's technique of assigning primes to the symbols and raising the primes to prime-powers, per Gödel's method.
Complications
Turing's proof is complicated by a large number of definitions, and confounded with what Martin Davis called "petty technical details" and "...technical details [that] are incorrect as given".[lower-alpha 3] Turing himself published "A Correction" in 1938: "The author is indebted to P. Bernays for pointing out these errors".[6]
Specifically, in its original form the third proof is badly marred by technical errors. And even after Bernays' suggestions and Turing's corrections, errors remained in the description of the universal machine. And confusingly, since Turing was unable to correct his original paper, some text within the body harks to Turing's flawed first effort.
Bernays' corrections may be found in Davis (1965), pp. 152–154; the original is to be found as "On Computable Numbers, with an Application to the Entscheidungsproblem. A Correction," Proceedings of the London Mathematical Society (2), 43 (1938), 544-546.
The on-line version of Turing's paper has these corrections in an addendum; however, corrections to the Universal Machine must be found in an analysis provided by Emil Post.
At first, the only mathematician to pay close attention to the details of the proof was Post (cf. Hodges p. 125) — mainly because he had arrived simultaneously at a similar reduction of "algorithm" to primitive machine-like actions, so he took a personal interest in the proof. Strangely (perhaps World War II intervened) it took Post some ten years to dissect it in the Appendix to his paper Recursive Unsolvability of a Problem of Thue, 1947.[lower-alpha 4]
Other problems present themselves: In his Appendix Post commented indirectly on the paper's difficulty and directly on its "outline nature"[lower-alpha 5] and "intuitive form" of the proofs.[lower-alpha 5] Post had to infer various points:
If our critique is correct, a machine is said to be circle-free if it is a Turing computing ... machine which prints an infinite number of 0s and 1s. And the two theorems of Turing's in question are really the following. There is no Turing ... machine which, when supplied with an arbitrary positive integer n, will determine whether n is the D.N of a Turing computing ... machine that is circle-free. [Secondly], There is no Turing convention-machine which, when supplied with an arbitrary positive integer n, will determine whether n is the D.N of a Turing computing ... machine that ever prints a given symbol (0 say).[lower-alpha 6]
Anyone who has ever tried to read the paper will understand Hodges' complaint:
The paper started attractively, but soon plunged (in typical Turing manner) into a thicket of obscure German Gothic type in order to develop his instruction table for the universal machine. The last people to give it a glance would be the applied mathematicians who had to resort to practical computation... (Hodges p. 124)
Glossary of terms used by Turing
1 computable number — a number whose decimal is computable by a machine (i.e., by finite means such as an algorithm)
2 M — a machine with a finite instruction table and a scanning/printing head. M moves an infinite tape divided into squares each “capable of bearing a symbol”. The machine-instructions are only the following: move one square left, move one square right, on the scanned square print symbol p, erase the scanned square, if the symbol is p then do instruction aaa, if the scanned symbol is not p then do instruction aaa, if the scanned symbol is none then do instruction aaa, if the scanned symbol is any do instruction aaa [where “aaa” is an instruction-identifier].
3 computing machine — an M that prints two kinds of symbols, symbols of the first type are called “figures” and are only binary symbols 1 and 0; symbols of the second type are any other symbols.
4 figures — symbols 1 and 0, a.k.a. “symbols of the first kind”
5 m-configuration — the instruction-identifier, either a symbol in the instruction table, or a string of symbols representing the instruction- number on the tape of the universal machine (e.g. "DAAAAA = #5")
6 symbols of the second kind — any symbols other than 1 and 0
7 circular — an unsuccessful computating machine. It fails to print, ad infinitum, the figures 0 or 1 that represent in binary the number it computes
8 circle-free — a successful computating machine. It prints, ad infinitum, the figures 0 or 1 that represent in binary the number it computes
9 sequence — as in “sequence computed by the machine”: symbols of the first kind a.k.a. figures a.k.a. symbols 0 and 1.
10 computable sequence — can be computed by a circle-free machine
11 S.D – Standard Description: a sequence of symbols A, C, D, L, R, N, “;” on a Turing machine tape
12 D.N — Description number: an S.D converted to a number: 1=A, 2=C, 3 =D, 4=L, 5=R, 6=N, 7=;
13 M(n) — a machine whose D.N is number “n”
14 satisfactory — a S.D or D.N that represents a circle-free machine
15 U — a machine equipped with a “universal” table of instructions. If U is “supplied with a tape on the beginning of which is written the S.D of some computing machine M, U will compute the same sequence as M.”
16 β’—“beta-primed”: A so-called “diagonal number” made up of the n-th figure (i.e. 0 or 1) of the n-th computable sequence [also: the computable number of H, see below]
17 u — an unsatisfactory, i.e. circular, S.D
18 s — satisfactory, i.e. circle-free S.D
19 D — a machine contained in H (see below). When supplied with the S.D of any computing machine M, D will test M's S.D and if circular mark it with “u” and if circle-free mark it with “s”
20 H — a computing machine. H computes B’, maintains R and N. H contains D and U and an unspecified machine (or process) that maintains N and R and provides D with the equivalent S.D of N. E also computes the figures of B’ and assembles the figures of B’.
21 R — a record, or tally, of the quantity of successful (circle-free) S.D tested by D
22 N — a number, starting with 1, to be converted into an S.D by machine E. E maintains N.
23 K — a number. The D.N of H.
Required for Proof #3
5 m-configuration — the instruction-identifier, either a symbol in the instruction table, or a string of symbols representing the instruction's number on the tape of the universal machine (e.g. "DAAAAA = instruction #5"). In Turing's S.D the m-configuration appears twice in each instruction, the left-most string is the "current instruction"; the right-most string is the next instruction.
24 complete configuration — the number (figure 1 or 0) of the scanned square, the complete sequence of all symbols on the tape, and the m-configuration (the instruction-identifier, either a symbol or a string of symbols representing a number, e.g. "instruction DAAAA = #5")
25 RSi(x, y) — "in the complete configuration x of M the symbol on square y is Si; "complete configuration" is definition #5
26 I(x, y) — "in the complete configuration x of M the square y is scanned"
27 Kqm(x) — "in the complete configuration x of M the machine-configuration (instruction number) is qm"
28 F(x,y) — "y is the immediate successor of x" (follows Gödel's use of "f" as the successor-function).
29 G(x,y) — "x precedes y", not necessarily immediately
30 Inst{qi, Sj Sk L ql} is an abbreviation, as are Inst{qi, Sj Sk R ql}, and Inst{qi, Sj Sk N ql}. See below.
Turing reduces his instruction set to three “canonical forms” – one for Left, Right, and No-movement. Si and Sk are symbols on the tape.
tape Final
m-configSymbolOperationsm-config
qiSiPSk, Lqm
qiSiPSk, Rqm
qiSiPSk, Nqm
For example, the operations in the first line are PSk = PRINT symbol Sk from the collection A, C, D, 0, 1, u, v, w, x, y, z, :, then move tape LEFT.
These he further abbreviated as: (N1) qi Sj Sk L qm (N2) qi Sj Sk R qm (N3) qi Sj Sk N qm
In Proof #3 he calls the first of these “Inst{qi Sj Sk L ql}”, and he shows how to write the entire machine S.D as the logical conjunction (logical OR): this string is called “Des(M)”, as in “Description-of-M”. i.e. if the machine prints 0 then 1's and 0's on alternate squares to the right ad infinitum it might have the table (a similar example appears on page 119):
q1, blank, P0, R, q2
q2, blank, P-blank, R, q3
q3, blank, P1, R, q4
q4, blank, P-blank, R, q1
(This has been reduced to canonical form with the “p-blank” instructions so it differs a bit from Turing's example.) If put them into the “ Inst( ) form” the instructions will be the following (remembering: S0 is blank, S1 = 0, S2 = 1):
Inst {q1 S0 S1 R q2}
Inst {q2 S0 S0 R q3}
Inst {q3 S0 S2 R q4}
Inst {q4 S0 S0 R q1}
The reduction to the Standard Description (S.D) will be:
; D A D D C R D A A ; D A A D D R D A A A ; D A A A D D C C R D A A A A ; D A A A A D D R D A ;
This agrees with his example in the book (there will be a blank between each letter and number). Universal machine U uses the alternate blank squares as places to put "pointers".
Notes
1. his italics, Davis (1965), p. 134
2. reprinted in Davis (1965), p. 5
3. Davis's commentary in Davis (1965), p. 145
4. reprinted in Davis (1965), p. 293
5. Post in Davis (1965), p. 299
6. Post in Davis (1965), p. 300
References
Citations
1. Davis (1965), p. 145.
2. Davis (1965), p. 132.
3. Davis (1965), p. 147.
4. Davis (1965), p. 148.
5. Davis (1965), p. 146.
6. Davis (1965), p. 152.
Works cited
• Davis, Martin (1965). The Undecidable, Basic Papers on Undecidable Propositions, Unsolvable Problems and Computable Functions. New York: Raven Press. The two papers of Post referenced above are included in this volume. Other papers include those by Gödel, Church, Rosser, and Kleene.
• Franzén, Torkel (2005). Gödel's Theorem: An Incomplete Guide to its Use and Abuse. A K Peters.
• Hodges, Andrew (1983). Alan Turing: The Enigma. New York: Simon and Schuster. Cf. Chapter "The Spirit of Truth" for a history leading to, and a discussion of, his proof.
• Reichenbach, Hans (1947). Elements of Symbolic Logic. New York: Dover Publications, Inc.
• Turing, A.M. (1937). "On Computable Numbers, with an Application to the Entscheidungsproblem". Proceedings of the London Mathematical Society. 2. 42 (1): 230–65. doi:10.1112/plms/s2-42.1.230. S2CID 73712.
• Turing, A.M. (1938). "On Computable Numbers, with an Application to the Entscheidungsproblem. A Correction". Proceedings of the London Mathematical Society. 2. 43 (6): 544–6. doi:10.1112/plms/s2-43.6.544. This is the epochal paper where Turing defines Turing machines, shows that the Entscheidungsproblem is unsolvable.
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Wikipedia
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Recursive language
In mathematics, logic and computer science, a formal language (a set of finite sequences of symbols taken from a fixed alphabet) is called recursive if it is a recursive subset of the set of all possible finite sequences over the alphabet of the language. Equivalently, a formal language is recursive if there exists a Turing machine that, when given a finite sequence of symbols as input, always halts and accepts it if it belongs to the language and halts and rejects it otherwise. In Theoretical computer science, such always-halting Turing machines are called total Turing machines or algorithms (Sipser 1997). Recursive languages are also called decidable.
This article is about a class of formal languages as they are studied in mathematics and theoretical computer science. For computer languages that allow a function to call itself recursively, see Recursion (computer science).
The concept of decidability may be extended to other models of computation. For example, one may speak of languages decidable on a non-deterministic Turing machine. Therefore, whenever an ambiguity is possible, the synonym used for "recursive language" is Turing-decidable language, rather than simply decidable.
The class of all recursive languages is often called R, although this name is also used for the class RP.
This type of language was not defined in the Chomsky hierarchy of (Chomsky 1959). All recursive languages are also recursively enumerable. All regular, context-free and context-sensitive languages are recursive.
Definitions
There are two equivalent major definitions for the concept of a recursive language:
1. A recursive formal language is a recursive subset in the set of all possible words over the alphabet of the language.
2. A recursive language is a formal language for which there exists a Turing machine that, when presented with any finite input string, halts and accepts if the string is in the language, and halts and rejects otherwise. The Turing machine always halts: it is known as a decider and is said to decide the recursive language.
By the second definition, any decision problem can be shown to be decidable by exhibiting an algorithm for it that terminates on all inputs. An undecidable problem is a problem that is not decidable.
Examples
As noted above, every context-sensitive language is recursive. Thus, a simple example of a recursive language is the set L={abc, aabbcc, aaabbbccc, ...}; more formally, the set
$L=\{\,w\in \{a,b,c\}^{*}\mid w=a^{n}b^{n}c^{n}{\mbox{ for some }}n\geq 1\,\}$
is context-sensitive and therefore recursive.
Examples of decidable languages that are not context-sensitive are more difficult to describe. For one such example, some familiarity with mathematical logic is required: Presburger arithmetic is the first-order theory of the natural numbers with addition (but without multiplication). While the set of well-formed formulas in Presburger arithmetic is context-free, every deterministic Turing machine accepting the set of true statements in Presburger arithmetic has a worst-case runtime of at least $2^{2^{cn}}$, for some constant c>0 (Fischer & Rabin 1974). Here, n denotes the length of the given formula. Since every context-sensitive language can be accepted by a linear bounded automaton, and such an automaton can be simulated by a deterministic Turing machine with worst-case running time at most $c^{n}$ for some constant c , the set of valid formulas in Presburger arithmetic is not context-sensitive. On positive side, it is known that there is a deterministic Turing machine running in time at most triply exponential in n that decides the set of true formulas in Presburger arithmetic (Oppen 1978). Thus, this is an example of a language that is decidable but not context-sensitive.
Closure properties
Recursive languages are closed under the following operations. That is, if L and P are two recursive languages, then the following languages are recursive as well:
• The Kleene star $L^{*}$
• The image φ(L) under an e-free homomorphism φ
• The concatenation $L\circ P$
• The union $L\cup P$
• The intersection $L\cap P$
• The complement of $L$
• The set difference $L-P$
The last property follows from the fact that the set difference can be expressed in terms of intersection and complement.
See also
• Recursively enumerable language
• Computable set
• Recursion
References
• Michael Sipser (1997). "Decidability". Introduction to the Theory of Computation. PWS Publishing. pp. 151–170. ISBN 978-0-534-94728-6.
• Chomsky, Noam (1959). "On certain formal properties of grammars". Information and Control. 2 (2): 137–167. doi:10.1016/S0019-9958(59)90362-6.
• Fischer, Michael J.; Rabin, Michael O. (1974). "Super-Exponential Complexity of Presburger Arithmetic". Proceedings of the SIAM-AMS Symposium in Applied Mathematics. 7: 27–41.
• Oppen, Derek C. (1978). "A 222pn Upper Bound on the Complexity of Presburger Arithmetic". J. Comput. Syst. Sci. 16 (3): 323–332. doi:10.1016/0022-0000(78)90021-1.
Automata theory: formal languages and formal grammars
Chomsky hierarchyGrammarsLanguagesAbstract machines
• Type-0
• —
• Type-1
• —
• —
• —
• —
• —
• Type-2
• —
• —
• Type-3
• —
• —
• Unrestricted
• (no common name)
• Context-sensitive
• Positive range concatenation
• Indexed
• —
• Linear context-free rewriting systems
• Tree-adjoining
• Context-free
• Deterministic context-free
• Visibly pushdown
• Regular
• —
• Non-recursive
• Recursively enumerable
• Decidable
• Context-sensitive
• Positive range concatenation*
• Indexed*
• —
• Linear context-free rewriting language
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Each category of languages, except those marked by a *, is a proper subset of the category directly above it. Any language in each category is generated by a grammar and by an automaton in the category in the same line.
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Wikipedia
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Newton Gateway to Mathematics
The Newton Gateway to Mathematics (formerly known as the Turing Gateway to Mathematics - TGM) is a knowledge exchange centre at the University of Cambridge in the UK. As a knowledge intermediary for the mathematical sciences, it is overseen by the Isaac Newton Institute and the Centre for Mathematical Sciences. The Newton Gateway to Mathematics is an intermediary for knowledge exchange for both professional and academic users of mathematics. Each year the Newton Gateway organises multiple events and workshops that feature expert speakers from various industries, governments and scientific organisations that discuss mathematical technical and models, presented by leaders from diverse backgrounds, such as the health care and finances.[1]
Newton Gateway to Mathematics
Turing Gateway - Faulks Gatehouse
MottoTo be a channel for the interchange of knowledge and ideas between academies and commercial users of modern mathematics.
TypeImpact Initiative at the University of Cambridge
Established2013
DirectorDavid Abrahams
Location
Cambridge
,
United Kingdom
CampusCentre for Mathematical Sciences (Cambridge)
AffiliationsIsaac Newton Institute, University of Cambridge
Websitegateway.newton.ac.uk
Goals
A primary function of the Newton Gateway to Mathematics is to provide a research site and knowledge pool for the transfer, translation, exchange and dissemination of mathematical knowledge and for specific problem solving.[2] It brings together individuals and groups with a deep interest in expanding knowledge about math and science. The organisation's events attract international attendees, from the UK and Europe as well as the USA. These events include programmes that can be applied to industrial operations, academic research and community projects.[3]
History
The Newton Gateway to Mathematics began in 2013 seed financed by the University of Cambridge's Higher Education Innovation Funding. Subsequent funding came from corporate and philanthropic partners. Originally named after the UK computer scientist and mathematician Alan Turing (1912–1954), the initiative was set up as a gateway to mathematics for people to gain a deeper insight on problem solving. It rebranded to become the Newton Gateway to Mathematics in January 2019.
Turing helped decrypt German military codes for the UK government during World War II. Following the war he designed the Automatic Computing Engine (ACE), an early electronic stored-programme computer based on a paper Turing wrote in 1945 called "Proposed Electronic Calculator."[4]
One of the first initiatives that set programmes in motion at the Turing Gateway to Mathematics was the 1st UK workshop on Optimisation in Space Engineering (OSE) in November 2013. Held in Birmingham, this workshop in association with the European Space Agency (ESA) and the University of Southampton, discussed issues for reaching technological solutions such as:
• interplanetary trajectory optimisation
• non-circular spacecraft orbits
• landing trajectories
A follow-up workshop on Optimisation in Space Engineering laid the groundwork for identifying challenges in the UK's aerospace industry. A third OSE workshop in September 2015 planned future workshops to further discuss problems and solutions for space engineering.
In 2014 and 2015 a programme was presented to explore solutions for the public sector, funded by the Engineering and Physical Sciences Research Council (EPSRC). Following a launch event at the Royal Society in London, subsequent events attracted delegates from government and academic organisations to focus on mathematics and public policy issues. The main themes consisted of the following:
• mathematics for future cities systems
• mathematical modelling of transport
• energy systems relating to modelling variability
• environment and climate change probability modelling
• health and society workshop associated with math insights
• understanding data from various perspectives
• optimisation of immunisation programmes
• ageing population preparation and shifting demographics
• health and disability modelling
The Newton Gateway to Mathematics began publishing its quarterly newsletter in January 2015. This publication keeps partners and other interested individuals up to data on its events, programmes and workshops. Throughout its inception, the Newton Gateway has played a key role in bringing together scientists to discuss various types of multimodal clinical imaging that has addressed the challenges involved with solutions for cancer, heart disease and antibiotic-resistant bacteria, among other major health concerns.
Multimodality analysis has also been applied to Big Data involving advanced concepts from biology to medicine at these gatherings.[5] Cutting edge imaging technologies have been showcased, such as multi-contrast magnetic resonance tomography (MRT), positron emission tomography (PET) and dynamic imaging. This technology targets engineers, mathematicians, biologists and other scientists who work with big data analytics.
Over the years the Newton Gateway to Mathematics has branched out into covering diverse themes that span the entire field of mathematics. With its acclaimed personnel and partners, it aspires to be one of the most intriguing and informative centres for mathematical and scientific knowledge sharing in the world. Its affiliation with the University of Cambridge, one of the longest-running campuses of all time, and the Isaac Newton Institute, give it a prestigious profile, which is why it attracts prominent experts in the fields of math and science for speaking opportunities.[6][1]
In 2017, events that took place included the Environmental Modelling in Industry Study Group,[7] Developments in Healthcare Imagine – Connecting with Academia, Data Sharing and Governance and the 2nd Edwards Symposium.
In January 2019, the organisation rebranded, changing its name to the Newton Gateway to Mathematics in order to avoid confusion with other organisations.[1]
Location and buildings
The Newton Gateway is located at 20 Clarkson Road, Cambridge, CB3, OEH, UK at the University of Cambridge, about 50 miles north of London. The Faulkes Gatehouse, located within the University of Cambridge's Centre for Mathematical Sciences Site, was financed by the Dill Faulkes Educational Trust (DFET). Construction of the building, which consists of a semi-circular room for seminars and three offices, was completed in June 2001. The seminar room has a capacity for 50 people. Two of the three offices are occupied by the Newton Gateway to Mathematics.[6][3]
Organisation and administration
The Newton Gateway employs three full-time staff members, in which the Manager is responsible to the Director of the Isaac Newton Institute. The Isaac Newton Institute's Management Committee oversees the budget for both the Gateway's short-term and long-term fiscal planning, while the Gateway to Mathematics' staff handles day-to-day work. Ultimately, the Isaac Newton Institute's Director is head of the entire operation. The current staff, as of 2018, includes Manager Jane Leeks, Knowledge Exchange Coordinator Clare Merritt and Events and Marketing Coordinator Lissie Hope.[4]
Profile and mission
A major goal of the Newton Gateway to Mathematics is to widen overall access to mathematics and accelerate learning curves for developing math skills. As a knowledge facilitator, the Gateway to Mathematics helps connect experts with knowledge seekers, particularly business executives who want to improve their companies with more efficient technology. The events cover a broad range of disciplines related to science and mathematics. In other words, every area of maths is considered in the Newton Gateway to Mathematics' planning for knowledge sharing events. These events are designed to help attendees learn solutions to organisational problems, frequently involving data management.[8][3] Among the contributors are renowned scientists like Onno Bokove, Carola-Bibiane Schönlieb, John Aston, Patrick Wolfe, Sofia Olhede, Sofia Olhede, Mike Cates, Mike Cates, Raphael Blumenfeld or Mark Warner, FRS.
While some meetings can be characterised as industry conferences, others are more casual receptions for networking. Annual dinners also give members opportunities to expand contacts and share information. As a catalyst to stimulate thinking on maths problems, the Newton Gateway to Mathematics looks for innovative approaches to make maths more understandable. This strategy, as an example, helps inject new ideas into analysis of UK economic issues and managing health care data. Types of professionals who may be asked as guest speakers include:
• accountants
• medical professionals
• science community researchers
• economists and financial analysts
• math professors
• investors and fund managers
• banking professionals
• government officials
• tax consultants and financial planners
• technology leaders
• aerospace engineers
The Newton Gateway to Mathematics sometimes partners with the external organisations to deliver Open for Business events. Another example of an event which involves lunches, dinners and hotel accommodations, is the tribute to 20th century renowned applied mathematics author Joseph "Joe" Keller in 2017. Supporters of programmes beyond Isaac Newton Institute and University of Cambridge include University of Oxford, Cambridge University Press, Schlumberger and the Institute of Mathematics and its Applications.
One of the main ways for organisations and individuals to get involved with the Newton Gateway to Mathematics and gain access to its knowledge sharing activities is to become a member of the Newton Gateway to Mathematics Partnership Scheme. Partners get to attend exclusive events where they can network with academic, government and corporate representatives to gain further insights on mathematical problem solving. The group organises about 15–20 events per year. Additionally, partners can promote their brands on the initiative's website and newsletters, as well as get visibility in the Newton Gateway to Mathematics' Annual Report.[1][9]
References
1. "About the Turing Gateway to Mathematics". Turing Gateway to Mathematics. Retrieved 8 February 2017.
2. "Isaac Newton Institute for Mathematical Sciences – Details of the Grant". Engineering and Physical Sciences Research Council. Retrieved 8 February 2017.
3. "Turing Gateway to Mathematics". Isaac Newton Institute. Retrieved 8 February 2017.
4. Turing, Alan M. "Proposed electronic calculator". Oxford Index. Retrieved 7 February 2017.
5. "Coping with Big Data – an Analytics and Computational Perspective". Institute of Mathematics and Its Applications (IMA). Retrieved 8 February 2017.
6. Cosper, Alex. "Turing Gateway to Mathematics - the knowledge transfer highway from academia to industry". Crossroads Today. Retrieved 8 February 2017.
7. "Sewer network challenge at MathsForesees study group 2017 | DARE: Data Assimilation for the REsilient City". blogs.reading.ac.uk. Retrieved 31 July 2018.
8. "Update from the Turing Gateway to Mathematics". Connect. Retrieved 8 February 2017.
9. Mackenzie, Kirill C. H. "Turing Gateway to Mathematics" (PDF). University of Sheffield. Retrieved 8 February 2017.
External links
• Official website
• Interactive map of the Mathematical Sciences site including links to the departments.
The European Mathematical Society
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• European Consortium for Mathematics in Industry
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• Catalan Society of Mathematics
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• Edinburgh Mathematical Society
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Academic Institutional Members
• Abdus Salam International Centre for Theoretical Physics
• Academy of Sciences of Moldova
• Bernoulli Center
• Centre de Recerca Matemàtica
• Centre International de Rencontres Mathématiques
• Centrum voor Wiskunde en Informatica
• Emmy Noether Research Institute for Mathematics
• Erwin Schrödinger International Institute for Mathematical Physics
• European Institute for Statistics, Probability and Operations Research
• Institut des Hautes Études Scientifiques
• Institut Henri Poincaré
• Institut Mittag-Leffler
• Institute for Mathematical Research
• International Centre for Mathematical Sciences
• Isaac Newton Institute for Mathematical Sciences
• Mathematisches Forschungsinstitut Oberwolfach
• Mathematical Research Institute
• Max Planck Institute for Mathematics in the Sciences
• Research Institute of Mathematics of the Voronezh State University
• Serbian Academy of Science and Arts
• Mathematical Society of Serbia
• Stefan Banach International Mathematical Center
• Thomas Stieltjes Institute for Mathematics
Institutional Members
• Central European University
• Faculty of Mathematics at the University of Barcelona
• Cellule MathDoc
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Wikipedia
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Turing machine examples
The following are examples to supplement the article Turing machine.
Turing machines
Machine
• Turing machine equivalents
• Turing machine examples
• Turing machine gallery
Variants
• Alternating Turing machine
• Neural Turing machine
• Nondeterministic Turing machine
• Quantum Turing machine
• Post–Turing machine
• Probabilistic Turing machine
• Multitape Turing machine
• Multi-track Turing machine
• Symmetric Turing machine
• Total Turing machine
• Unambiguous Turing machine
• Universal Turing machine
• Zeno machine
Science
• Alan Turing
• Category:Turing machine
Turing's very first example
The following table is Turing's very first example (Alan Turing 1937):
"1. A machine can be constructed to compute the sequence 0 1 0 1 0 1..." (0 <blank> 1 <blank> 0...) (Undecidable p. 119)
Configuration Behavior
m-configuration
(state)
Tape symbol Tape operations Final m-configuration
(state)
b blank P0, R c
c blank R e
e blank P1, R f
f blank R b
With regard to what actions the machine actually does, Turing (1936) (Undecidable p. 121) states the following:
"This [example] table (and all succeeding tables of the same kind) is to be understood to mean that for a configuration described in the first two columns the operations in the third column are carried out successively, and the machine then goes over into the m-configuration in the final column." (Undecidable p. 121)
He makes this very clear when he reduces the above table to a single instruction called "b" (Undecidable p. 120), but his instruction consists of 3 lines. Instruction "b" has three different symbol possibilities {None, 0, 1}. Each possibility is followed by a sequence of actions until we arrive at the rightmost column, where the final m-configuration is "b":
Current m-configuration (instruction) Tape symbol Operations on the tape Final m-configuration (instruction)
b None P0 b
b 0 R, R, P1 b
b 1 R, R, P0 b
As observed by a number of commentators including Turing (1937) himself, (e.g., Post (1936), Post (1947), Kleene (1952), Wang (1954)) the Turing instructions are not atomic — further simplifications of the model can be made without reducing its computational power; see more at Post–Turing machine.
As stated in the article Turing machine, Turing proposed that his table be further atomized by allowing only a single print/erase followed by a single tape movement L/R/N. He gives us this example of the first little table converted (Undecidable, p. 127):
Current m-configuration (Turing state) Tape symbol Print-operation Tape-motion Final m-configuration (Turing state)
q1 blank P0 R q2
q2 blank P blank, i.e. E R q3
q3 blank P1 R q4
q4 blank P blank, i.e. E R q1
Turing's statement still implies five atomic operations. At a given instruction (m-configuration) the machine:
1. observes the tape-symbol underneath the head
2. based on the observed symbol goes to the appropriate instruction-sequence to use
3. prints symbol Sj or erases or does nothing
4. moves tape left, right or not at all
5. goes to the final m-configuration for that symbol
Because a Turing machine's actions are not atomic, a simulation of the machine must atomize each 5-tuple into a sequence of simpler actions. One possibility — used in the following examples of "behaviors" of his machine — is as follows:
(qi) Test tape-symbol under head: If the symbol is S0 go to qi.01, if symbol S1 go to qi.11, if symbol S2 go to qi.21, etc.
(qi.01) print symbol Sj0 or erase or do nothing then go to qi.02
(qi.02) move tape left or right nor not at all then go to qm0
(qi.11) print symbol Sj1 or erase or do nothing then go to qi.12
(qi.12) move tape left or right nor not at all then go to qm1
(qi.21) print symbol Sj2 or erase or do nothing then go to qi.22
(qi.22) move tape left or right nor not at all then go to qm2
(etc — all symbols must be accounted for)
So-called "canonical" finite state machines do the symbol tests "in parallel"; see more at microprogramming.
In the following example of what the machine does, we will note some peculiarities of Turing's models:
"The convention of writing the figures only on alternate squares is very useful: I shall always make use of it." (Undecidable p. 121)
Thus when printing he skips every other square. The printed-on squares are called F-squares; the blank squares in between may be used for "markers" and are called "E-squares" as in "liable to erasure." The F-squares in turn are his "Figure squares" and will only bear the symbols 1 or 0 — symbols he called "figures" (as in "binary numbers").
In this example the tape starts out "blank", and the "figures" are then printed on it. For brevity only the TABLE-states are shown here:
Sequence Instruction identifier Head
. . . . . . . . . . . . . . . . . .
1 1 . . . . . . . . . . . . . . . . . .
2 2 . . . . . 0 . . . . . . . . . . . .
3 3 . . . . . . 0 . . . . . . . . . . .
4 4 . . . . . 1 . 0 . . . . . . . . . .
5 1 . . . . . . 1 . 0 . . . . . . . . .
6 2 . . . . . 0 . 1 . 0 . . . . . . . .
7 3 . . . . . . 0 . 1 . 0 . . . . . . .
8 4 . . . . . 1 . 0 . 1 . 0 . . . . . .
9 1 . . . . . . 1 . 0 . 1 . 0 . . . . .
10 2 . . . . . 0 . 1 . 0 . 1 . 0 . . . .
11 3 . . . . . . 0 . 1 . 0 . 1 . 0 . . .
12 4 . . . . . 1 . 0 . 1 . 0 . 1 . 0 . .
13 1 . . . . . . 1 . 0 . 1 . 0 . 1 . 0 .
14 2 . . . . . 0 . 1 . 0 . 1 . 0 . 1 . 0
The same "run" with all the intermediate tape-printing and movements is shown here:
A close look at the table reveals certain problems with Turing's own example—not all the symbols are accounted for.
For example, suppose his tape was not initially blank. What would happen? The Turing machine would read different values than the intended values.
A copy subroutine
This is a very important subroutine used in the "multiply" routine.
The example Turing machine handles a string of 0s and 1s, with 0 represented by the blank symbol. Its task is to double any series of 1s encountered on the tape by writing a 0 between them. For example, when the head reads "111", it will write a 0, then "111". The output will be "1110111".
In order to accomplish its task, this Turing machine will need only 5 states of operation, which are called {s1, s2, s3, s4, s5}. Each state does 4 actions:
1. Read the symbol under the head
2. Write the output symbol decided by the state
3. Move the tape to the left or to the right decided by the state
4. Switch to the following state decided by the current state
Initial m-configuration (current instruction) Tape symbol Print operation Tape motion Final m-configuration (next instruction)
s1 0 N N H
s1 1 E R s2
s2 0 E R s3
s2 1 P1 R s2
s3 0 P1 L s4
s3 1 P1 R s3
s4 0 E L s5
s4 1 P1 L s4
s5 0 P1 R s1
s5 1 P1 L s5
H — — —
A "run" of the machine sequences through 16 machine-configurations (aka Turing states):
Sequence Instruction identifier Head
1 s1 0 0 0 0 1 1 0 0 0 0 0
2 s2 0 0 0 0 0 1 0 0 0 0 0
3 s2 0 0 0 0 0 0 1 0 0 0 0
4 s3 0 0 0 0 0 0 0 1 0 0 0
5 s4 0 0 0 0 1 0 1 0 0 0 0
6 s5 0 0 0 1 0 1 0 0 0 0 0
7 s5 0 0 1 0 1 0 0 0 0 0 0
8 s1 0 0 0 1 0 1 1 0 0 0 0
9 s2 0 0 0 0 1 0 0 1 0 0 0
10 s3 0 0 0 0 0 1 0 0 1 0 0
11 s3 0 0 0 0 0 0 1 0 0 1 0
12 s4 0 0 0 0 1 1 0 0 1 0 0
13 s4 0 0 0 1 1 0 0 1 0 0 0
14 s5 0 0 1 1 0 0 1 0 0 0 0
15 s1 0 0 0 1 1 0 1 1 0 0 0
16 H 0 0 0 1 1 0 1 1 0 0 0
The behavior of this machine can be described as a loop: it starts out in s1, replaces the first 1 with a 0, then uses s2 to move to the right, skipping over 1s and the first 0 encountered. s3 then skips over the next sequence of 1s (initially there are none) and replaces the first 0 it finds with a 1. s4 moves back to the left, skipping over 1s until it finds a 0 and switches to s5. s5 then moves to the left, skipping over 1s until it finds the 0 that was originally written by s1.
It replaces that 0 with a 1, moves one position to the right and enters s1 again for another round of the loop.
This continues until s1 finds a 0 (this is the 0 in the middle of the two strings of 1s) at which time the machine halts.
Alternative description
Another description sees the problem as how to keep track of how many "1"s there are. We can't use one state for each possible number (a state for each of 0,1,2,3,4,5,6 etc), because then we'd need infinite states to represent all the natural numbers, and the state machine is finite - we'll have to track this using the tape in some way.
The basic way it works is by copying each "1" to the other side, by moving back and forth - it is intelligent enough to remember which part of the trip it is on. In more detail, it carries each "1" across to the other side, by recognizing the separating "0" in the middle, and recognizing the "0" on the other side to know it's reached the end. It comes back using the same method, detecting the middle "0", and then the "0" on the original side. This "0" on the original side is the key to the puzzle of how it keeps track of the number of 1's.
The trick is that before carrying the "1", it marks that digit as "taken" by replacing it with an "0". When it returns, it fills that "0" back in with a "1", then moves on to the next one, marking it with an "0" and repeating the cycle, carrying that "1" across and so on. With each trip across and back, the marker "0" moves one step closer to the centre. This is how it keeps track of how many "1"'s it has taken across.
When it returns, the marker "0" looks like the end of the collection of "1"s to it - any "1"s that have already been taken across are invisible to it (on the other side of the marker "0") and so it is as if it is working on an (N-1) number of "1"s - similar to a proof by mathematical induction.
A full "run" showing the results of the intermediate "motions". To see it better click on the image, then click on the higher resolution download:
3-state Busy Beaver
The following Turing table of instructions was derived from Peterson (1988) page 198, Figure 7.15. Peterson moves the head; in the following model the tape moves.
Tape symbol Current state A Current state B Current state C
Write symbol Move tape Next state Write symbol Move tape Next state Write symbol Move tape Next state
0 1 R B 1 L A 1 L B
1 1 L C 1 R B 1 N HALT
The "state" drawing of the 3-state busy beaver shows the internal sequences of events required to actually perform "the state". As noted above Turing (1937) makes it perfectly clear that this is the proper interpretation of the 5-tuples that describe the instruction (Undecidable, p. 119). For more about the atomization of Turing 5-tuples see Post–Turing machine:
The following table shows the "compressed" run — just the Turing states:
Sequence Instruction identifier Head
1 b 0 0 0 0 0 0 0 0 0 0 0 0 0 0
2 B 0 0 0 0 0 0 0 1 0 0 0 0 0 0
3 A 0 0 0 0 0 1 1 0 0 0 0 0 0 0
4 C 0 0 0 0 1 1 0 0 0 0 0 0 0 0
5 B 0 0 0 1 1 1 0 0 0 0 0 0 0 0
6 A 0 0 1 1 1 1 0 0 0 0 0 0 0 0
7 B 0 0 0 1 1 1 1 1 0 0 0 0 0 0
8 B 0 0 0 0 1 1 1 1 1 0 0 0 0 0
9 B 0 0 0 0 0 1 1 1 1 1 0 0 0 0
10 B 0 0 0 0 0 0 1 1 1 1 1 0 0 0
11 B 0 0 0 0 0 0 0 1 1 1 1 1 0 0
12 A 0 0 0 0 0 1 1 1 1 1 1 0 0 0
13 C 0 0 0 0 1 1 1 1 1 1 0 0 0 0
14 H 0 0 0 0 1 1 1 1 1 1 0 0 0 0
The full "run" of the 3-state busy beaver. The resulting Turing-states (what Turing called the "m-configurations" — "machine-configurations") are shown highlighted in grey in column A, and also under the machine's instructions (columns AF-AU)):
References
For complete references see Turing machine.
• Ivars Peterson, 1988, The Mathematical Tourist: Snapshots of Modern Mathematics, W. H. Freeman and Company, New York, ISBN 0-7167-2064-7 (pbk.). Turing machines are described on pp. 194ff, the busy beaver example is in Figure 7.15 on page 198.
• Martin Davis editor, 1965, The Undecidable: Basic Papers on Undecidable Propositions, Unsolvable Problems and Computable Functions, Raven Press, New York, no ISBN, no card catalog number.
• Alan Turing, 1937, On Computable Numbers, with an Application to the Entscheidungsproblem, pp. 116ff, with brief commentary by Davis on page 115.
• Alan Turing, 1937, On Computable Numbers, with an Application to the Entscheidungsproblem. A Correction, p. 152-154.
Automata theory: formal languages and formal grammars
Chomsky hierarchyGrammarsLanguagesAbstract machines
• Type-0
• —
• Type-1
• —
• —
• —
• —
• —
• Type-2
• —
• —
• Type-3
• —
• —
• Unrestricted
• (no common name)
• Context-sensitive
• Positive range concatenation
• Indexed
• —
• Linear context-free rewriting systems
• Tree-adjoining
• Context-free
• Deterministic context-free
• Visibly pushdown
• Regular
• —
• Non-recursive
• Recursively enumerable
• Decidable
• Context-sensitive
• Positive range concatenation*
• Indexed*
• —
• Linear context-free rewriting language
• Tree-adjoining
• Context-free
• Deterministic context-free
• Visibly pushdown
• Regular
• Star-free
• Finite
• Turing machine
• Decider
• Linear-bounded
• PTIME Turing Machine
• Nested stack
• Thread automaton
• restricted Tree stack automaton
• Embedded pushdown
• Nondeterministic pushdown
• Deterministic pushdown
• Visibly pushdown
• Finite
• Counter-free (with aperiodic finite monoid)
• Acyclic finite
Each category of languages, except those marked by a *, is a proper subset of the category directly above it. Any language in each category is generated by a grammar and by an automaton in the category in the same line.
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Turing machine gallery
The following article is a supplement to the article Turing machine.
Turing machine as a mechanical device
The Turing machine shown here consists of a special paper tape that can be erased as well as written with a "tally mark". Perhaps the TABLE is made out of a similar "read only" paper tape reader, or perhaps it reads punched cards. Turing's biographer Andrew Hodges (1983) has written that Turing as a child liked typewriters. A "'miraculous machine' -- a mechanical process which could work on Hilbert's decision problem" (Hodges p. 98) had been suggested by G. H. Hardy, one of Turing's teachers. Nevertheless, "His machine had no obvious model in anything that existed in 1936, except in general terms of the new electrical industries, with their teleprinters, television 'scanning', and automatic telephone exchange connections. It was his own invention." (Hodges p. 109)
Davis (2000) says that Turing built a binary multiplier out of electromechanical relays (p. 170). As noted in the history section of algorithm punched or printed paper tape and punched paper cards were commonplace in the 1930s.
Boolos and Jeffrey (1974, 1999) note that "being in one state or another might be a matter of having one or another cog of a certain gear uppermost..." (p. 21).
Turing machine as a "poor mug" inside a box pulling the box along a rail
"We are not concerned with the mechanics or the electronics of the matter. Perhaps the simplest way to picture the thing is quite crudely: Inside the box there is a man, who does all the reading and writing and erasing and moving. (The box has no bottom: the poor mug just walks along between the ties, pulling the box with him.) The man has a list of m instructions written down on a piece of paper. He is in state qi when he is carrying out instruction number i [etc.]" (Boolos and Jeffrey (1974, 1999) p.21)
This description closely follows Emil Post's "Formulation I" for a "finite combinatory process": a man, equipped with and following a "fixed unalterable set of instructions", moving left or right through "an infinite sequence of spaces or boxes" and while in a box either printing on a piece of paper a single "vertical stroke" or erasing it (Post 1936 reprinted in Undecidable p. 289). Post's formulation was the first of its type to be published; it preceded Turing's by a matter of a few months.
Both descriptions—Post's, and Boolos and Jeffrey's — use simpler 4-tuples rather than 5-tuples to define the 'm-configurations' (instructions) of their processes.
A robot carries out the instructions
This model was suggested by Stone (1972):
"Let us adopt the point of view that a computer is a robot that will perform any task that can be described as a sequence of instructions" (p. 3).
Stone uses the robot to develop his notion of algorithm. This in turn leads him to his description of the Turing machine and his statement:
"The evidence seems to indicate that every algorithm for any computing device has an equivalent Turing machine algorithm ... if [Church's thesis] is true, it is certainly remarkable that Turing machines, with their extremely primitive operations, are capable of performing any computation that any other device can perform, regardless of how complex a device we choose." (p. 13)
Other images
• An artistic representation of a Turing Machine
• Model of a Turing machine
References
See the main article Turing machine for references.
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Description number
Description numbers are numbers that arise in the theory of Turing machines. They are very similar to Gödel numbers, and are also occasionally called "Gödel numbers" in the literature. Given some universal Turing machine, every Turing machine can, given its encoding on that machine, be assigned a number. This is the machine's description number. These numbers play a key role in Alan Turing's proof of the undecidability of the halting problem, and are very useful in reasoning about Turing machines as well.
An example of a description number
Say we had a Turing machine M with states q1, ... qR, with a tape alphabet with symbols s1, ... sm, with the blank denoted by s0, and transitions giving the current state, current symbol, and actions performed (which might be to overwrite the current tape symbol and move the tape head left or right, or maybe not move it at all), and the next state. Under the original universal machine described by Alan Turing, this machine would be encoded as input to it as follows:
1. The state qi is encoded by the letter 'D' followed by the letter 'A' repeated i times (a unary encoding)
2. The tape symbol sj is encoded by the letter 'D' followed by the letter 'C' repeated j times
3. The transitions are encoded by giving the state, input symbol, symbol to write on the tape, direction to move (expressed by the letters 'L', 'R', or 'N', for left, right, or no movement), and the next state to enter, with states and symbols encoded as above.
The UTM's input thus consists of the transitions separated by semicolons, so its input alphabet consists of the seven symbols, 'D', 'A', 'C', 'L', 'R', 'N', and ';'. For example, for a very simple Turing machine that alternates printing 0 and 1 on its tape forever:
1. State: q1, input symbol: blank, action: print 1, move right, next state: q2
2. State: q2, input symbol: blank, action: print 0, move right, next state: q1
Letting the blank be s0, '0' be s1 and '1' be s2, the machine would be encoded by the UTM as:
DADDCCRDAA;DAADDCRDA;
But then, if we replaced each of the seven symbols 'A' by 1, 'C' by 2, 'D' by 3, 'L' by 4, 'R' by 5, 'N' by 6, and ';' by 7, we would have an encoding of the Turing machine as a natural number: this is the description number of that Turing machine under Turing's universal machine. The simple Turing machine described above would thus have the description number 313322531173113325317. There is an analogous process for every other type of universal Turing machine. It is usually not necessary to actually compute a description number in this way: the point is that every natural number may be interpreted as the code for at most one Turing machine, though many natural numbers may not be the code for any Turing machine (or to put it another way, they represent Turing machines that have no states). The fact that such a number always exists for any Turing machine is generally the important thing.
Application to undecidability proofs
Description numbers play a key role in many undecidability proofs, such as the proof that the halting problem is undecidable. In the first place, the existence of this direct correspondence between natural numbers and Turing machines shows that the set of all Turing machines is denumerable, and since the set of all partial functions is uncountably infinite, there must certainly be many functions that cannot be computed by Turing machines.
By making use of a technique similar to Cantor's diagonal argument, it is possible exhibit such an uncomputable function, for example, that the halting problem in particular is undecidable. First, let us denote by U(e, x) the action of the universal Turing machine given a description number e and input x, returning 0 if e is not the description number of a valid Turing machine. Now, supposing that there were some algorithm capable of settling the halting problem, i.e. a Turing machine TEST(e) which given the description number of some Turing machine would return 1 if the Turing machine halts on every input, or 0 if there are some inputs that would cause it to run forever. By combining the outputs of these machines, it should be possible to construct another machine δ(k) that returns U(k, k) + 1 if TEST(k) is 1 and 0 if TEST(k) is 0. From this definition δ is defined for every input and must naturally be total recursive. Since δ is built up from what we have assumed are Turing machines as well then it too must have a description number, call it e. So, we can feed the description number e to the UTM again, and by definition, δ(k) = U(e, k), so δ(e) = U(e, e). But since TEST(e) is 1, by our other definition, δ(e) = U(e, e) + 1, leading to a contradiction. Thus, TEST(e) cannot exist, and in this way we have settled the halting problem as undecidable.
See also
• Gödel number
• Universal Turing machine
• Church numeral
• Halting problem
References
• John Hopcroft and Jeffrey D. Ullman (1979). Introduction to Automata Theory, Languages, and Computation (1st ed.). Addison-Wesley. ISBN 0-201-44124-1. (the Cinderella book)
• Turing, A. M. "On computable numbers, with an application to the Entscheidungsproblem", Proc. Roy. Soc. London, 2(42), 1936, pp. 230–265.
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Turing reduction
In computability theory, a Turing reduction from a decision problem $A$ to a decision problem $B$ is an oracle machine which decides problem $A$ given an oracle for $B$ (Rogers 1967, Soare 1987). It can be understood as an algorithm that could be used to solve $A$ if it had available to it a subroutine for solving $B$. The concept can be analogously applied to function problems.
If a Turing reduction from $A$ to $B$ exists, then every algorithm for $B$[lower-alpha 1] can be used to produce an algorithm for $A$, by inserting the algorithm for $B$ at each place where the oracle machine computing $A$ queries the oracle for $B$. However, because the oracle machine may query the oracle a large number of times, the resulting algorithm may require more time asymptotically than either the algorithm for $B$ or the oracle machine computing $A$. A Turing reduction in which the oracle machine runs in polynomial time is known as a Cook reduction.
The first formal definition of relative computability, then called relative reducibility, was given by Alan Turing in 1939 in terms of oracle machines. Later in 1943 and 1952 Stephen Kleene defined an equivalent concept in terms of recursive functions. In 1944 Emil Post used the term "Turing reducibility" to refer to the concept.
Definition
Given two sets $A,B\subseteq \mathbb {N} $ of natural numbers, we say $A$ is Turing reducible to $B$ and write
$A\leq _{T}B$
if and only if there is an oracle machine that computes the characteristic function of A when run with oracle B. In this case, we also say A is B-recursive and B-computable.
If there is an oracle machine that, when run with oracle B, computes a partial function with domain A, then A is said to be B-recursively enumerable and B-computably enumerable.
We say $A$ is Turing equivalent to $B$ and write $A\equiv _{T}B\,$ if both $A\leq _{T}B$ and $B\leq _{T}A.$ The equivalence classes of Turing equivalent sets are called Turing degrees. The Turing degree of a set $X$ is written ${\textbf {deg}}(X)$.
Given a set ${\mathcal {X}}\subseteq {\mathcal {P}}(\mathbb {N} )$, a set $A\subseteq \mathbb {N} $ is called Turing hard for ${\mathcal {X}}$ if $X\leq _{T}A$ for all $X\in {\mathcal {X}}$. If additionally $A\in {\mathcal {X}}$ then $A$ is called Turing complete for ${\mathcal {X}}$.
Relation of Turing completeness to computational universality
Turing completeness, as just defined above, corresponds only partially to Turing completeness in the sense of computational universality. Specifically, a Turing machine is a universal Turing machine if its halting problem (i.e., the set of inputs for which it eventually halts) is many-one complete. Thus, a necessary but insufficient condition for a machine to be computationally universal, is that the machine's halting problem be Turing-complete for the set ${\mathcal {X}}$ of recursively enumerable sets. Insufficient because it may still be the case that, the language accepted by the machine is not itself recursively enumerable.
Example
Let $W_{e}$ denote the set of input values for which the Turing machine with index e halts. Then the sets $A=\{e\mid e\in W_{e}\}$ and $B=\{(e,n)\mid n\in W_{e}\}$ are Turing equivalent (here $(-,-)$ denotes an effective pairing function). A reduction showing $A\leq _{T}B$ can be constructed using the fact that $e\in A\Leftrightarrow (e,e)\in B$. Given a pair $(e,n)$, a new index $i(e,n)$ can be constructed using the smn theorem such that the program coded by $i(e,n)$ ignores its input and merely simulates the computation of the machine with index e on input n. In particular, the machine with index $i(e,n)$ either halts on every input or halts on no input. Thus $i(e,n)\in A\Leftrightarrow (e,n)\in B$ holds for all e and n. Because the function i is computable, this shows $B\leq _{T}A$. The reductions presented here are not only Turing reductions but many-one reductions, discussed below.
Properties
• Every set is Turing equivalent to its complement.
• Every computable set is Turing reducible to every other set. Because any computable set can be computed with no oracle, it can be computed by an oracle machine that ignores the given oracle.
• The relation $\leq _{T}$ is transitive: if $A\leq _{T}B$ and $B\leq _{T}C$ then $A\leq _{T}C$. Moreover, $A\leq _{T}A$ holds for every set A, and thus the relation $\leq _{T}$ is a preorder (it is not a partial order because $A\leq _{T}B$ and $B\leq _{T}A$ does not necessarily imply $A=B$).
• There are pairs of sets $(A,B)$ such that A is not Turing reducible to B and B is not Turing reducible to A. Thus $\leq _{T}$ is not a total order.
• There are infinite decreasing sequences of sets under $\leq _{T}$. Thus this relation is not well-founded.
• Every set is Turing reducible to its own Turing jump, but the Turing jump of a set is never Turing reducible to the original set.
The use of a reduction
Since every reduction from a set $B$ to a set $A$ has to determine whether a single element is in $A$ in only finitely many steps, it can only make finitely many queries of membership in the set $B$. When the amount of information about the set $B$ used to compute a single bit of $A$ is discussed, this is made precise by the use function. Formally, the use of a reduction is the function that sends each natural number $n$ to the largest natural number $m$ whose membership in the set B was queried by the reduction while determining the membership of $n$ in $A$.
Stronger reductions
There are two common ways of producing reductions stronger than Turing reducibility. The first way is to limit the number and manner of oracle queries.
• Set $A$ is many-one reducible to $B$ if there is a total computable function $f$ such that an element $n$ is in $A$ if and only if $f(n)$ is in $B$. Such a function can be used to generate a Turing reduction (by computing $f(n)$, querying the oracle, and then interpreting the result).
• A truth table reduction or a weak truth table reduction must present all of its oracle queries at the same time. In a truth table reduction, the reduction also gives a boolean function (a truth table) which, when given the answers to the queries, will produce the final answer of the reduction. In a weak truth table reduction, the reduction uses the oracle answers as a basis for further computation depending on the given answers (but not using the oracle). Equivalently, a weak truth table reduction is one for which the use of the reduction is bounded by a computable function. For this reason, weak truth table reductions are sometimes called "bounded Turing" reductions.
The second way to produce a stronger reducibility notion is to limit the computational resources that the program implementing the Turing reduction may use. These limits on the computational complexity of the reduction are important when studying subrecursive classes such as P. A set A is polynomial-time reducible to a set $B$ if there is a Turing reduction of $A$ to $B$ that runs in polynomial time. The concept of log-space reduction is similar.
These reductions are stronger in the sense that they provide a finer distinction into equivalence classes, and satisfy more restrictive requirements than Turing reductions. Consequently, such reductions are harder to find. There may be no way to build a many-one reduction from one set to another even when a Turing reduction for the same sets exists.
Weaker reductions
According to the Church–Turing thesis, a Turing reduction is the most general form of an effectively calculable reduction. Nevertheless, weaker reductions are also considered. Set $A$ is said to be arithmetical in $B$ if $A$ is definable by a formula of Peano arithmetic with $B$ as a parameter. The set $A$ is hyperarithmetical in $B$ if there is a recursive ordinal $\alpha $ such that $A$ is computable from $B^{(\alpha )}$, the α-iterated Turing jump of $B$. The notion of relative constructibility is an important reducibility notion in set theory.
See also
• Karp reduction
Notes
1. It is possible that B is an undecidable problem for which no algorithm exists.
References
• M. Davis, ed., 1965. The Undecidable—Basic Papers on Undecidable Propositions, Unsolvable Problems and Computable Functions, Raven, New York. Reprint, Dover, 2004. ISBN 0-486-43228-9.
• S. C. Kleene, 1952. Introduction to Metamathematics. Amsterdam: North-Holland.
• S. C. Kleene and E. L. Post, 1954. "The upper semi-lattice of degrees of recursive unsolvability". Annals of Mathematics v. 2 n. 59, 379–407.
• Post, E. L. (1944). "Recursively enumerable sets of positive integers and their decision problems" (PDF). Bulletin of the American Mathematical Society. 50 (5): 284–316. doi:10.1090/s0002-9904-1944-08111-1. Retrieved 2015-12-17.
• A. Turing, 1939. "Systems of logic based on ordinals." Proceedings of the London Mathematics Society, ser. 2 v. 45, pp. 161–228. Reprinted in "The Undecidable", M. Davis ed., 1965.
• H. Rogers, 1967. Theory of recursive functions and effective computability. McGraw-Hill.
• R. Soare, 1987. Recursively enumerable sets and degrees, Springer.
• Davis, Martin (November 2006). "What is...Turing Reducibility?" (PDF). Notices of the American Mathematical Society. 53 (10): 1218–1219. Retrieved 2008-01-16.
External links
• NIST Dictionary of Algorithms and Data Structures: Turing reduction
• University of Cambridge, Andrew Pitts, Tobias Kohn: Computation Theory
• Prof. Jean Gallier’s Homepage
Authority control: National
• Germany
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Turing switch
In theoretical network science, the Turing switch is a logical construction modeling the operation of the network switch, just as in theoretical computer science a Turing machine models the operation of a computer. Both are named in honor of the English logician Alan Turing, although the research in Turing switches is not based on Turing's research. Some introductory research on the Turing switch was started at the University of Cambridge by Jon Crowcroft (Homepage).
In essence, Crowcroft suggests that instead of using general-purpose computers to do packet switching, the required operations should be reduced to application specific logic and then that application specific logic should be implemented using optical components. The work is not actually based on Turing's research.
A Turing switch consists of a switched fabric, one or more ingress interfaces (also referred to as sources), one or more egress interfaces (sinks), and a decision procedure to determine an egress interface given an ingress interface. Interfaces are sometimes referred to as ports. A packet (cell or switched unit) arrives at an ingress interface, the appropriate egress interface is determined by the decision procedure, and the packet is then transported across the switching fabric to the egress interface. A packet is a symbol or sequence of 1's and 0's. An ingress interface is connected to an ingress line and an egress interface to an egress line. The ingress line is said to feed the ingress interface; the egress interface feeds the egress line.[1]
See also
• Network switch
• Software-defined networking
References
1. Jon Crowcroft Turing Switches. Turing machines for all-optical Internet routing UCAM-CL-TR-556 ISSN 1476-2986 January 2003
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Turkish Mathematical Society
The Turkish Mathematical Society (Turkish: Türk Matematik Derneği, TMD) is a Turkish organization dedicated to the development of mathematics in Turkey. Its members are individual mathematicians living in Turkey or Turkish mathematicians living abroad.
Turkish Mathematical Society
Türk Matematik Derneği
Formation1948
HeadquartersIstanbul, Turkey
Location
• Turkey
Official language
Turkish
President
Burak Özbağcı
Websitetmd.org.tr
Goals
The Society seeks to serve mathematicians particularly in universities, research institutes and other forms of higher education. Its aims are to
1. Promote mathematical research in Turkey,
2. Concern itself with the broader relations of mathematics to society,
3. Assist and advise on problems of mathematical education,
4. Try to establish solidarity among mathematicians,
5. Cooperate with or join national and international institutions with common aims,
6. Raise public awareness of mathematics.
The TMD is a Member of the International Mathematical Union (IMU),[1] the European Mathematical Society (EMS)[2] and the Mathematical Society of South Eastern Europe (MASSEE).[3] In 2012, the society published a report of education and research policy in Turkey in relation to mathematics, and has attempted to influence national policy since.[4][5]
History
The Turkish Mathematical Society was founded in 1948, by eminent researchers of the Istanbul University and Istanbul Technical University, including Cahit Arf, Mustafa İnan, and Nazım Terzioğlu.[6] It became a full member of IMU in 1960 and was raised to Group II in 2016.[7] TMD became a member of the EMS in 2008 and of the MASSEE in 2014. The Society is located in Istanbul and has had a branch in Ankara since 1992. An annual symposium has been organized every year for the last 30 years.[8] A popular quarterly mathematics magazine, Matematik Dünyası, has been published since 1991.
Presidents
• Until 1976: Nazim Terzioglu
• 1976–1982: Cahit Arf
• 1982–1986: Fikret Kortel
• 1986–1989: Cahit Arf
• 1989–2008: Tosun Terzioğlu
• 2008–2010: Ali Ülger
• 2010–2016: Betül Tanbay
• 2016–present: Attila Aşkar
Structure and governance
The governing body of the TMD is its General Assembly, consisting of all full members. The General Assembly meets every two years, and appoints the Executive Committee members who are responsible for the running of the society.
See also
• List of mathematical societies
References
1. "Turkey". International Mathematical Union. Retrieved 2018-01-31.
2. "Corporate Members". European Mathematical Society. Retrieved 2018-01-31.
3. "Members and Management Board". Mathematical Society of South Eastern Europe. Retrieved 2018-01-31.
4. "Matematikte vizyon ve strateji yok". www.haberturk.com (in Turkish). Retrieved 2018-03-18.
5. "'Analizsiz matematik evrimsiz biyoloji ezberle felsefe olmaz'". Hürriyet (in Turkish). Retrieved 2018-03-18.
6. "'Cumhuriyetin öncü kadınları' sergide buluşacak". Radikal (in Turkish). Retrieved 2018-03-18.
7. the_technician. "International Mathematical Union (IMU): Group II". imuweb.mathunion.org. Retrieved 2018-01-31.
8. "TMD – Türk Matematik Derneği". tmd.org.tr. Retrieved 2018-01-31.
External links
• The Turkish Mathematical Society
• The European Mathematical Society Homepage
• The International Mathematical Union Homepage
• The Mathematical Society of South-Eastern Europe Homepage
• Matematik Dünyası (quarterly mathematics magazine - in Turkish)
Authority control
• VIAF
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Turnstile (symbol)
In mathematical logic and computer science the symbol ⊢ ($\vdash $) has taken the name turnstile because of its resemblance to a typical turnstile if viewed from above. It is also referred to as tee and is often read as "yields", "proves", "satisfies" or "entails".
Interpretations
The turnstile represents a binary relation. It has several different interpretations in different contexts:
• In epistemology, Per Martin-Löf (1996) analyzes the $\vdash $ symbol thus: "...[T]he combination of Frege's Urteilsstrich, judgement stroke [ | ], and Inhaltsstrich, content stroke [—], came to be called the assertion sign."[1] Frege's notation for a judgement of some content A
$\vdash A$
can then be read
I know A is true.[2]
In the same vein, a conditional assertion
$P\vdash Q$
can be read as:
From P, I know that Q
• In metalogic, the study of formal languages; the turnstile represents syntactic consequence (or "derivability"). This is to say, that it shows that one string can be derived from another in a single step, according to the transformation rules (i.e. the syntax) of some given formal system.[3] As such, the expression
$P\vdash Q$
means that Q is derivable from P in the system.
Consistent with its use for derivability, a "⊢" followed by an expression without anything preceding it denotes a theorem, which is to say that the expression can be derived from the rules using an empty set of axioms. As such, the expression
$\vdash Q$
means that Q is a theorem in the system.
• In proof theory, the turnstile is used to denote "provability" or "derivability". For example, if T is a formal theory and S is a particular sentence in the language of the theory then
$T\vdash S$
means that S is provable from T.[4] This usage is demonstrated in the article on propositional calculus. The syntactic consequence of provability should be contrasted to semantic consequence, denoted by the double turnstile symbol $\models $. One says that $S$ is a semantic consequence of $T$, or $T\models S$, when all possible valuations in which $T$ is true, $S$ is also true. For propositional logic, it may be shown that semantic consequence $\models $ and derivability $\vdash $ are equivalent to one-another. That is, propositional logic is sound ($\vdash $ implies $\models $) and complete ($\models $ implies $\vdash $)[5]
• In sequent calculus, the turnstile is used to denote a sequent. A sequent $A_{1},\,\dots ,A_{m}\,\vdash \,B_{1},\,\dots ,B_{n}$ asserts that, if all the antecedents $A_{1},\,\dots ,A_{m}$ are true, then at least one of the consequents $B_{1},\,\dots ,B_{n}$ must be true.
• In the typed lambda calculus, the turnstile is used to separate typing assumptions from the typing judgment.[6][7]
• In category theory, a reversed turnstile ($\dashv $), as in $F\dashv G$, is used to indicate that the functor F is left adjoint to the functor G.[8] More rarely, a turnstile ($\vdash $), as in $G\vdash F$, is used to indicate that the functor G is right adjoint to the functor F.[9]
• In APL the symbol is called "right tack" and represents the ambivalent right identity function where both X⊢Y and ⊢Y are Y. The reversed symbol "⊣" is called "left tack" and represents the analogous left identity where X⊣Y is X and ⊣Y is Y.[10][11]
• In combinatorics, $\lambda \vdash n$ means that λ is a partition of the integer n.[12]
• In Hewlett-Packard's HP-41C/CV/CX and HP-42S series of calculators, the symbol (at code point 127 in the FOCAL character set) is called "Append character" and is used to indicate that the following characters will be appended to the alpha register rather than replacing the existing contents of the register. The symbol is also supported (at code point 148) in a modified variant of the HP Roman-8 character set used by other HP calculators.
• On the Casio fx-92 Collège 2D and fx-92+ Spéciale Collège calculators,[13] the symbol represents the modulo operator; entering $5\vdash 2$ will produce an answer of $Q=2;R=1$, where Q is the quotient and R is the remainder. On other Casio calculators (such as on the Belgian variants—the fx-92B Spéciale Collège and fx-92B Collège 2D calculators[14]—where the decimal separator is represented as a dot instead of a comma), the modulo operator is represented by ÷R instead.
Typography
In TeX, the turnstile symbol $\vdash $ is obtained from the command \vdash.
In Unicode, the turnstile symbol (⊢) is called right tack and is at code point U+22A2.[15] (Code point U+22A6 is named assertion sign (⊦).)
• U+22A2 ⊢ RIGHT TACK (⊢, ⊢)
• = turnstile
• = proves, implies, yields
• = reducible
• U+22A3 ⊣ LEFT TACK (⊣, ⊣)
• = reverse turnstile
• = non-theorem, does not yield
• U+22AC ⊬ DOES NOT PROVE (⊬)
• ≡ 22A2⊢ 0338$̸
On a typewriter, a turnstile can be composed from a vertical bar (|) and a dash (–).
In LaTeX there is a turnstile package which issues this sign in many ways, and is capable of putting labels below or above it, in the correct places.[16]
Similar graphemes
• ꜔ (U+A714) Modifier Letter Mid Left-Stem Tone Bar
• ├ (U+251C) Box Drawings Light Vertical And Right
• ㅏ (U+314F) Hangul Letter A
• Ͱ (U+0370) Greek Capital Letter Heta
• ͱ (U+0371) Greek Small Letter Heta
• Ⱶ (U+2C75) Latin Capital Letter Half H
• ⱶ (U+2C76) Latin Small Letter Half H
• ⎬ (U+23AC) Right Curly Bracket Middle Piece
See also
• Double turnstile $\models $
• List of logic symbols
• List of mathematical symbols
Notes
1. Martin-Löf 1996, pp. 6, 15
2. Martin-Löf 1996, p. 15
3. "Chapter 6, Formal Language Theory" (PDF).
4. Troelstra & Schwichtenberg 2000
5. Dirk van Dalen, Logic and Structure (1980), Springer, ISBN 3-540-20879-8. See Chapter 1, section 1.5.
6. "Peter Selinger, Lecture Notes on the Lambda Calculus" (PDF).
7. Schmidt 1994
8. "adjoint functor in nLab". ncatlab.org.
9. @FunctorFact (5 July 2016). "Functor Fact on Twitter" (Tweet) – via Twitter.
10. "A Dictionary of APL". www.jsoftware.com.
11. Iverson 1987
12. Stanley, Richard P. (1999). Enumerative Combinatorics. Vol. 2 (1st ed.). Cambridge: Cambridge University Press. p. 287.
13. fx-92 Spéciale Collège Mode d'emploi (PDF). Casio. 2015. p. 12.
14. "Remainder Calculations - Casio fx-92B User Manual [Page 13] | ManualsLib". www.manualslib.com. Retrieved 2020-12-24.
15. "Unicode standard" (PDF).
16. "CTAN: /tex-archive/macros/latex/contrib/turnstile". ctan.org.
References
• Frege, Gottlob (1879). "Begriffsschrift: Eine der arithmetischen nachgebildete Formelsprache des reinen Denkens". Halle. {{cite journal}}: Cite journal requires |journal= (help)
• Iverson, Kenneth (1987). "A Dictionary of APL". {{cite journal}}: Cite journal requires |journal= (help)
• Martin-Löf, Per (1996). "On the meanings of the logical constants and the justifications of the logical laws" (PDF). Nordic Journal of Philosophical Logic. 1 (1): 11–60. (Lecture notes to a short course at Università degli Studi di Siena, April 1983.)
• Schmidt, David (1994). The Structure of Typed Programming Languages. MIT Press. ISBN 0-262-19349-3.
• Troelstra, A. S.; Schwichtenberg, H. (2000). Basic Proof Theory (2nd ed.). Cambridge University Press. ISBN 978-0-521-77911-1.
Common logical symbols
∧ or &
and
∨
or
¬ or ~
not
→
implies
⊃
implies,
superset
↔ or ≡
iff
|
nand
∀
universal
quantification
∃
existential
quantification
⊤
true,
tautology
⊥
false,
contradiction
⊢
entails,
proves
⊨
entails,
therefore
∴
therefore
∵
because
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Turán's inequalities
In mathematics, Turán's inequalities are some inequalities for Legendre polynomials found by Pál Turán (1950) (and first published by Szegö (1948)). There are many generalizations to other polynomials, often called Turán's inequalities, given by (E. F. Beckenbach, W. Seidel & Otto Szász 1951) and other authors.
If $P_{n}$ is the $n$th Legendre polynomial, Turán's inequalities state that
$\,\!P_{n}(x)^{2}>P_{n-1}(x)P_{n+1}(x)\ {\text{for}}\ -1<x<1.$
For $H_{n}$, the $n$th Hermite polynomial, Turán's inequalities are
$H_{n}(x)^{2}-H_{n-1}(x)H_{n+1}(x)=(n-1)!\cdot \sum _{i=0}^{n-1}{\frac {2^{n-i}}{i!}}H_{i}(x)^{2}>0,$
whilst for Chebyshev polynomials they are
$T_{n}(x)^{2}-T_{n-1}(x)T_{n+1}(x)=1-x^{2}>0\ {\text{for}}\ -1<x<1.$
See also
• Askey–Gasper inequality
• Sturm Chain
References
• Beckenbach, E. F.; Seidel, W.; Szász, Otto (1951), "Recurrent determinants of Legendre and of ultraspherical polynomials", Duke Math. J., 18: 1–10, doi:10.1215/S0012-7094-51-01801-7, MR 0040487
• Szegö, G. (1948), "On an inequality of P. Turán concerning Legendre polynomials", Bull. Amer. Math. Soc., 54 (4): 401–405, doi:10.1090/S0002-9904-1948-09017-6, MR 0023954
• Turán, Paul (1950), "On the zeros of the polynomials of Legendre", Časopis Pěst. Mat. Fys., 75 (3): 113–122, doi:10.21136/CPMF.1950.123879, MR 0041284
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Turán's method
In mathematics, Turán's method provides lower bounds for exponential sums and complex power sums. The method has been applied to problems in equidistribution.
The method applies to sums of the form
$s_{\nu }=\sum _{n=1}^{N}b_{n}z_{n}^{\nu }\ $
where the b and z are complex numbers and ν runs over a range of integers. There are two main results, depending on the size of the complex numbers z.
Turán's first theorem
The first result applies to sums sν where $|z_{n}|\geq 1$ for all n. For any range of ν of length N, say ν = M + 1, ..., M + N, there is some ν with |sν| at least c(M, N)|s0| where
$c(M,N)=\left({\sum _{k=0}^{N-1}{\binom {M+k}{k}}2^{k}}\right)^{-1}\ .$
The sum here may be replaced by the weaker but simpler $\left({\frac {N}{2e(M+N)}}\right)^{N-1}$.
We may deduce the Fabry gap theorem from this result.
Turán's second theorem
The second result applies to sums sν where $|z_{n}|\leq 1$ for all n. Assume that the z are ordered in decreasing absolute value and scaled so that |z1| = 1. Then there is some ν with
$|s_{\nu }|\geq 2\left({\frac {N}{8e(M+N)}}\right)^{N}\min _{1\leq j\leq N}\left\vert {\sum _{n=1}^{j}b_{n}}\right\vert \ .$
See also
• Turán's theorem in graph theory
References
• Montgomery, Hugh L. (1994). Ten lectures on the interface between analytic number theory and harmonic analysis. Regional Conference Series in Mathematics. Vol. 84. Providence, RI: American Mathematical Society. ISBN 0-8218-0737-4. Zbl 0814.11001.
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Turán sieve
In number theory, the Turán sieve is a technique for estimating the size of "sifted sets" of positive integers which satisfy a set of conditions which are expressed by congruences. It was developed by Pál Turán in 1934.
Description
In terms of sieve theory the Turán sieve is of combinatorial type: deriving from a rudimentary form of the inclusion–exclusion principle. The result gives an upper bound for the size of the sifted set.
Let A be a set of positive integers ≤ x and let P be a set of primes. For each p in P, let Ap denote the set of elements of A divisible by p and extend this to let Ad be the intersection of the Ap for p dividing d, when d is a product of distinct primes from P. Further let A1 denote A itself. Let z be a positive real number and P(z) denote the product of the primes in P which are ≤ z. The object of the sieve is to estimate
$S(A,P,z)=\left\vert A\setminus \bigcup _{p\mid P(z)}A_{p}\right\vert .$
We assume that |Ad| may be estimated, when d is a prime p by
$\left\vert A_{p}\right\vert ={\frac {1}{f(p)}}X+R_{p}$
and when d is a product of two distinct primes d = p q by
$\left\vert A_{pq}\right\vert ={\frac {1}{f(p)f(q)}}X+R_{p,q}$
where X = |A| and f is a function with the property that 0 ≤ f(d) ≤ 1. Put
$U(z)=\sum _{p\mid P(z)}f(p).$
Then
$S(A,P,z)\leq {\frac {X}{U(z)}}+{\frac {2}{U(z)}}\sum _{p\mid P(z)}\left\vert R_{p}\right\vert +{\frac {1}{U(z)^{2}}}\sum _{p,q\mid P(z)}\left\vert R_{p,q}\right\vert .$
Applications
• The Hardy–Ramanujan theorem that the normal order of ω(n), the number of distinct prime factors of a number n, is log(log(n));
• Almost all integer polynomials (taken in order of height) are irreducible.
References
• Alina Carmen Cojocaru; M. Ram Murty. An introduction to sieve methods and their applications. London Mathematical Society Student Texts. Vol. 66. Cambridge University Press. pp. 47–62. ISBN 0-521-61275-6.
• Greaves, George (2001). Sieves in number theory. Springer-Verlag. ISBN 3-540-41647-1.
• Halberstam, Heini; Richert, H.-E. (1974). Sieve Methods. London Mathematical Society Monographs. Vol. 4. Academic Press. ISBN 0-12-318250-6. MR 0424730. Zbl 0298.10026.
• Christopher Hooley (1976). Applications of sieve methods to the theory of numbers. Cambridge University Press. p. 21. ISBN 0-521-20915-3.
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Turán number
In mathematics, the Turán number T(n,k,r) for r-uniform hypergraphs of order n is the smallest number of r-edges such that every induced subgraph on k vertices contains an edge. This number was determined for r = 2 by Turán (1941), and the problem for general r was introduced in Turán (1961). The paper (Sidorenko 1995) gives a survey of Turán numbers.
Definitions
Fix a set X of n vertices. For given r, an r-edge or block is a set of r vertices. A set of blocks is called a Turán (n,k,r) system (n ≥ k ≥ r) if every k-element subset of X contains a block. The Turán number T(n,k,r) is the minimum size of such a system.
Example
The complements of the lines of the Fano plane form a Turán (7,5,4)-system. T(7,5,4) = 7.[1]
Relations to other combinatorial designs
It can be shown that
$T(n,k,r)\geq {\binom {n}{r}}{\binom {k}{r}}^{-1}.$
Equality holds if and only if there exists a Steiner system S(n - k, n - r, n).[2]
An (n,r,k,r)-lotto design is an (n, k, r)-Turán system. Thus, T(n,k, r) = L(n,r,k,r).[3]
See also
• Forbidden subgraph problem
• Combinatorial design
References
1. Colbourn & Dinitz 2007, pg. 649, Example 61.3
2. Colbourn & Dinitz 2007, pg. 649, Remark 61.4
3. Colbourn & Dinitz 2007, pg. 513, Proposition 32.12
Bibliography
• Colbourn, Charles J.; Dinitz, Jeffrey H. (2007), Handbook of Combinatorial Designs (2nd ed.), Boca Raton: Chapman & Hall/ CRC, ISBN 1-58488-506-8
• Godbole, A. P. (2001) [1994], "Turán number", Encyclopedia of Mathematics, EMS Press
• Sidorenko, A. (1995), "What we know and what we do not know about Turán numbers", Graphs and Combinatorics, 11 (2): 179–199, doi:10.1007/BF01929486
• Turán, P (1941), "Egy gráfelméleti szélsőértékfeladatról (Hungarian. An extremal problem in graph theory.)", Mat. Fiz. Lapok (in Hungarian), 48: 436–452
• Turán, P. (1961), "Research problems", Magyar Tud. Akad. Mat. Kutato Int. Közl., 6: 417–423
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Turán's theorem
In graph theory, Turán's theorem bounds the number of edges that can be included in an undirected graph that does not have a complete subgraph of a given size. It is one of the central results of extremal graph theory, an area studying the largest or smallest graphs with given properties, and is a special case of the forbidden subgraph problem on the maximum number of edges in a graph that does not have a given subgraph.
Not to be confused with Turán's method in analytic number theory.
An example of an $n$-vertex graph that does not contain any $(r+1)$-vertex clique $K_{r+1}$ may be formed by partitioning the set of $n$ vertices into $r$ parts of equal or nearly equal size, and connecting two vertices by an edge whenever they belong to two different parts. The resulting graph is the Turán graph $T(n,r)$. Turán's theorem states that the Turán graph has the largest number of edges among all Kr+1-free n-vertex graphs.
Turán's theorem, and the Turán graphs giving its extreme case, were first described and studied by Hungarian mathematician Pál Turán in 1941.[1] The special case of the theorem for triangle-free graphs is known as Mantel's theorem; it was stated in 1907 by Willem Mantel, a Dutch mathematician.[2]
Statement
Turán's theorem states that every graph $G$ with $n$ vertices that does not contain $K_{r+1}$ as a subgraph has at most as many edges as the Turán graph $T(n,r)$. For a fixed value of $r$, this graph has
$\left(1-{\frac {1}{r}}+o(1)\right){\frac {n^{2}}{2}}$
edges, using little-o notation. Intuitively, this means that as $n$ gets larger, the fraction of edges included in $T(n,r)$ gets closer and closer to $1-{\frac {1}{r}}$. Many of the following proofs only give the upper bound of $\left(1-{\frac {1}{r}}\right){\frac {n^{2}}{2}}$.[3]
Proofs
Aigner & Ziegler (2018) list five different proofs of Turán's theorem.[3] Many of the proofs involve reducing to the case where the graph is a complete multipartite graph, and showing that the number of edges is maximized when there are $r$ parts of size as close as possible to equal.
Induction
This was Turán's original proof. Take a $K_{r+1}$-free graph on $n$ vertices with the maximal number of edges. Find a $K_{r}$ (which exists by maximality), and partition the vertices into the set $A$ of the $r$ vertices in the $K_{r}$ and the set $B$ of the $n-r$ other vertices.
Now, one can bound edges above as follows:
• There are exactly ${\binom {r}{2}}$ edges within $A$.
• There are at most $(r-1)|B|=(r-1)(n-r)$ edges between $A$ and $B$, since no vertex in $B$ can connect to all of $A$.
• The number of edges within $B$ is at most the number of edges of $T(n-r,r)$ by the inductive hypothesis.
Adding these bounds gives the result.[1][3]
Maximal Degree Vertex
This proof is due to Paul Erdős. Take the vertex $v$ of largest degree. Consider the set $A$ of vertices not adjacent to $v$ and the set $B$ of vertices adjacent to $v$.
Now, delete all edges within $A$ and draw all edges between $A$ and $B$. This increases the number of edges by our maximality assumption and keeps the graph $K_{r+1}$-free. Now, $B$ is $K_{r}$-free, so the same argument can be repeated on $B$.
Repeating this argument eventually produces a graph in the same form as a Turán graph, which is a collection of independent sets, with edges between each two vertices from different independent sets. A simple calculation shows that the number of edges of this graph is maximized when all independent set sizes are as close to equal as possible.[3][4]
Complete Multipartite Optimization
This proof, as well as the Zykov Symmetrization proof, involve reducing to the case where the graph is a complete multipartite graph, and showing that the number of edges is maximized when there are $r$ independent sets of size as close as possible to equal. This step can be done as follows:
Let $S_{1},S_{2},\ldots ,S_{r}$ be the independent sets of the multipartite graph. Since two vertices have an edge between them if and only if they are not in the same independent set, the number of edges is
$\sum _{i\neq j}\left|S_{i}\right|\left|S_{j}\right|={\frac {1}{2}}\left(n^{2}-\sum _{i}\left|S_{i}\right|^{2}\right),$
where the left hand side follows from direct counting, and the right hand side follows from complementary counting. To show the $\left(1-{\frac {1}{r}}\right){\frac {n^{2}}{2}}$ bound, applying the Cauchy–Schwarz inequality to the $ \sum \limits _{i}\left|S_{i}\right|^{2}$ term on the right hand side suffices, since $ \sum \limits _{i}\left|S_{i}\right|=n$.
To prove the Turán Graph is optimal, one can argue that no two $S_{i}$ differ by more than one in size. In particular, supposing that we have $\left|S_{i}\right|\geq \left|S_{j}\right|+2$ for some $i\neq j$, moving one vertex from $S_{j}$ to $S_{i}$ (and adjusting edges accordingly) would increase the value of the sum. This can be seen by examining the changes to either side of the above expression for the number of edges, or by noting that the degree of the moved vertex increases.
Lagrangian
This proof is due to Motzkin & Straus (1965). They begin by considering a $K_{r+1}$ free graph with vertices labelled $1,2,\ldots ,n$, and considering maximizing the function
$f(x_{1},x_{2},\ldots ,x_{n})=\sum _{i,j\ {\text{adjacent}}}x_{i}x_{j}$
over all nonnegative $x_{1},x_{2},\ldots ,x_{n}$ with sum $1$. This function is known as the Lagrangian of the graph and its edges. The idea behind their proof is that if $x_{i},x_{j}$ are both nonzero while $i,j$ are not adjacent in the graph, the function
$f(x_{1},\ldots ,x_{i}-t,\ldots ,x_{j}+t,\ldots ,x_{n})$
is linear in $t$. Hence, one can either replace $(x_{i},x_{j})$ with either $(x_{i}+x_{j},0)$ or $(0,x_{i}+x_{j})$ without decreasing the value of the function. Hence, there is a point with at most $r$ nonzero variables where the function is maximized.
Now, the Cauchy–Schwarz inequality gives that the maximal value is at most ${\frac {1}{2}}\left(1-{\frac {1}{r}}\right)$. Plugging in $x_{i}={\frac {1}{n}}$ for all $i$ gives that the maximal value is at least ${\frac {|E|}{n^{2}}}$, giving the desired bound.[3][5]
Probabilistic Method
The key claim in this proof was independently found by Caro and Wei. This proof is due to Noga Alon and Joel Spencer, from their book The Probabilistic Method. The proof shows that every graph with degrees $d_{1},d_{2},\ldots ,d_{n}$ has an independent set of size at least
$S={\frac {1}{d_{1}+1}}+{\frac {1}{d_{2}+1}}+\cdots +{\frac {1}{d_{n}+1}}.$
The proof attempts to find such an independent set as follows:
• Consider a random permutation of the vertices of a $K_{r+1}$-free graph
• Select every vertex that is adjacent to none of the vertices before it.
A vertex of degree $d$ is included in this with probability ${\frac {1}{d+1}}$, so this process gives an average of $S$ vertices in the chosen set.
Applying this fact to the complement graph and bounding the size of the chosen set using the Cauchy–Schwarz inequality proves Turán's theorem.[3] See Method of conditional probabilities § Turán's theorem for more.
Zykov Symmetrization
Aigner and Ziegler call the final one of their five proofs "the most beautiful of them all". Its origins are unclear, but the approach is often referred to as Zykov Symmetrization as it was used in Zykov's proof of a generalization of Turán's Theorem [6]. This proof goes by taking a $K_{r+1}$-free graph, and applying steps to make it more similar to the Turán Graph while increasing edge count.
In particular, given a $K_{r+1}$-free graph, the following steps are applied:
• If $u,v$ are non-adjacent vertices and $u$ has a higher degree than $v$, replace $v$ with a copy of $u$. Repeat this until all non-adjacent vertices have the same degree.
• If $u,v,w$ are vertices with $u,v$ and $v,w$ non-adjacent but $u,w$ adjacent, then replace both $u$ and $w$ with copies of $v$.
All of these steps keep the graph $K_{r+1}$ free while increasing the number of edges.
Now, non-adjacency forms an equivalence relation. The equivalence classes give that any maximal graph the same form as a Turán graph. As in the maximal degree vertex proof, a simple calculation shows that the number of edges is maximized when all independent set sizes are as close to equal as possible.[3]
Mantel's theorem
The special case of Turán's theorem for $r=2$ is Mantel's theorem: The maximum number of edges in an $n$-vertex triangle-free graph is $\lfloor n^{2}/4\rfloor .$[2] In other words, one must delete nearly half of the edges in $K_{n}$ to obtain a triangle-free graph.
A strengthened form of Mantel's theorem states that any Hamiltonian graph with at least $n^{2}/4$ edges must either be the complete bipartite graph $K_{n/2,n/2}$ or it must be pancyclic: not only does it contain a triangle, it must also contain cycles of all other possible lengths up to the number of vertices in the graph.[7]
Another strengthening of Mantel's theorem states that the edges of every $n$-vertex graph may be covered by at most $\lfloor n^{2}/4\rfloor $ cliques which are either edges or triangles. As a corollary, the graph's intersection number (the minimum number of cliques needed to cover all its edges) is at most $\lfloor n^{2}/4\rfloor $.[8]
Generalizations
Other Forbidden Subgraphs
Turán's theorem shows that the largest number of edges in a $K_{r+1}$-free graph is $\left(1-{\frac {1}{r}}+o(1)\right){\frac {n^{2}}{2}}$. The Erdős–Stone theorem finds the number of edges up to a $o(n^{2})$ error in all other graphs:
(Erdős–Stone) Suppose $H$ is a graph with chromatic number $\chi (H)$. The largest possible number of edges in a graph where $H$ does not appear as a subgraph is
$\left(1-{\frac {1}{\chi (H)-1}}+o(1)\right){\frac {n^{2}}{2}}$
where the $o(1)$ constant only depends on $H$.
One can see that the Turán graph $T(n,\chi (H)-1)$ cannot contain any copies of $H$, so the Turán graph establishes the lower bound. As a $K_{r+1}$ has chromatic number $r+1$, Turán's theorem is the special case in which $H$ is a $K_{r+1}$.
The general question of how many edges can be included in a graph without a copy of some $H$ is the forbidden subgraph problem.
Maximizing Other Quantities
Another natural extension of Turán's theorem is the following question: if a graph has no $K_{r+1}$s, how many copies of $K_{a}$ can it have? Turán's theorem is the case where $a=2$. Zykov's Theorem answers this question:
(Zykov's Theorem) The graph on $n$ vertices with no $K_{r+1}$s and the largest possible number of $K_{a}$s is the Turán graph $T(n,r)$
This was first shown by Zykov (1949) using Zykov Symmetrization[1][3]. Since the Turán Graph contains $r$ parts with size around ${\frac {n}{r}}$, the number of $K_{a}$s in $T(n,r)$ is around ${\binom {r}{a}}\left({\frac {n}{r}}\right)^{a}$. A paper by Alon and Shikhelman in 2016 gives the following generalization, which is similar to the Erdos-Stone generalization of Turán's theorem:
(Alon-Shikhelman, 2016) Let $H$ be a graph with chromatic number $\chi (H)>a$. The largest possible number of $K_{a}$s in a graph with no copy of $H$ is $(1+o(1)){\binom {\chi (H)-1}{a}}\left({\frac {n}{\chi (H)-1}}\right)^{a}.$[9]
As in Erdős–Stone, the Turán graph $T(n,\chi (H)-1)$ attains the desired number of copies of $K_{a}$.
Edge-Clique region
Turan's theorem states that if a graph has edge homomorphism density strictly above $1-{\frac {1}{r-1}}$, it has a nonzero number of $K_{r}$s. One could ask the far more general question: if you are given the edge density of a graph, what can you say about the density of $K_{r}$s?
An issue with answering this question is that for a given density, there may be some bound not attained by any graph, but approached by some infinite sequence of graphs. To deal with this, weighted graphs or graphons are often considered. In particular, graphons contain the limit of any infinite sequence of graphs.
For a given edge density $d$, the construction for the largest $K_{r}$ density is as follows:
Take a number of vertices $N$ approaching infinity. Pick a set of ${\sqrt {d}}N$ of the vertices, and connect two vertices if and only if they are in the chosen set.
This gives a $K_{r}$ density of $d^{k/2}.$ The construction for the smallest $K_{r}$ density is as follows:
Take a number of vertices approaching infinity. Let $t$ be the integer such that $1-{\frac {1}{t-1}}<d\leq 1-{\frac {1}{t}}$. Take a $t$-partite graph where all parts but the unique smallest part have the same size, and sizes of the parts are chosen such that the total edge density is $d$.
For $d\leq 1-{\frac {1}{r-1}}$, this gives a graph that is $(r-1)$-partite and hence gives no $K_{r}$s.
The lower bound was proven by Razborov (2008)[10] for the case of triangles, and was later generalized to all cliques by Reiher (2016)[11]. The upper bound is a consequence of the Kruskal–Katona theorem [12].
See also
• Erdős–Stone theorem, a generalization of Turán's theorem from forbidden cliques to forbidden Turán graphs
References
1. Turán, Paul (1941), "On an extremal problem in graph theory", Matematikai és Fizikai Lapok (in Hungarian), 48: 436–452
2. Mantel, W. (1907), "Problem 28 (Solution by H. Gouwentak, W. Mantel, J. Teixeira de Mattes, F. Schuh and W. A. Wythoff)", Wiskundige Opgaven, 10: 60–61
3. Aigner, Martin; Ziegler, Günter M. (2018), "Chapter 41: Turán's graph theorem", Proofs from THE BOOK (6th ed.), Springer-Verlag, pp. 285–289, doi:10.1007/978-3-662-57265-8_41, ISBN 978-3-662-57265-8
4. Erdős, Pál (1970), "Turán Pál gráf tételéről" [On the graph theorem of Turán] (PDF), Matematikai Lapok (in Hungarian), 21: 249–251, MR 0307975
5. Motzkin, T. S.; Straus, E. G. (1965), "Maxima for graphs and a new proof of a theorem of Turán", Canadian Journal of Mathematics, 17: 533–540, doi:10.4153/CJM-1965-053-6, MR 0175813, S2CID 121387797
6. Zykov, A. (1949), "On some properties of linear complexes", Mat. Sb., New Series (in Russian), 24: 163–188
7. Bondy, J. A. (1971), "Pancyclic graphs I", Journal of Combinatorial Theory, Series B, 11 (1): 80–84, doi:10.1016/0095-8956(71)90016-5
8. Erdős, Paul; Goodman, A. W.; Pósa, Louis (1966), "The representation of a graph by set intersections" (PDF), Canadian Journal of Mathematics, 18 (1): 106–112, doi:10.4153/CJM-1966-014-3, MR 0186575, S2CID 646660
9. Alon, Noga; Shikhelman, Clara (2016), "Many T copies in H-free graphs", Journal of Combinatorial Theory, Series B, 121: 146–172, arXiv:1409.4192, doi:10.1016/j.jctb.2016.03.004, S2CID 5552776
10. Razborov, Alexander (2008). "On the minimal density of triangles in graphs" (PDF). Combinatorics, Probability and Computing. 17 (4): 603–618. doi:10.1017/S0963548308009085. S2CID 26524353 – via MathSciNet (AMS).
11. Reiher, Christian (2016), "The clique density theorem", Annals of Mathematics, 184 (3): 683–707, arXiv:1212.2454, doi:10.4007/annals.2016.184.3.1, S2CID 59321123
12. Lovász, László, Large networks and graph limits
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Turán–Kubilius inequality
The Turán–Kubilius inequality is a mathematical theorem in probabilistic number theory. It is useful for proving results about the normal order of an arithmetic function.[1]: 305–308 The theorem was proved in a special case in 1934 by Pál Turán and generalized in 1956 and 1964 by Jonas Kubilius.[1]: 316
Statement of the theorem
This formulation is from Tenenbaum.[1]: 302 Other formulations are in Narkiewicz[2]: 243 and in Cojocaru & Murty.[3]: 45–46
Suppose f is an additive complex-valued arithmetic function, and write p for an arbitrary prime and ν for an arbitrary positive integer. Write
$A(x)=\sum _{p^{\nu }\leq x}f(p^{\nu })p^{-\nu }(1-p^{-1})$
and
$B(x)^{2}=\sum _{p^{\nu }\leq x}\left|f(p^{\nu })\right|^{2}p^{-\nu }.$
Then there is a function ε(x) that goes to zero when x goes to infinity, and such that for x ≥ 2 we have
${\frac {1}{x}}\sum _{n\leq x}|f(n)-A(x)|^{2}\leq (2+\varepsilon (x))B(x)^{2}.$
Applications of the theorem
Turán developed the inequality to create a simpler proof of the Hardy–Ramanujan theorem about the normal order of the number ω(n) of distinct prime divisors of an integer n.[1]: 316 There is an exposition of Turán's proof in Hardy & Wright, §22.11.[4] Tenenbaum[1]: 305–308 gives a proof of the Hardy–Ramanujan theorem using the Turán–Kubilius inequality and states without proof several other applications.
Notes
1. Tenenbaum, Gérald (1995). Introduction to Analytic and Probabilistic Number Theory. Cambridge studies in advanced mathematics. Vol. 46. Cambridge University Press. ISBN 0-521-41261-7.
2. Narkiewicz, Władysław (1983). Number Theory. Singapore: World Scientific. ISBN 978-9971-950-13-2.
3. Cojocaru, Alina Carmen; Murty, M. Ram (2005). An Introduction to Sieve Methods and Their Applications. London Mathematical Society Student Texts. Vol. 66. Cambridge University Press. ISBN 0-521-61275-6.
4. Hardy, G. H.; Wright, E. M. (2008) [First edition 1938]. An Introduction to the Theory of Numbers. Revised by D. R. Heath-Brown and Joseph H. Silverman (Sixth ed.). Oxford, Oxfordshire: Oxford University Press. ISBN 978-0-19-921986-5.
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Tutte 12-cage
In the mathematical field of graph theory, the Tutte 12-cage or Benson graph[1] is a 3-regular graph with 126 vertices and 189 edges named after W. T. Tutte.[2]
Tutte 12-cage
The Tutte 12-cage
Named afterW. T. Tutte
Vertices126
Edges189
Radius6
Diameter6
Girth12
Automorphisms12096
Chromatic number2
Chromatic index3
PropertiesCubic
Cage
Hamiltonian
Semi-symmetric
Bipartite
Table of graphs and parameters
The Tutte 12-cage is the unique (3-12)-cage (sequence A052453 in the OEIS). It was discovered by C. T. Benson in 1966.[3] It has chromatic number 2 (bipartite), chromatic index 3, girth 12 (as a 12-cage) and diameter 6. Its crossing number is known to be less than 165, see Wolfram MathWorld.[4][5]
Construction
The Tutte 12-cage is a cubic Hamiltonian graph and can be defined by the LCF notation [17, 27, –13, –59, –35, 35, –11, 13, –53, 53, –27, 21, 57, 11, –21, –57, 59, –17]7.[6]
There are, up to isomorphism, precisely two generalized hexagons of order (2,2) as proved by Cohen and Tits. They are the split Cayley hexagon H(2) and its point-line dual. Clearly both of them have the same incidence graph, which is in fact isomorphic to the Tutte 12-cage.[1]
The Balaban 11-cage can be constructed by excision from the Tutte 12-cage by removing a small subtree and suppressing the resulting vertices of degree two.[7]
Algebraic properties
The automorphism group of the Tutte 12-cage is of order 12,096 and is a semi-direct product of the projective special unitary group PSU(3,3) with the cyclic group Z/2Z.[1] It acts transitively on its edges but not on its vertices, making it a semi-symmetric graph, a regular graph that is edge-transitive but not vertex-transitive. In fact, the automorphism group of the Tutte 12-cage preserves the bipartite parts and acts primitively on each part. Such graphs are called bi-primitive graphs and only five cubic bi-primitive graphs exist; they are named the Iofinova-Ivanov graphs and are of order 110, 126, 182, 506 and 990.[8]
All the cubic semi-symmetric graphs on up to 768 vertices are known. According to Conder, Malnič, Marušič and Potočnik, the Tutte 12-cage is the unique cubic semi-symmetric graph on 126 vertices and is the fifth smallest possible cubic semi-symmetric graph after the Gray graph, the Iofinova–Ivanov graph on 110 vertices, the Ljubljana graph and a graph on 120 vertices with girth 8.[9]
The characteristic polynomial of the Tutte 12-cage is
$(x-3)x^{28}(x+3)(x^{2}-6)^{21}(x^{2}-2)^{27}.\ $
It is the only graph with this characteristic polynomial; therefore, the 12-cage is determined by its spectrum.
Gallery
• The chromatic number of the Tutte 12-cage is 2.
• The chromatic index of the Tutte 12-cage is 3.
References
1. Geoffrey Exoo & Robert Jajcay, Dynamic cage survey, Electr. J. Combin. 15 (2008).
2. Weisstein, Eric W. "Tutte 12-cage". MathWorld.
3. Benson, C. T. "Minimal Regular Graphs of Girth 8 and 12." Can. J. Math. 18, 1091–1094, 1966.
4. Exoo, G. "Rectilinear Drawings of Famous Graphs".
5. Pegg, E. T. and Exoo, G. "Crossing Number Graphs." Mathematica J. 11, 2009.
6. Polster, B. A Geometrical Picture Book. New York: Springer, p. 179, 1998.
7. Balaban, A. T. "Trivalent Graphs of Girth Nine and Eleven and Relationships Among the Cages." Rev. Roumaine Math 18, 1033–1043, 1973.
8. Iofinova, M. E. and Ivanov, A. A. "Bi-Primitive Cubic Graphs." In Investigations in the Algebraic Theory of Combinatorial Objects. pp. 123–134, 2002. (Vsesoyuz. Nauchno-Issled. Inst. Sistem. Issled., Moscow, pp. 137–152, 1985.)
9. Conder, Marston; Malnič, Aleksander; Marušič, Dragan; Potočnik, Primož (2006), "A census of semisymmetric cubic graphs on up to 768 vertices", Journal of Algebraic Combinatorics, 23: 255–294, doi:10.1007/s10801-006-7397-3.
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Tutte–Coxeter graph
In the mathematical field of graph theory, the Tutte–Coxeter graph or Tutte eight-cage or Cremona–Richmond graph is a 3-regular graph with 30 vertices and 45 edges. As the unique smallest cubic graph of girth 8, it is a cage and a Moore graph. It is bipartite, and can be constructed as the Levi graph of the generalized quadrangle W2 (known as the Cremona–Richmond configuration). The graph is named after William Thomas Tutte and H. S. M. Coxeter; it was discovered by Tutte (1947) but its connection to geometric configurations was investigated by both authors in a pair of jointly published papers (Tutte 1958; Coxeter 1958a).
Not to be confused with Coxeter graph.
Tutte–Coxeter graph
Named afterW. T. Tutte
H. S. M. Coxeter
Vertices30
Edges45
Radius4
Diameter4
Girth8
Automorphisms1440 (Aut(S6))
Chromatic number2
Chromatic index3
Book thickness3
Queue number2
PropertiesCubic
Cage
Moore graph
Symmetric
Distance-regular
Distance-transitive
Bipartite
Table of graphs and parameters
All the cubic distance-regular graphs are known.[1] The Tutte–Coxeter is one of the 13 such graphs.
It has crossing number 13,[2][3] book thickness 3 and queue number 2.[4]
Constructions and automorphisms
The Tutte–Coxeter graph is the bipartite Levi graph connecting the 15 perfect matchings of a 6-vertex complete graph K6 to its 15 edges, as described by Coxeter (1958b), based on work by Sylvester (1844). Each vertex corresponds to an edge or a perfect matching, and connected vertices represent the incidence structure between edges and matchings.
Based on this construction, Coxeter showed that the Tutte–Coxeter graph is a symmetric graph; it has a group of 1440 automorphisms, which may be identified with the automorphisms of the group of permutations on six elements (Coxeter 1958b). The inner automorphisms of this group correspond to permuting the six vertices of the K6 graph; these permutations act on the Tutte–Coxeter graph by permuting the vertices on each side of its bipartition while keeping each of the two sides fixed as a set. In addition, the outer automorphisms of the group of permutations swap one side of the bipartition for the other. As Coxeter showed, any path of up to five edges in the Tutte–Coxeter graph is equivalent to any other such path by one such automorphism.
The Tutte–Coxeter graph as a building
This graph is the spherical building associated to the symplectic group $Sp_{4}(\mathbb {F} _{2})$ (there is an exceptional isomorphism between this group and the symmetric group $S_{6}$). More specifically, it is the incidence graph of a generalized quadrangle.
Concretely, the Tutte-Coxeter graph can be defined from a 4-dimensional symplectic vector space $V$ over $\mathbb {F} _{2}$ as follows:
• vertices are either nonzero vectors, or isotropic 2-dimensional subspaces,
• there is an edge between a nonzero vector v and an isotropic 2-dimensional subspace $W\subset V$ if and only if $v\in W$.
Gallery
• The chromatic number of the Tutte–Coxeter graph is 2.
• The chromatic index of the Tutte–Coxeter graph is 3.
References
1. Brouwer, A. E.; Cohen, A. M.; and Neumaier, A. Distance-Regular Graphs. New York: Springer-Verlag, 1989.
2. Pegg, E. T.; Exoo, G. (2009). "Crossing Number Graphs". Mathematica Journal. 11 (2). doi:10.3888/tmj.11.2-2.
3. Exoo, G. "Rectilinear Drawings of Famous Graphs".
4. Wolz, Jessica; Engineering Linear Layouts with SAT. Master Thesis, University of Tübingen, 2018
• Coxeter, H. S. M. (1958a). "The chords of the non-ruled quadric in PG(3,3)". Can. J. Math. 10: 484–488. doi:10.4153/CJM-1958-047-0.
• Coxeter, H. S. M. (1958b). "Twelve points in PG(5,3) with 95040 self-transformations". Proceedings of the Royal Society A. 247 (1250): 279–293. doi:10.1098/rspa.1958.0184. JSTOR 100667. S2CID 121676627.
• Sylvester, J. J. (1844). "Elementary researches in the analysis of combinatorial aggregation". Phil. Mag. Series 3. 24: 285–295. doi:10.1080/14786444408644856.
• Tutte, W. T. (1947). "A family of cubical graphs". Proc. Cambridge Philos. Soc. 43 (4): 459–474. doi:10.1017/S0305004100023720.
• Tutte, W. T. (1958). "The chords of the non-ruled quadric in PG(3,3)". Can. J. Math. 10: 481–483. doi:10.4153/CJM-1958-046-3.
External links
• François Labelle. "3D Model of Tutte's 8-cage".
• Weisstein, Eric W. "Levi Graph". MathWorld.
• Exoo, G. "Rectilinear Drawings of Famous Graphs."
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Tutte embedding
In graph drawing and geometric graph theory, a Tutte embedding or barycentric embedding of a simple, 3-vertex-connected, planar graph is a crossing-free straight-line embedding with the properties that the outer face is a convex polygon and that each interior vertex is at the average (or barycenter) of its neighbors' positions. If the outer polygon is fixed, this condition on the interior vertices determines their position uniquely as the solution to a system of linear equations. Solving the equations geometrically produces a planar embedding. Tutte's spring theorem, proven by W. T. Tutte (1963), states that this unique solution is always crossing-free, and more strongly that every face of the resulting planar embedding is convex.[1] It is called the spring theorem because such an embedding can be found as the equilibrium position for a system of springs representing the edges of the graph.
Example
Let G be the graph of a cube, and (selecting one of its quadrilateral faces as the outer face) fix the four vertices of the outer face at the four corners of a unit square, the points whose x and y coordinates are all four combinations of zero and one. Then, if the remaining four vertices are placed at the four points whose x and y coordinates are combinations of 1/3 and 2/3, as in the figure, the result will be a Tutte embedding. For, at each interior vertex v of the embedding, and for each of the two coordinates, the three neighbors of v have coordinate values that are equal to v, smaller by 1/3, and larger by 1/3; the average of these values is the same as the coordinate value of v itself.
System of linear equations
The condition that a vertex v be at the average of its neighbors' positions may be expressed as two linear equations, one for the x coordinate of v and another for the y coordinate of v. For a graph with n vertices, h of which are fixed in position on the outer face, there are two equations for each interior vertex and also two unknowns (the coordinates) for each interior vertex. Therefore, this gives a system of linear equations with 2(n − h) equations in 2(n − h) unknowns, the solution to which is a Tutte embedding. As Tutte (1963) showed, for 3-vertex-connected planar graphs, this system is non-degenerate. Therefore, it has a unique solution, and (with the outer face fixed) the graph has a unique Tutte embedding. This embedding can be found in polynomial time by solving the system of equations, for instance by using Gaussian elimination.[2]
Polyhedral representation
By Steinitz's theorem, the 3-connected planar graphs to which Tutte's spring theorem applies coincide with the polyhedral graphs, the graphs formed by the vertices and edges of a convex polyhedron. According to the Maxwell–Cremona correspondence, a two-dimensional embedding of a planar graph forms the vertical projection of a three-dimensional convex polyhedron if and only if the embedding has an equilibrium stress, an assignment of forces to each edge (affecting both endpoints in equal and opposite directions parallel to the edge) such that the forces cancel at every vertex. For a Tutte embedding, assigning to each edge an attractive force proportional to its length (like a spring) makes the forces cancel at all interior vertices, but this is not necessarily an equilibrium stress at the vertices of the outer polygon. However, when the outer polygon is a triangle, it is possible to assign repulsive forces to its three edges to make the forces cancel there, too. In this way, Tutte embeddings can be used to find Schlegel diagrams of every convex polyhedron. For every 3-connected planar graph G, either G itself or the dual graph of G has a triangle, so either this gives a polyhedral representation of G or of its dual; in the case that the dual graph is the one with the triangle, polarization gives a polyhedral representation of G itself.[2]
Applications in geometry processing
In geometry processing, Tutte's embedding is used for 2D uv-parametrization ${\mathcal {S}}^{*}$of 3D surfaces ${\mathcal {S}}$ most commonly for the cases where the topology of the surface remains the same across ${\mathcal {S}}$ and ${\mathcal {S}}^{*}$(disk topology). Tutte's method minimizes the total distortion energy of the parametrized space by considering each transformed vertex as a point mass, and edges across the corresponding vertices as springs. The tightness of each spring is determined by the length of the edges in the original 3D surface to preserve the shape. Since it is reasonable to have smaller parametrized edge lengths for the smaller edges of ${\mathcal {S}}$, and larger parametrized edge lengths for the larger edges of ${\mathcal {S}}$, the spring constants $w_{ij}$are usually taken as the inverse of the absolute distance between the vertices $i,~j$ in the 3D space.
${\text{Distortion Energy,}}~E=\Sigma _{(i,j)\in E({\mathcal {S}})}~w_{ij}~\|{\textbf {u}}_{\text{i}}-{\textbf {u}}_{\text{j}}\|^{2}{\text{, where }}w_{ij}={\frac {1}{\|{\textbf {v}}_{\text{i}}-{\textbf {v}}_{\text{j}}\|}}{\text{, }}\{{\textbf {u}}_{\text{i}}\}\in {\mathcal {S}}^{*}{\text{, }}\{{\textbf {v}}_{\text{i}}\}\in {\mathcal {S}}$
where $E({\mathcal {S}})$ represents the set of edges in the original 3D surface. When the weights $w_{ij}$ are positive (as is the case above), it is guaranteed that the mapping is bijective without any inversions. But when no further constraints are applied, the solution that minimizes the distortion energy trivially collapses to a single point in the parametrized space.
Therefore, one must provide boundary conditions where a set of known vertices of the 3D surface are mapped to known points in the 2D parametrized space. One common way of choosing such boundary conditions is to consider the vertices on the largest boundary loop of the original 3D surface which can be then constrained to be mapped to the outer ring of a unit disk in the 2D parametrized space. Note that if the 3D surface is a manifold, the boundary edges can be detected by verifying that they belong to only one face of the mesh.
Applications of parametrization in graphics and animation include texture mapping, among many others.
Generalizations
Colin de Verdière (1991) generalized Tutte's spring theorem to graphs on higher-genus surfaces with non-positive curvature, where edges are represented by geodesics;[3] this result was later independently rediscovered by Hass & Scott (2015).[4] Analogous results for graphs embedded on a torus were independently proved by Delgado-Friedrichs (2005),[5] by Gortler, Gotsman & Thurston (2006),[6] and by Lovász (2019).[7]
Chilakamarri, Dean & Littman (1995) investigate three-dimensional graph embeddings of the graphs of four-dimensional polytopes, formed by the same method as Tutte's embedding: choose one facet of the polytope as being the outer face of a three-dimensional embedding, and fix its vertices into place as the vertices of a three-dimensional polyhedron in space. Let each remaining vertex of the polytope be free to move in space, and replace each edge of the polytope by a spring. Then, find the minimum-energy configuration of the system of springs. As they show, the system of equations obtained in this way is again non-degenerate, but it is unclear under what conditions this method will find an embedding that realizes all facets of the polytope as convex polyhedra.[8]
Related results
The result that every simple planar graph can be drawn with straight line edges is called Fáry's theorem.[9] The Tutte spring theorem proves this for 3-connected planar graphs, but the result is true more generally for planar graphs regardless of connectivity. Using the Tutte spring system for a graph that is not 3-connected may result in degeneracies, in which subgraphs of the given graph collapse onto a point or a line segment; however, an arbitrary planar graph may be drawn using the Tutte embedding by adding extra edges to make it 3-connected, drawing the resulting 3-connected graph, and then removing the extra edges.
A graph is k-vertex-connected, but not necessarily planar, if and only if it has a convex embedding into (k −1)-dimensional space in which an arbitrary k-tuple of vertices are placed at the vertices of a simplex and, for each remaining vertex v, the convex hull of the neighbors of v is full-dimensional with v in its interior. If such an embedding exists, one can be found by fixing the locations of the chosen k vertices and solving a system of equations that places each vertex at the average of its neighbors, just as in Tutte's planar embedding.[10]
In finite element mesh generation, Laplacian smoothing is a common method for postprocessing a generated mesh to improve the quality of its elements;[11] it is particularly popular for quadrilateral meshes, for which other methods such as Lloyd's algorithm for triangular mesh smoothing are less applicable. In this method, each vertex is moved to or towards the average of its neighbors' positions, but this motion is only performed for a small number of iterations, to avoid large distortions of element sizes or (in the case of non-convex mesh domains) tangled non-planar meshes.
Force-directed graph drawing systems continue to be a popular method for visualizing graphs, but these systems typically use more complicated systems of forces that combine attractive forces on graph edges (as in Tutte's embedding) with repulsive forces between arbitrary pairs of vertices. These additional forces may cause the system to have many locally stable configurations rather than, as in Tutte's embedding, a single global solution.[12]
References
1. Tutte, W. T. (1963), "How to draw a graph", Proceedings of the London Mathematical Society, 13: 743–767, doi:10.1112/plms/s3-13.1.743, MR 0158387.
2. Rote, Günter (2012), "Realizing planar graphs as convex polytopes", Graph Drawing: 19th International Symposium, GD 2011, Eindhoven, The Netherlands, September 21–23, 2011, Revised Selected Papers, Lecture Notes in Computer Science, vol. 7034, Springer, pp. 238–241, doi:10.1007/978-3-642-25878-7_23.
3. Colin de Verdière, Yves. (1991), "Comment rendre géodésique une triangulation d'une surface ?", L'Enseignement Mathématique, 37 (3–4): 201–212, doi:10.5169/seals-58738, MR 1151746.
4. Hass, Joel; Scott, Peter (2015), "Simplicial energy and simplicial harmonic maps", Asian Journal of Mathematics, 19 (4): 593–636, doi:10.4310/AJM.2015.v19.n4.a2, MR 3423736, S2CID 15606779.
5. Delgado-Friedrichs, Olaf (2005), "Equilibrium placement of periodic graphs and convexity of plane tilings", Discrete & Computational Geometry, 33 (1): 67–81, doi:10.1007/s00454-004-1147-x, MR 2105751
6. Gortler, Steven J.; Gotsman, Craig; Thurston, Dylan (2006), "Discrete one-forms on meshes and applications to 3D mesh parameterization", Computer Aided Geometric Design, 23 (2): 83–112, doi:10.1016/j.cagd.2005.05.002, MR 2189438, S2CID 135438.
7. Lovász, Lázsló (2019), Graphs and Geometry (PDF), American Mathematics Society, p. 98, ISBN 978-1-4704-5087-8, retrieved 18 July 2019
8. Chilakamarri, Kiran; Dean, Nathaniel; Littman, Michael (1995), "Three-dimensional Tutte embedding", Proceedings of the Twenty-Sixth Southeastern International Conference on Combinatorics, Graph Theory and Computing (Boca Raton, FL, 1995), Congressus Numerantium, 107: 129–140, MR 1369261.
9. For the relation between Tutte's and Fáry's theorem, and the history of rediscovery of Fáry's theorem, see Felsner, Stefan (2012), Geometric Graphs and Arrangements: Some Chapters from Combinatorial Geometry, Advanced Lectures in Mathematics, Springer, p. 37, ISBN 9783322803030.
10. Linial, N.; Lovász, L.; Wigderson, A. (1988), "Rubber bands, convex embeddings and graph connectivity", Combinatorica, 8 (1): 91–102, doi:10.1007/BF02122557, MR 0951998, S2CID 6164458.
11. Herrmann, Leonard R. (1976), "Laplacian-isoparametric grid generation scheme", Journal of the Engineering Mechanics Division, 102 (5): 749–907, doi:10.1061/JMCEA3.0002158.
12. Kobourov, Stephen G. (2012), Spring Embedders and Force-Directed Graph Drawing Algorithms, arXiv:1201.3011, Bibcode:2012arXiv1201.3011K.
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Wikipedia
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Tutte graph
In the mathematical field of graph theory, the Tutte graph is a 3-regular graph with 46 vertices and 69 edges named after W. T. Tutte.[1] It has chromatic number 3, chromatic index 3, girth 4 and diameter 8.
Tutte graph
Tutte graph
Named afterW. T. Tutte
Vertices46
Edges69
Radius5
Diameter8
Girth4
Automorphisms3 (Z/3Z)
Chromatic number3
Chromatic index3
PropertiesCubic
Planar
Polyhedral
Table of graphs and parameters
The Tutte graph is a cubic polyhedral graph, but is non-hamiltonian. Therefore, it is a counterexample to Tait's conjecture that every 3-regular polyhedron has a Hamiltonian cycle.[2]
Published by Tutte in 1946, it is the first counterexample constructed for this conjecture.[3] Other counterexamples were found later, in many cases based on Grinberg's theorem.
Construction
From a small planar graph called the Tutte fragment, W. T. Tutte constructed a non-Hamiltonian polyhedron, by putting together three such fragments. The "compulsory" edges of the fragments, that must be part of any Hamiltonian path through the fragment, are connected at the central vertex; because any cycle can use only two of these three edges, there can be no Hamiltonian cycle.
The resulting graph is 3-connected and planar, so by Steinitz' theorem it is the graph of a polyhedron. It has 25 faces.
It can be realized geometrically from a tetrahedron (the faces of which correspond to the four large nine-sided faces in the drawing, three of which are between pairs of fragments and the fourth of which forms the exterior) by multiply truncating three of its vertices.
Algebraic properties
The automorphism group of the Tutte graph is Z/3Z, the cyclic group of order 3.
The characteristic polynomial of the Tutte graph is :
$(x-3)(x^{15}-22x^{13}+x^{12}+184x^{11}-26x^{10}-731x^{9}+199x^{8}+1383x^{7}-576x^{6}-1061x^{5}+561x^{4}+233x^{3}-151x^{2}+4x+4)^{2}$
$(x^{15}+3x^{14}-16x^{13}-50x^{12}+94x^{11}+310x^{10}-257x^{9}-893x^{8}+366x^{7}+1218x^{6}-347x^{5}-717x^{4}+236x^{3}+128x^{2}-56x+4).$
Related graphs
Although the Tutte graph is the first 3-regular non-Hamiltonian polyhedral graph to be discovered, it is not the smallest such graph.
In 1965 Lederberg found the Barnette–Bosák–Lederberg graph on 38 vertices.[4][5] In 1968, Grinberg constructed additional small counterexamples to the Tait's conjecture – the Grinberg graphs on 42, 44 and 46 vertices.[6] In 1974 Faulkner and Younger published two more graphs – the Faulkner–Younger graphs on 42 and 44 vertices.[7]
Finally Holton and McKay showed there are exactly six 38-vertex non-Hamiltonian polyhedra that have nontrivial three-edge cuts. They are formed by replacing two of the vertices of a pentagonal prism by the same fragment used in Tutte's example.[8]
References
1. Weisstein, Eric W. "Tutte's Graph". MathWorld.
2. Tait, P. G. (1884), "Listing's Topologie", Philosophical Magazine, 5th Series, 17: 30–46. Reprinted in Scientific Papers, Vol. II, pp. 85–98.
3. Tutte, W. T. (1946), "On Hamiltonian circuits" (PDF), Journal of the London Mathematical Society, 21 (2): 98–101, doi:10.1112/jlms/s1-21.2.98.
4. Lederberg, J. "DENDRAL-64: A System for Computer Construction, Enumeration and Notation of Organic Molecules as Tree Structures and Cyclic Graphs. Part II. Topology of Cyclic Graphs." Interim Report to the National Aeronautics and Space Administration. Grant NsG 81-60. December 15, 1965. .
5. Weisstein, Eric W. "Barnette-Bosák-Lederberg Graph". MathWorld.
6. Grinberg, E. J. "Plane Homogeneous Graphs of Degree Three without Hamiltonian Circuits." Latvian Math. Yearbook, Izdat. Zinatne, Riga 4, 51–58, 1968.
7. Faulkner, G. B. and Younger, D. H. "Non-Hamiltonian Cubic Planar Maps." Discrete Math. 7, 67–74, 1974.
8. Holton, D. A.; McKay, B. D. (1988), "The smallest non-Hamiltonian 3-connected cubic planar graphs have 38 vertices", Journal of Combinatorial Theory, Series B, 45 (3): 305–319, doi:10.1016/0095-8956(88)90075-5.
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Wikipedia
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Tutte homotopy theorem
In mathematics, the Tutte homotopy theorem, introduced by Tutte (1958), generalises the concept of "path" from graphs to matroids, and states roughly that closed paths can be written as compositions of elementary closed paths, so that in some sense they are homotopic to the trivial closed path.
Statement
A matroid on a set Q is specified by a class of non-empty subsets M of Q, called circuits, such that no element of M contains another, and if X and Y are in M, a is in X and Y, b is in X but not in Y, then there is some Z in M containing b but not a and contained in X∪Y.
The subsets of Q that are unions of circuits are called flats (this is the language used in Tutte's original paper, however in modern usage the flats of a matroid mean something different). The elements of M are called 0-flats, the minimal non-empty flats that are not 0-flats are called 1-flats, the minimal nonempty flats that are not 0-flats or 1-flats are called 2-flats, and so on.
A path is a finite sequence of 0-flats such that any two consecutive elements of the path lie in some 1-flat.
An elementary path is one of the form (X,Y,X), or (X,Y,Z,X) with X,Y,Z all lying in some 2-flat.
Two paths P and Q such that the last 0-flat of P is the same as the first 0-flat of Q can be composed in the obvious way to give a path PQ.
Two paths are called homotopic if one can be obtained from the other by the operations of adding or removing elementary paths inside a path, in other words changing a path PR to PQR or vice versa, where Q is elementary.
A weak form of Tutte's homotopy theorem states that any closed path is homotopic to the trivial path. A stronger form states a similar result for paths not meeting certain "convex" subsets.
References
• Tutte, William Thomas (1958), "A homotopy theorem for matroids. I", Transactions of the American Mathematical Society, 88 (1): 144–160, doi:10.2307/1993243, ISSN 0002-9947, JSTOR 1993243, MR 0101526
• Tutte, William Thomas (1958), "A homotopy theorem for matroids. II", Transactions of the American Mathematical Society, 88 (1): 161–174, doi:10.2307/1993244, ISSN 0002-9947, JSTOR 1993244, MR 0101526
• Tutte, W.T. (1971), Introduction to the theory of matroids, Modern Analytic and Computational Methods in Science and Mathematics, vol. 37, New York: American Elsevier Publishing Company, pp. 72–77, Zbl 0231.05027
• White, Neil (1987), "Unimodular matroids", in White, Neil (ed.), Combinatorial geometries, Encyclopedia of Mathematics and its Applications, vol. 29, Cambridge University Press, pp. 40–52, doi:10.1017/CBO9781107325715, ISBN 978-0-521-33339-9, MR 0921064
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Wikipedia
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Tutte matrix
In graph theory, the Tutte matrix A of a graph G = (V, E) is a matrix used to determine the existence of a perfect matching: that is, a set of edges which is incident with each vertex exactly once.
If the set of vertices is $V=\{1,2,\dots ,n\}$ then the Tutte matrix is an n × n matrix A with entries
$A_{ij}={\begin{cases}x_{ij}\;\;{\mbox{if}}\;(i,j)\in E{\mbox{ and }}i<j\\-x_{ji}\;\;{\mbox{if}}\;(i,j)\in E{\mbox{ and }}i>j\\0\;\;\;\;{\mbox{otherwise}}\end{cases}}$
where the xij are indeterminates. The determinant of this skew-symmetric matrix is then a polynomial (in the variables xij, i < j ): this coincides with the square of the pfaffian of the matrix A and is non-zero (as a polynomial) if and only if a perfect matching exists. (This polynomial is not the Tutte polynomial of G.)
The Tutte matrix is named after W. T. Tutte, and is a generalisation of the Edmonds matrix for a balanced bipartite graph.
References
• R. Motwani, P. Raghavan (1995). Randomized Algorithms. Cambridge University Press. p. 167.
• Allen B. Tucker (2004). Computer Science Handbook. CRC Press. p. 12.19. ISBN 1-58488-360-X.
• W.T. Tutte (April 1947). "The factorization of linear graphs" (PDF). J. London Math. Soc. 22 (2): 107–111. doi:10.1112/jlms/s1-22.2.107. Retrieved 2008-06-15.
Matrix classes
Explicitly constrained entries
• Alternant
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• Conference
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With specific applications
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Used in statistics
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Used in graph theory
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Related terms
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• Mathematics portal
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Wikipedia
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Tutte theorem
In the mathematical discipline of graph theory the Tutte theorem, named after William Thomas Tutte, is a characterization of finite graphs with perfect matchings. It is a generalization of Hall's marriage theorem from bipartite to arbitrary graphs. It is a special case of the Tutte–Berge formula.
Not to be confused with Tutte homotopy theorem or Tutte's spring theorem.
Intuition
Our goal is to characterize all graphs that do not have a perfect matching. Let us start with the most obvious case of a graph without a perfect matching: a graph with an odd number of vertices. In such a graph, any matching leaves at least one unmatched vertex, so it cannot be perfect.
A slightly more general case is a disconnected graph in which one or more components have an odd number of vertices (even if the total number of vertices is even). Let us call such components odd components. In any matching, each vertex can only be matched to vertices in the same component. Therefore, any matching leaves at least one unmatched vertex in every odd component, so it cannot be perfect.
Next, consider a graph G with a vertex u such that, if we remove from G the vertex u and its adjacent edges, the remaining graph (denoted G − u) has two or more odd components. As above, any matching leaves, in every odd component, at least one vertex that is unmatched to other vertices in the same component. Such a vertex can only be matched to u. But since there are two or more unmatched vertices, and only one of them can be matched to u, at least one other vertex remains unmatched, so the matching is not perfect.
Finally, consider a graph G with a set of vertices U such that, if we remove from G the vertices in U and all edges adjacent to them, the remaining graph (denoted G − U) has more than |U| odd components. As explained above, any matching leaves at least one unmatched vertex in every odd component, and these can be matched only to vertices of U - but there are not enough vertices on U for all these unmatched vertices, so the matching is not perfect.
We have arrived at a necessary condition: if G has a perfect matching, then for every vertex subset U in G, the graph G − U has at most |U| odd components.
Tutte's theorem says that this condition is both necessary and sufficient for the existence of perfect matching.
Tutte's theorem
A graph, G = (V, E), has a perfect matching if and only if for every subset U of V, the subgraph G − U has at most |U| odd components (connected components having an odd number of vertices).[1]
Proof
First we write the condition:
$(*)\qquad \forall U\subseteq V,\quad odd(G-U)\leq |U|$
where $odd(X)$ denotes the number of odd components of the subgraph induced by $X$.
Necessity of (∗): Consider a graph G, with a perfect matching. Let U be an arbitrary subset of V. Delete U. Let C be an arbitrary odd component in G − U. Since G had a perfect matching, at least one vertex in C must be matched to a vertex in U. Hence, each odd component has at least one vertex matched with a vertex in U. Since each vertex in U can be in this relation with at most one connected component (because of it being matched at most once in a perfect matching), odd(G − U) ≤ |U|.[2]
Sufficiency of (∗): Let G be an arbitrary graph with no perfect matching. We will find a Tutte violator, that is, a subset S of V such that |S| < odd(G − S). We can suppose that G is edge-maximal, i.e., G + e has a perfect matching for every edge e not present in G already. Indeed, if we find a Tutte violator S in edge-maximal graph G, then S is also a Tutte violator in every spanning subgraph of G, as every odd component of G − S will be split into possibly more components at least one of which will again be odd.
We define S to be the set of vertices with degree |V| − 1. First we consider the case where all components of G − S are complete graphs. Then S has to be a Tutte violator, since if odd(G − S) ≤ |S|, then we could find a perfect matching by matching one vertex from every odd component with a vertex from S and pairing up all other vertices (this will work unless |V| is odd, but then ∅ is a Tutte violator).
Now suppose that K is a component of G − S and x, y ∈ K are vertices such that xy ∉ E. Let x, a, b ∈ K be the first vertices on a shortest x,y-path in K. This ensures that xa, ab ∈ E and xb ∉ E. Since a ∉ S, there exists a vertex c such that ac ∉ E. From the edge-maximality of G, we define M1 as a perfect matching in G + xb and M2 as a perfect matching in G + ac. Observe that surely xb ∈ M1 and ac ∈ M2.
Let P be the maximal path in G that starts from c with an edge from M1 and whose edges alternate between M1 and M2. How can P end? Unless we arrive at 'special' vertex such as x, a or b, we can always continue: c is M2-matched by ca, so the first edge of P is not in M2, therefore the second vertex is M2-matched by a different edge and we continue in this manner.
Let v denote the last vertex of P. If the last edge of P is in M1, then v has to be a, since otherwise we could continue with an edge from M2 (even to arrive at x or b). In this case we define C:=P + ac. If the last edge of P is in M2, then surely v ∈ {x, b} for analogous reason and we define C:=P + va + ac.
Now C is a cycle in G + ac of even length with every other edge in M2. We can now define M:=M2 Δ C (where Δ is symmetric difference) and we obtain a perfect matching in G, a contradiction.
Equivalence to the Tutte-Berge formula
The Tutte–Berge formula says that the size of a maximum matching of a graph $G=(V,E)$ equals $\min _{U\subseteq V}\left(|U|-\operatorname {odd} (G-U)+|V|\right)/2$. Equivalently, the number of unmatched vertices in a maximum matching equals $\max _{U\subseteq V}\left(\operatorname {odd} (G-U)-|U|\right)$.
This formula follows from Tutte's theorem, together with the observation that $G$ has a matching of size $k$ if and only if the graph $G^{(k)}$ obtained by adding $|V|-2k$ new vertices, each joined to every original vertex of $G$, has a perfect matching. Since any set $X$ which separates $G^{(k)}$ into more than $|X|$ components must contain all the new vertices, (*) is satisfied for $G^{(k)}$ if and only if $k\leq \min _{U\subseteq V}\left(|U|-\operatorname {odd} (G-U)+|V|\right)/2$.
In infinite graphs
For connected infinite graphs that are locally finite (every vertex has finite degree), a generalization of Tutte's condition holds: such graphs have perfect matchings if and only if there is no finite subset, the removal of which creates a number of finite odd components larger than the size of the subset.[3]
See also
• Bipartite matching
• Hall's marriage theorem
• Petersen's theorem
Notes
1. Lovász & Plummer (1986), p. 84; Bondy & Murty (1976), Theorem 5.4, p. 76.
2. Bondy & Murty (1976), pp. 76–78.
3. Tutte, W. T. (1950). "The factorization of locally finite graphs". Canadian Journal of Mathematics. 2: 44–49. doi:10.4153/cjm-1950-005-2. MR 0032986. S2CID 124434131.
References
• Bondy, J. A.; Murty, U. S. R. (1976). Graph theory with applications. New York: American Elsevier Pub. Co. ISBN 0-444-19451-7.
• Lovász, László; Plummer, M. D. (1986). Matching theory. Amsterdam: North-Holland. ISBN 0-444-87916-1.
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Wikipedia
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Tutte–Berge formula
In the mathematical discipline of graph theory the Tutte–Berge formula is a characterization of the size of a maximum matching in a graph. It is a generalization of Tutte theorem on perfect matchings, and is named after W. T. Tutte (who proved Tutte's theorem) and Claude Berge (who proved its generalization).
Statement
The theorem states that the size of a maximum matching of a graph $G=(V,E)$ equals
${\frac {1}{2}}\min _{U\subseteq V}\left(|U|-\operatorname {odd} (G-U)+|V|\right),$
where $\operatorname {odd} (H)$ counts how many of the connected components of the graph $H$ have an odd number of vertices.
Equivalently, the number of unmatched vertices in a maximum matching equals
$\max _{U\subseteq V}\left(\operatorname {odd} (G-U)-|U|\right)$.
Explanation
Intuitively, for any subset $U$ of the vertices, the only way to completely cover an odd component of $G-U$ by a matching is for one of the matched edges covering the component to be incident to $U$. If, instead, some odd component had no matched edge connecting it to $U$, then the part of the matching that covered the component would cover its vertices in pairs, but since the component has an odd number of vertices it would necessarily include at least one leftover and unmatched vertex. Therefore, if some choice of $U$ has few vertices but its removal creates a large number of odd components, then there will be many unmatched vertices, implying that the matching itself will be small. This reasoning can be made precise by stating that the size of a maximum matching is at most equal to the value given by the Tutte–Berge formula.
The characterization of Tutte and Berge proves that this is the only obstacle to creating a large matching: the size of the optimal matching will be determined by the subset $U$ with the biggest difference between its numbers of odd components outside $U$ and vertices inside $U$. That is, there always exists a subset $U$ such that deleting $U$ creates the correct number of odd components needed to make the formula true. One way to find such a set $U$ is to choose any maximum matching $M$, and to let $X$ be the set of vertices that are either unmatched in $M$, or that can be reached from an unmatched vertex by an alternating path that ends with a matched edge. Then, let $U$ be the set of vertices that are matched by $M$ to vertices in $X$. No two vertices in $X$ can be adjacent, for if they were then their alternating paths could be concatenated to give a path by which the matching could be increased, contradicting the maximality of $M$. Every neighbor of a vertex $x$ in $X$ must belong to $U$, for otherwise we could extend an alternating path to $x$ by one more pair of edges, through the neighbor, causing the neighbor to become part of $U$. Therefore, in $G-U$, every vertex of $X$ forms a single-vertex component, which is odd. There can be no other odd components, because all other vertices remain matched after deleting $U$. So with this construction the size of $U$ and the number of odd components created by deleting $U$ are what they need to be to make the formula be true.
Relation to Tutte's theorem
Tutte's theorem characterizes the graphs with perfect matchings as being the ones for which deleting any subset $U$ of vertices creates at most $|U|$ odd components. (A subset $U$ that creates at least $|U|$ odd components can always be found in the empty set.) In this case, by the Tutte–Berge formula, the size of the matching is $|V|/2$; that is, the maximum matching is a perfect matching. Thus, Tutte's theorem can be derived as a corollary of the Tutte–Berge formula, and the formula can be seen as a generalization of Tutte's theorem.
See also
• Graph toughness, a problem of creating many connected components by removing a small set of vertices without regard to the parity of the components
• Hall's marriage theorem
References
• Berge, C. (1958). "Sur le couplage maximum d'un graphe". Comptes rendus hebdomadaires des séances de l'Académie des sciences. 247: 258–259.
• Berge, C. (1962). The Theory of Graphs. Methuen. Theorem 5, p. 181. Reprinted by Dover Publications, 2001.
• Bondy, J. A.; Murty, U. S. R. (2007). Graph Theory: An Advanced Course. Graduate Texts in Mathematics. Springer-Verlag. p. 428. ISBN 978-1-84628-969-9.
• Bondy, J. A.; Murty, U. S. R. (1976). Graph Theory with Applications. New York: North Holland. Exercise 5.3.4, p. 80. ISBN 0-444-19451-7.{{cite book}}: CS1 maint: url-status (link)
• Brualdi, Richard A. (2006). Combinatorial matrix classes. Encyclopedia of Mathematics and Its Applications. Vol. 108. Cambridge: Cambridge University Press. p. 360. ISBN 0-521-86565-4. Zbl 1106.05001.
• Lovász, László; Plummer, M. D. (1986). Matching Theory. Annals of Discrete Mathematics. Vol. 29. North-Holland. pp. 90–91. ISBN 0-444-87916-1. MR 0859549.
• Schrijver, Alexander (2003). Combinatorial Optimization: Polyhedra and Efficiency. Springer-Verlag. p. 413. ISBN 3-540-44389-4.
• Tutte, W. T. (1947). "The factorization of linear graphs". Journal of the London Mathematical Society. Series 1. 22 (2): 107–111. doi:10.1112/jlms/s1-22.2.107. hdl:10338.dmlcz/128215.
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Wikipedia
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Tuttminx
A Tuttminx (/ˈtʊtmɪŋks/ or /ˈtʌtmɪŋks/) is a Rubik's Cube-like twisty puzzle, in the shape of a truncated icosahedron. It was invented by Lee Tutt in 2005.[1] It has a total of 150 movable pieces to rearrange, compared to 20 movable pieces of the Rubik's Cube.
Description
The Tuttminx has a total of 32 face centre pieces (12 pentagon and 20 hexagon), 60 corner pieces, and 90 edge pieces. The face centres each have a single colour, which identifies the colour of that face in the solved state. The edge pieces have two colours, and the corner pieces have three colours. Each hexagonal face contains a centre piece, 6 corner pieces, and 6 edge pieces, while each pentagonal face contains a centre piece, 5 corner pieces, and 5 edge pieces.
The puzzle twists around the faces: each twist rotates one face centre piece and moves all edge and corner pieces surrounding it. The pentagonal faces can be twisted 72° in either direction, while the hexagonal faces can be rotated 120°.
The purpose of the puzzle is to scramble the colours, and then restore it to its original state of having one colour per face.
Number of combinations
The puzzle has 150 movable pieces: 60 corner pieces, 60 edge pieces that are adjacent to a pentagonal face (so-called pentagonal edges) and 30 edge pieces that are not (non-pentagonal edges). Only even permutations of all three types of pieces are possible (i.e. it is impossible to have only one pair of identical pieces swapped). Thus, there are 60!/2 possible ways to arrange the corner pieces, 60!/2 ways to arrange the pentagonal edges and 30!/2 ways to arrange the non-pentagonal edges.
All corner pieces have only one possible orientation, as do all pentagonal edge pieces. The non-pentagonal edge pieces all have 2 possible orientations each. Only even orientations of those are possible (meaning that it is impossible to have only one edge piece flipped over). This means there are 229 ways to orientate the edge pieces.
The number of possible combinations on the Tuttminx is therefore equal to
${\frac {60!\times 60!\times 30!\times 2^{29}}{8}}\approx 1.2325\times 10^{204}$
The full number is 1 232 507 756 161 568 013 733 174 639 895 750 813 761 087 074 840 896 182 396 140 424 396 146 760 158 229 902 239 889 099 665 575 990 049 299 860 175 851 176 152 712 039 950 335 697 389 221 704 074 672 278 055 758 253 470 515 200 000 000 000 000 000 000 000 000 000 000 000 (about 1.2325 septensexagintillion on the short scale and 1.2325 quattuortrigintillion on the long scale).
Variations
There are a few variations of the Tuttminx that have been made. The most popular ones include:
• Void Tuttminx, which is a regular Tuttminx but without the face centre pieces;[2]
• Rayminx (/ˈreɪmɪŋks/, also called Giga Tuttminx), which is a higher-order version of the Tuttminx;[3]
• Futtminx, which was invented by Oskar van Deventer and has been designed so that the hexagonal faces can be rotated 60° and mix up with the pentagonal faces.[4]
• Love bird, a 2x2x2 version of the Tuttminx.
• Tuttminx ball, also sticker variations of that.
• Chamfered dodecahedron tuttminx. It looks similar to the Tuttminx, but hexagons are located differently, they are distorted, and there are more of them (30 instead of 20).
See also
• Rubik's Cube
• Megaminx
• Helicopter Cube
• Pyraminx Crystal
• Goldberg Polyhedron
References
1. Twisty Puzzles: Tuttminx, 2005
2. Twisty Puzzles: Void Tuttminx, 2012
3. Twisty Puzzles: Rayminx, 2013
4. Futtminx on YouTube
Rubik's Cube
Puzzle inventors
• Ernő Rubik
• Larry Nichols
• Uwe Mèffert
• Tony Fisher
• Panagiotis Verdes
• Oskar van Deventer
Rubik's Cubes
• Overview
• Rubik's family cubes of all sizes
• 2×2×2 (Pocket Cube)
• 3×3×3 (Rubik's Cube)
• 4×4×4 (Rubik's Revenge)
• 5×5×5 (Professor's Cube)
• 6×6×6 (V-Cube 6)
• 7×7×7 (V-Cube 7)
• 8×8×8 (V-Cube 8)
Cubic variations
• Helicopter Cube
• Skewb
• Dino Cube
• Square 1
• Sudoku Cube
• Nine-Colour Cube
• Gear Cube
• Void Cube
Non-cubic
variations
Tetrahedron
• Pyraminx
• Pyraminx Duo
• Pyramorphix
• BrainTwist
Octahedron
• Skewb Diamond
Dodecahedron
• Megaminx
• Pyraminx Crystal
• Skewb Ultimate
Icosahedron
• Impossiball
• Dogic
Great dodecahedron
• Alexander's Star
Truncated icosahedron
• Tuttminx
Cuboid
• Rubik's Domino (2x3x3)
Virtual variations
(>3D)
• MagicCube4D
• MagicCube5D
• MagicCube7D
• Magic 120-cell
Derivatives
• Missing Link
• Rubik's 360
• Rubik's Clock
• Rubik's Magic
• Master Edition
• Rubik's Revolution
• Rubik's Snake
• Rubik's Triamid
Renowned solvers
• Yu Nakajima
• Édouard Chambon
• Bob Burton, Jr.
• Jessica Fridrich
• Chris Hardwick
• Kevin Hays
• Rowe Hessler
• Leyan Lo
• Shotaro Makisumi
• Toby Mao
• Prithveesh K. Bhat
• Krishnam Raju Gadiraju
• Tyson Mao
• Frank Morris
• Lars Petrus
• Gilles Roux
• David Singmaster
• Ron van Bruchem
• Eric Limeback
• Anthony Michael Brooks
• Mats Valk
• Feliks Zemdegs
• Collin Burns
• Max Park
Solutions
Speedsolving
• Speedcubing
Methods
• Layer by Layer
• CFOP method
• Optimal
Mathematics
• God's algorithm
• Superflip
• Thistlethwaite's algorithm
• Rubik's Cube group
Official organization
• World Cube Association
Related articles
• Rubik's Cube in popular culture
• Rubik, the Amazing Cube
• The Simple Solution to Rubik's Cube
• 1982 World Rubik's Cube Championship
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Tuza's conjecture
Tuza's conjecture is an unsolved problem in graph theory, a branch of mathematics, concerning triangles in undirected graphs.
Statement
In any graph $G$, one can define two quantities $\nu (G)$ and $\tau (G)$ based on the triangles in $G$. The quantity $\nu (G)$ is the "triangle packing number", the largest number of edge-disjoint triangles that it is possible to find in $G$.[1] It can be computed in polynomial time as a special case of the matroid parity problem.[2] The quantity $\tau (G)$ is the size of the smallest "triangle-hitting set", a set of edges that touches at least one edge from each triangle.[1]
Clearly, $\nu (G)\leq \tau (G)\leq 3\nu (G)$. For the first inequality, $\nu (G)\leq \tau (G)$, any triangle-hitting set must include at least one edge from each triangle of the optimal packing, and none of these edges can be shared between two or more of these triangles because the triangles are disjoint. For the second inequality, $\tau (G)\leq 3\nu (G)$, one can construct a triangle-hitting set of size $3\nu (G)$ by choosing all edges of the triangles of an optimal packing. This must hit all triangles in $G$, even the ones not in the packing, because otherwise the packing could be made larger by adding any unhit triangle.[1]
Tuza's conjecture asserts that the second inequality is not tight, and can be replaced by $\tau (G)\leq 2\nu (G)$. That is, according to this unproven conjecture, every undirected graph $G$ has a triangle-hitting set whose size is at most twice the number of triangles in an optimal packing.[1]
History and partial results
Zsolt Tuza formulated Tuza's conjecture in 1981.[1][3] If true, it would be best possible: there are infinitely many graphs for which $\tau (G)=2\nu (G)$, including all of the block graphs whose blocks are cliques of 2, 4, or 5 vertices.[1]
The conjecture is known to hold for planar graphs,[1] and more generally for sparse graphs of degeneracy at most six.[4] (Planar graphs have degeneracy at most five.) It is also known to hold for graphs of treewidth at most six,[5] for threshold graphs,[6] for sufficiently dense graphs, and for chordal graphs that contain a large clique.[1] For random graphs in the Erdős–Rényi–Gilbert model, it is true with high probability.[7]
Although Tuza's conjecture remains unproven, the bound $\tau (G)\leq 3\nu (G)$ can be improved, for all graphs, to $\tau (G)\leq (3-{\tfrac {3}{23}})\nu (G)\approx 2.8695\nu (G)$.[8]
See also
• Mantel's theorem
• Triangle removal lemma
References
1. Tuza, Zsolt (1990), "A conjecture on triangles of graphs", Graphs and Combinatorics, 6 (4): 373–380, doi:10.1007/BF01787705, MR 1092587
2. Lawler, Eugene L. (1976), "Chapter 9: The Matroid Parity Problem", Combinatorial Optimization: Networks and Matroids, New York: Holt, Rinehart and Winston, pp. 356–367, MR 0439106
3. Tuza, Zsolt (1984), "Conjecture", in Hajnal, A.; Lovász, L.; Sós, V. T. (eds.), Finite and Infinite Sets: Proceedings of the sixth Hungarian combinatorial colloquium held in Eger, July 6–11, 1981, Colloquia Mathematica Societatis János Bolyai, vol. 37, p. 888, ISBN 0-444-86763-5, MR 0818224
4. Puleo, Gregory J. (2015), "Tuza's conjecture for graphs with maximum average degree less than 7", European Journal of Combinatorics, 49: 134–152, arXiv:1308.2211, doi:10.1016/j.ejc.2015.03.006, MR 3349530
5. Botler, Fábio; Fernandes, Cristina G.; Gutiérrez, Juan (2021), "On Tuza's conjecture for triangulations and graphs with small treewidth", Discrete Mathematics, 344 (4), Paper No. 112281, arXiv:2002.07925, doi:10.1016/j.disc.2020.112281, MR 4204419
6. Bonamy, Marthe; Bożyk, Łukasz; Grzesik, Andrzej; Hatzel, Meike; Masařík, Tomáš; Novotná, Jana; Okrasa, Karolina (2022), "Tuza's conjecture for threshold graphs", Discrete Mathematics & Theoretical Computer Science, 24 (1): P24:1–P24:14, arXiv:2105.09871, MR 4471222
7. Kahn, Jeff; Park, Jinyoung (2022), "Tuza's conjecture for random graphs", Random Structures & Algorithms, 61 (2): 235–249, MR 4456027
8. Haxell, P. E. (1999), "Packing and covering triangles in graphs", Discrete Mathematics, 195 (1–3): 251–254, doi:10.1016/S0012-365X(98)00183-6, MR 1663859
External links
• van der Pol, Jorn (March 6, 2023), "Triangles, arcs, and ovals", The Matroid Union
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Alexander Tuzhilin
Alexander Sergei Tuzhilin (born 1957) is a Professor of Data Science and Information Systems and the Leonard N. Stern Endowed Professor of Business at New York University's Stern School of Business. He also serves as the Dean of Computer Science at the University of the People on the pro bono basis.
Alexander Tuzhilin
Born
Alexander Sergei Tuzhilin
1957 (age 65–66)
Alma materNew York University (S.B. 1980)
Stanford University (S.B. 1981)
Courant Institute of Mathematical Sciences (Ph.D. 1989)
Scientific career
FieldsInformation Systems, data mining
InstitutionsNew York University Stern School of Business
University of the People
ThesisUsing relational discrete event systems and models for prediction of future behavior of databases (1989)
Doctoral advisorZvi Kedem
Professor Tuzhilin is known for his work on personalization, recommender systems, machine learning and AI, where he has made several contributions, including being instrumental in developing the area of Context-Aware Recommender Systems (CARS), proposing novel methods of providing unexpected and cross-domain recommendations based on the principles of deep-learning, developing novel approaches to customer segmentation, and discovery of unexpected patterns in data.
Education
Tuzhilin received his B.A. in Mathematics from the New York University in 1980, M.S. in Engineering Economics from the Department of Management Science and Engineering at Stanford University in 1981, and Ph.D. in computer science from NYU's Courant Institute of Mathematical Sciences in 1989, his doctoral advisor being Zvi Kedem.[1][2]
Career
Tuzhilin joined the faculty at the New York University Stern School of Business in 1989 as an Assistant Professor of Information Systems. He is currently the Leonard N. Stern Professor of Business. He is also the Dean of Computer Science at the University of the People.[3]
Research
Tuzhilin researches data mining in databases, personalization, recommender systems, and customer relationship management.[2]
In 2006, Tuzhilin was hired by Google and given access to its monitoring systems to do a study on click fraud. This was part of a class-action settlement requiring Google to offer advertisers up to $60 million in refunds. Tuzhilin concluded that defining and tracking click fraud will be difficult, because it is often not possible to decipher whether Web surfers were clicking on an advertising link out of malice or as part of an innocent online excursion.[4][5][6][7]
Patents
In 2001, Tuzhilin patented a method of building customer profiles and using them to recommend products and services. Tuzhilin said of the patent, 'It's very broad and very general, and occupies some prime real estate in this space. It essentially covers technologies that are crucial for implementation of customer relationship management.' He added that the patent was careful not to stipulate that the technology was designed for Internet applications. Others pointed out that there were legal exceptions to business methods patents. Any individuals or companies that can show they have been engaged in a business practice for at least a year before a patent application for that practice was filed may be able to circumvent the patent.[8]
In March 2012, Yahoo sued Facebook for violating 10 of its patents. Facebook countersued Yahoo, claiming that it violated Facebook patents that covered 80% of the Yahoo's 2011 revenues. Three of Facebook's patents were originally granted to Tuzhilin.[9][10][11]
References
1. "Alexander Tuzhilin". NYU Stern. Retrieved 2020-04-19.
2. Tuzhilin, Alexander. "CV" (PDF). NYU Stern.
3. Dr. Alexander Tuzhilin Dean, Computer Science, University of the People
4. Search giants team up to combat 'click fraud', New York Times, August 2, 2006
5. Expert's Report Backs Google in Lawsuit Over Click Fraud, Los Angeles Times, July 22, 2006
6. Report: Google Tries to Fight Click Fraud, Washington Post, July 21, 2006
7. So Many Hits, So Few Sales, Wall Street Journal, November 13, 2006
8. A method of collecting consumer data renews questions about patents on business practices, New York Times, July 30, 2001
9. Facebook Accuses Yahoo of Infringing on Patents, New York Times, April 3, 2012
10. How Facebook’s Winning The War Against Yahoo, Patent By Patent, TechCrunch, Apr 4, 2012
11. How Facebook built its legal defence against Yahoo, Financial Times, Apr 03 2012
External links
• Alexander Tuzhilin publications indexed by Google Scholar
• Alexander Tuzhilin at the Mathematics Genealogy Project
Authority control
International
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• VIAF
National
• Norway
• Germany
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• United States
• Netherlands
Academics
• DBLP
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• Scopus
Other
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Tverberg's theorem
In discrete geometry, Tverberg's theorem, first stated by Helge Tverberg in 1966,[1] is the result that sufficiently many points in d-dimensional Euclidean space can be partitioned into subsets with intersecting convex hulls. Specifically, for any positive integers d, r and any set of
$(d+1)(r-1)+1\ $
points there exists a point x (not necessarily one of the given points) and a partition of the given points into r subsets, such that x belongs to the convex hull of all of the subsets. The partition resulting from this theorem is known as a Tverberg partition.
The special case r = 2 was proved earlier by Radon, and it is known as Radon's theorem.
Examples
The case d = 1 states that any 2r-1 points on the real line can be partitioned into r subsets with intersecting convex hulls. Indeed, if the points are x1 < x2 < ... < x2r < x2r-1, then the partition into Ai = {xi, x2r-i} for i in 1,...,r satisfies this condition (and it is unique).
For r = 2, Tverberg's theorem states that any d + 2 points may be partitioned into two subsets with intersecting convex hulls. This is known as Radon's theorem. In this case, for points in general position, the partition is unique.
The case r = 3 and d = 2 states that any seven points in the plane may be partitioned into three subsets with intersecting convex hulls. The illustration shows an example in which the seven points are the vertices of a regular heptagon. As the example shows, there may be many different Tverberg partitions of the same set of points; these seven points may be partitioned in seven different ways that differ by rotations of each other.
Topological Tverberg Theorem
An equivalent formulation of Tverberg's theorem is:
Let d, r be positive integers, and let N := (d+1)(r-1). If ƒ is any affine function from an N-dimensional simplex ΔN to Rd, then there are r pairwise-disjoint faces of ΔN whose images under ƒ intersect. That is: there exist faces F1,...,Fr of ΔN such that $\forall i,j\in [r]:F_{i}\cap F_{j}=\emptyset $ and $f(F_{1})\cap \cdots \cap f(F_{r})\neq \emptyset $.
They are equivalent because any affine function on a simplex is uniquely determined by the images of its vertices. Formally, let ƒ be an affine function from ΔN to Rd. Let $v_{1},v_{2},\dots ,v_{N+1}$ be the vertices of ΔN, and let $x_{1},x_{2},\dots ,x_{N+1}$ be their images under ƒ. By the original formulation, the $x_{1},x_{2},\dots ,x_{N+1}$ can be partitioned into r disjoint subsets, e.g. ((xi)i in Aj)j in [r] with overlapping convex hull. Because f is affine, the convex hull of (xi)i in Aj is the image of the face spanned by the vertices (vi)i in Aj for all j in [r]. These faces are pairwise-disjoint, and their images under f intersect - as claimed by the new formulation. The topological Tverberg theorem generalizes this formluation. It allows f to be any continuous function - not necessarily affine. But, currently it is proved only for the case where r is a prime power:
Let d be a positive integer, and let r be a power of a prime number. Let N := (d+1)(r-1). If ƒ is any continuous function from an N-dimensional simplex ΔN to Rd, then there are r pairwise-disjoint faces of ΔN whose images under ƒ intersect. That is: there exist faces F1,...,Fr of ΔN such that $\forall i,j\in [r]:F_{i}\cap F_{j}=\emptyset $ and $f(F_{1})\cap \cdots \cap f(F_{r})\neq \emptyset $.
Proofs
The topological Tverberg theorem was proved for prime r by Barany, Shlosman and Szucs.[2] Matousek[3]: 162--163 presents a proof using deleted joins.
The theorem was proved for r a prime-power by Ozaydin,[4] and later by Volovikov[5] and Sarkaria.[6]
See also
• Rota's basis conjecture
• Tverberg-type theorems and the Fractional Helly property.[7]
References
1. Tverberg, H. (1966), "A generalization of Radon's theorem" (PDF), Journal of the London Mathematical Society, 41: 123–128, doi:10.1112/jlms/s1-41.1.123
2. Bárány, I.; Shlosman, S. B.; Szücs, A. (1981-02-01). "On a Topological Generalization of a Theorem of Tverberg". Journal of the London Mathematical Society. s2-23 (1): 158–164. doi:10.1112/jlms/s2-23.1.158.
3. Matoušek, Jiří (2007). Using the Borsuk-Ulam Theorem: Lectures on Topological Methods in Combinatorics and Geometry (2nd ed.). Berlin-Heidelberg: Springer-Verlag. ISBN 978-3-540-00362-5. Written in cooperation with Anders Björner and Günter M. Ziegler , Section 4.3
4. Ozaydin, Murad (1987). "Equivariant Maps for the Symmetric Group". {{cite journal}}: Cite journal requires |journal= (help)
5. Volovikov, A. Yu. (1996-03-01). "On a topological generalization of the Tverberg theorem". Mathematical Notes. 59 (3): 324–326. doi:10.1007/BF02308547. ISSN 1573-8876.
6. Sarkaria, K. S. (2000-11-01). "Tverberg partitions and Borsuk–Ulam theorems". Pacific Journal of Mathematics. 196 (1): 231–241. ISSN 0030-8730.
7. Hell, S. (2006), Tverberg-type theorems and the Fractional Helly property, Dissertation, TU Berlin, doi:10.14279/depositonce-1464
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Twelfth root of two
The twelfth root of two or ${\sqrt[{12}]{2}}$ (or equivalently $2^{1/12}$) is an algebraic irrational number, approximately equal to 1.0594631. It is most important in Western music theory, where it represents the frequency ratio (musical interval) of a semitone (Play ) in twelve-tone equal temperament. This number was proposed for the first time in relationship to musical tuning in the sixteenth and seventeenth centuries. It allows measurement and comparison of different intervals (frequency ratios) as consisting of different numbers of a single interval, the equal tempered semitone (for example, a minor third is 3 semitones, a major third is 4 semitones, and perfect fifth is 7 semitones).[lower-alpha 1] A semitone itself is divided into 100 cents (1 cent = ${\sqrt[{1200}]{2}}=2^{1/1200}$).
Octaves (12 semitones) increase exponentially when measured on a linear frequency scale (Hz).
Octaves are equally spaced when measured on a logarithmic scale (cents).
Numerical value
The twelfth root of two to 20 significant figures is 1.0594630943592952646.[2] Fraction approximations in increasing order of accuracy include 18/17, 89/84, 196/185, 1657/1564, and 18904/17843.
As of December 2013, its numerical value has been computed to at least twenty billion decimal digits.[3]
The equal-tempered chromatic scale
A musical interval is a ratio of frequencies and the equal-tempered chromatic scale divides the octave (which has a ratio of 2:1) into twelve equal parts. Each note has a frequency that is 21⁄12 times that of the one below it.
Applying this value successively to the tones of a chromatic scale, starting from A above middle C (known as A4) with a frequency of 440 Hz, produces the following sequence of pitches:
Note Standard interval name(s)
relating to A 440
Frequency
(Hz)
Multiplier Coefficient
(to six places)
Just intonation
ratio
AUnison440.0020⁄121.000000 1
A♯/B♭Minor second/Half step/Semitone466.1621⁄121.059463 ≈ 16⁄15
BMajor second/Full step/Whole tone493.8822⁄121.122462 ≈ 9⁄8
CMinor third523.2523⁄121.189207 ≈ 6⁄5
C♯/D♭Major third554.3724⁄121.259921 ≈ 5⁄4
DPerfect fourth587.3325⁄121.334839 ≈ 4⁄3
D♯/E♭Augmented fourth/Diminished fifth/Tritone622.2526⁄121.414213 ≈ 7⁄5
EPerfect fifth659.2627⁄121.498307 ≈ 3⁄2
FMinor sixth698.4628⁄121.587401 ≈ 8⁄5
F♯/G♭Major sixth739.9929⁄121.681792 ≈ 5⁄3
GMinor seventh783.99210⁄121.781797 ≈ 16⁄9
G♯/A♭Major seventh830.61211⁄121.887748 ≈ 15⁄8
AOctave880.00212⁄122.000000 2
The final A (A5: 880 Hz) is exactly twice the frequency of the lower A (A4: 440 Hz), that is, one octave higher.
Other tuning scales
Other tuning scales use slightly different interval ratios:
• The just or Pythagorean perfect fifth is 3/2, and the difference between the equal tempered perfect fifth and the just is a grad, the twelfth root of the Pythagorean comma (12√531441/524288).
• The equal tempered Bohlen–Pierce scale uses the interval of the thirteenth root of three (13√3).
• Stockhausen's Studie II (1954) makes use of the twenty-fifth root of five (25√5), a compound major third divided into 5×5 parts.
• The delta scale is based on ≈50√3/2.
• The gamma scale is based on ≈20√3/2.
• The beta scale is based on ≈11√3/2.
• The alpha scale is based on ≈9√3/2.
Pitch adjustment
Since the frequency ratio of a semitone is close to 106% ($1.05946\times 100=105.946$), increasing or decreasing the playback speed of a recording by 6% will shift the pitch up or down by about one semitone, or "half-step". Upscale reel-to-reel magnetic tape recorders typically have pitch adjustments of up to ±6%, generally used to match the playback or recording pitch to other music sources having slightly different tunings (or possibly recorded on equipment that was not running at quite the right speed). Modern recording studios utilize digital pitch shifting to achieve similar results, ranging from cents up to several half-steps (note that reel-to-reel adjustments also affect the tempo of the recorded sound, while digital shifting does not).
History
Historically this number was proposed for the first time in relationship to musical tuning in 1580 (drafted, rewritten 1610) by Simon Stevin.[4] In 1581 Italian musician Vincenzo Galilei may be the first European to suggest twelve-tone equal temperament.[1] The twelfth root of two was first calculated in 1584 by the Chinese mathematician and musician Zhu Zaiyu using an abacus to reach twenty four decimal places accurately,[1] calculated circa 1605 by Flemish mathematician Simon Stevin,[1] in 1636 by the French mathematician Marin Mersenne and in 1691 by German musician Andreas Werckmeister.[5]
See also
• Fret
• Just intonation § Practical difficulties
• Music and mathematics
• Piano key frequencies
• Scientific pitch notation
• Twelve-tone technique
• The Well-Tempered Clavier
Notes
1. "The smallest interval in an equal-tempered scale is the ratio $r^{n}=p$, so $r={\sqrt[{n}]{p}}$, where the ratio r divides the ratio p (= 2/1 in an octave) into n equal parts."[1]
References
1. Joseph, George Gheverghese (2010). The Crest of the Peacock: Non-European Roots of Mathematics, p.294-5. Third edition. Princeton. ISBN 9781400836369.
2. Sloane, N. J. A. (ed.). "Sequence A010774 (Decimal expansion of 12th root of 2)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
3. Komsta, Łukasz. "Computations page". Komsta.net. Retrieved 23 December 2016.
4. Christensen, Thomas (2002), The Cambridge History of Western Music Theory, p. 205, ISBN 978-0521686983
5. Goodrich, L. Carrington (2013). A Short History of the Chinese People, [unpaginated]. Courier. ISBN 9780486169231. Cites: Chu Tsai-yü (1584). New Remarks on the Study of Resonant Tubes.
Further reading
• Barbour, J. M. (1933). "A Sixteenth Century Chinese Approximation for π". American Mathematical Monthly. 40 (2): 69–73. doi:10.2307/2300937. JSTOR 2300937.
• Ellis, Alexander; Helmholtz, Hermann (1954). On the Sensations of Tone. Dover Publications. ISBN 0-486-60753-4.
• Partch, Harry (1974). Genesis of a Music. Da Capo Press. ISBN 0-306-80106-X.
Algebraic numbers
• Algebraic integer
• Chebyshev nodes
• Constructible number
• Conway's constant
• Cyclotomic field
• Eisenstein integer
• Gaussian integer
• Golden ratio (φ)
• Perron number
• Pisot–Vijayaraghavan number
• Quadratic irrational number
• Rational number
• Root of unity
• Salem number
• Silver ratio (δS)
• Square root of 2
• Square root of 3
• Square root of 5
• Square root of 6
• Square root of 7
• Doubling the cube
• Twelfth root of two
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Irrational numbers
• Chaitin's (Ω)
• Liouville
• Prime (ρ)
• Omega
• Cahen
• Logarithm of 2
• Gauss's (G)
• Twelfth root of 2
• Apéry's (ζ(3))
• Plastic (ρ)
• Square root of 2
• Supergolden ratio (ψ)
• Erdős–Borwein (E)
• Golden ratio (φ)
• Square root of 3
• Square root of pi (√π)
• Square root of 5
• Silver ratio (δS)
• Square root of 6
• Square root of 7
• Euler's (e)
• Pi (π)
• Schizophrenic
• Transcendental
• Trigonometric
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Tetrahedron
In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the ordinary convex polyhedra.[1]
Not to be confused with tetrahedroid, Tetraedron, or Tetrahedron (journal).
Regular tetrahedron
(Click here for rotating model)
TypePlatonic solid
ElementsF = 4, E = 6
V = 4 (χ = 2)
Faces by sides4{3}
Conway notationT
Schläfli symbols{3,3}
h{4,3}, s{2,4}, sr{2,2}
Face configurationV3.3.3
Wythoff symbol3 | 2 3
| 2 2 2
Coxeter diagram =
SymmetryTd, A3, [3,3], (*332)
Rotation groupT, [3,3]+, (332)
ReferencesU01, C15, W1
Propertiesregular, convexdeltahedron
Dihedral angle70.528779° = arccos(1⁄3)
3.3.3
(Vertex figure)
Self-dual
(dual polyhedron)
Net
Tetrahedral objects
The tetrahedron is the three-dimensional case of the more general concept of a Euclidean simplex, and may thus also be called a 3-simplex.
The tetrahedron is one kind of pyramid, which is a polyhedron with a flat polygon base and triangular faces connecting the base to a common point. In the case of a tetrahedron the base is a triangle (any of the four faces can be considered the base), so a tetrahedron is also known as a "triangular pyramid".
Like all convex polyhedra, a tetrahedron can be folded from a single sheet of paper. It has two such nets.[1]
For any tetrahedron there exists a sphere (called the circumsphere) on which all four vertices lie, and another sphere (the insphere) tangent to the tetrahedron's faces.[2]
Regular tetrahedron
A regular tetrahedron is a tetrahedron in which all four faces are equilateral triangles. It is one of the five regular Platonic solids, which have been known since antiquity.
In a regular tetrahedron, all faces are the same size and shape (congruent) and all edges are the same length.
Regular tetrahedra alone do not tessellate (fill space), but if alternated with regular octahedra in the ratio of two tetrahedra to one octahedron, they form the alternated cubic honeycomb, which is a tessellation. Some tetrahedra that are not regular, including the Schläfli orthoscheme and the Hill tetrahedron, can tessellate.
The regular tetrahedron is self-dual, which means that its dual is another regular tetrahedron. The compound figure comprising two such dual tetrahedra form a stellated octahedron or stella octangula.
Coordinates for a regular tetrahedron
The following Cartesian coordinates define the four vertices of a tetrahedron with edge length 2, centered at the origin, and two level edges:
$\left(\pm 1,0,-{\frac {1}{\sqrt {2}}}\right)\quad {\mbox{and}}\quad \left(0,\pm 1,{\frac {1}{\sqrt {2}}}\right)$
Expressed symmetrically as 4 points on the unit sphere, centroid at the origin, with lower face parallel to the $xy$ plane, the vertices are:
$v_{1}=\left({\sqrt {\frac {8}{9}}},0,-{\frac {1}{3}}\right)$
$v_{2}=\left(-{\sqrt {\frac {2}{9}}},{\sqrt {\frac {2}{3}}},-{\frac {1}{3}}\right)$
$v_{3}=\left(-{\sqrt {\frac {2}{9}}},-{\sqrt {\frac {2}{3}}},-{\frac {1}{3}}\right)$
$v_{4}=(0,0,1)$
with the edge length of ${\sqrt {\frac {8}{3}}}$.
Still another set of coordinates are based on an alternated cube or demicube with edge length 2. This form has Coxeter diagram and Schläfli symbol h{4,3}. The tetrahedron in this case has edge length 2√2. Inverting these coordinates generates the dual tetrahedron, and the pair together form the stellated octahedron, whose vertices are those of the original cube.
Tetrahedron: (1,1,1), (1,−1,−1), (−1,1,−1), (−1,−1,1)
Dual tetrahedron: (−1,−1,−1), (−1,1,1), (1,−1,1), (1,1,−1)
Angles and distances
For a regular tetrahedron of edge length a:
Face area $A_{0}={\frac {\sqrt {3}}{4}}a^{2}\,$
Surface area[3] $A=4\,A_{0}={\sqrt {3}}a^{2}\,$
Height of pyramid[4] $h={\frac {\sqrt {6}}{3}}a={\sqrt {\frac {2}{3}}}\,a\,$
Centroid to vertex distance ${\frac {3}{4}}\,h={\frac {\sqrt {6}}{4}}\,a={\sqrt {\frac {3}{8}}}\,a\,$
Edge to opposite edge distance $l={\frac {1}{\sqrt {2}}}\,a\,$
Volume[3] $V={\frac {1}{3}}A_{0}h={\frac {\sqrt {2}}{12}}a^{3}={\frac {a^{3}}{6{\sqrt {2}}}}\,$
Face-vertex-edge angle $\arccos \left({\frac {1}{\sqrt {3}}}\right)=\arctan \left({\sqrt {2}}\right)\,$
(approx. 54.7356°)
Face-edge-face angle, i.e., "dihedral angle"[3] $\arccos \left({\frac {1}{3}}\right)=\arctan \left(2{\sqrt {2}}\right)\,$
(approx. 70.5288°)
Vertex-Center-Vertex angle,[5] the angle between lines from the tetrahedron center to any two vertices. It is also the angle between Plateau borders at a vertex. In chemistry it is called the tetrahedral bond angle. This angle (in radians) is also the length of the circular arc on the unit sphere resulting from centrally projecting one edge of the tetrahedron to the sphere. $\arccos \left(-{\frac {1}{3}}\right)=2\arctan \left({\sqrt {2}}\right)\,$
(approx. 109.4712°)
Solid angle at a vertex subtended by a face $\arccos \left({\frac {23}{27}}\right)$
(approx. 0.55129 steradians)
(approx. 1809.8 square degrees)
Radius of circumsphere[3] $R={\frac {\sqrt {6}}{4}}a={\sqrt {\frac {3}{8}}}\,a\,$
Radius of insphere that is tangent to faces[3] $r={\frac {1}{3}}R={\frac {a}{\sqrt {24}}}\,$
Radius of midsphere that is tangent to edges[3] $r_{\mathrm {M} }={\sqrt {rR}}={\frac {a}{\sqrt {8}}}\,$
Radius of exspheres $r_{\mathrm {E} }={\frac {a}{\sqrt {6}}}\,$
Distance to exsphere center from the opposite vertex $d_{\mathrm {VE} }={\frac {\sqrt {6}}{2}}a={\sqrt {\frac {3}{2}}}a\,$
With respect to the base plane the slope of a face (2√2) is twice that of an edge (√2), corresponding to the fact that the horizontal distance covered from the base to the apex along an edge is twice that along the median of a face. In other words, if C is the centroid of the base, the distance from C to a vertex of the base is twice that from C to the midpoint of an edge of the base. This follows from the fact that the medians of a triangle intersect at its centroid, and this point divides each of them in two segments, one of which is twice as long as the other (see proof).
For a regular tetrahedron with side length a, radius R of its circumscribing sphere, and distances di from an arbitrary point in 3-space to its four vertices, we have[6]
${\begin{aligned}{\frac {d_{1}^{4}+d_{2}^{4}+d_{3}^{4}+d_{4}^{4}}{4}}+{\frac {16R^{4}}{9}}&=\left({\frac {d_{1}^{2}+d_{2}^{2}+d_{3}^{2}+d_{4}^{2}}{4}}+{\frac {2R^{2}}{3}}\right)^{2};\\4\left(a^{4}+d_{1}^{4}+d_{2}^{4}+d_{3}^{4}+d_{4}^{4}\right)&=\left(a^{2}+d_{1}^{2}+d_{2}^{2}+d_{3}^{2}+d_{4}^{2}\right)^{2}.\end{aligned}}$
Isometries of the regular tetrahedron
The vertices of a cube can be grouped into two groups of four, each forming a regular tetrahedron (see above, and also animation, showing one of the two tetrahedra in the cube). The symmetries of a regular tetrahedron correspond to half of those of a cube: those that map the tetrahedra to themselves, and not to each other.
The tetrahedron is the only Platonic solid that is not mapped to itself by point inversion.
The regular tetrahedron has 24 isometries, forming the symmetry group Td, [3,3], (*332), isomorphic to the symmetric group, S4. They can be categorized as follows:
• T, [3,3]+, (332) is isomorphic to alternating group, A4 (the identity and 11 proper rotations) with the following conjugacy classes (in parentheses are given the permutations of the vertices, or correspondingly, the faces, and the unit quaternion representation):
• identity (identity; 1)
• rotation about an axis through a vertex, perpendicular to the opposite plane, by an angle of ±120°: 4 axes, 2 per axis, together 8 ((1 2 3), etc.; 1 ± i ± j ± k/2)
• rotation by an angle of 180° such that an edge maps to the opposite edge: 3 ((1 2)(3 4), etc.; i, j, k)
• reflections in a plane perpendicular to an edge: 6
• reflections in a plane combined with 90° rotation about an axis perpendicular to the plane: 3 axes, 2 per axis, together 6; equivalently, they are 90° rotations combined with inversion (x is mapped to −x): the rotations correspond to those of the cube about face-to-face axes
Orthogonal projections of the regular tetrahedron
The regular tetrahedron has two special orthogonal projections, one centered on a vertex or equivalently on a face, and one centered on an edge. The first corresponds to the A2 Coxeter plane.
Orthographic projection
Centered by Face/vertex Edge
Image
Projective
symmetry
[3] [4]
Cross section of regular tetrahedron
The two skew perpendicular opposite edges of a regular tetrahedron define a set of parallel planes. When one of these planes intersects the tetrahedron the resulting cross section is a rectangle.[7] When the intersecting plane is near one of the edges the rectangle is long and skinny. When halfway between the two edges the intersection is a square. The aspect ratio of the rectangle reverses as you pass this halfway point. For the midpoint square intersection the resulting boundary line traverses every face of the tetrahedron similarly. If the tetrahedron is bisected on this plane, both halves become wedges.
This property also applies for tetragonal disphenoids when applied to the two special edge pairs.
Spherical tiling
The tetrahedron can also be represented as a spherical tiling, and projected onto the plane via a stereographic projection. This projection is conformal, preserving angles but not areas or lengths. Straight lines on the sphere are projected as circular arcs on the plane.
Orthographic projection Stereographic projection
Helical stacking
Regular tetrahedra can be stacked face-to-face in a chiral aperiodic chain called the Boerdijk–Coxeter helix.
In four dimensions, all the convex regular 4-polytopes with tetrahedral cells (the 5-cell, 16-cell and 600-cell) can be constructed as tilings of the 3-sphere by these chains, which become periodic in the three-dimensional space of the 4-polytope's boundary surface.
Irregular tetrahedra
Tetrahedral symmetry subgroup relations
Tetrahedral symmetries shown in tetrahedral diagrams
Tetrahedra which do not have four equilateral faces are categorized and named by the symmetries they do possess.
If all three pairs of opposite edges of a tetrahedron are perpendicular, then it is called an orthocentric tetrahedron. When only one pair of opposite edges are perpendicular, it is called a semi-orthocentric tetrahedron.
An isodynamic tetrahedron is one in which the cevians that join the vertices to the incenters of the opposite faces are concurrent.
An isogonic tetrahedron has concurrent cevians that join the vertices to the points of contact of the opposite faces with the inscribed sphere of the tetrahedron.
Trirectangular tetrahedron
In a trirectangular tetrahedron the three face angles at one vertex are right angles, as at the corner of a cube.
Kepler discovered the relationship between the cube, regular tetrahedron and trirectangular tetrahedron.[8]
Disphenoid
Main article: Disphenoid
A disphenoid is a tetrahedron with four congruent triangles as faces; the triangles necessarily have all angles acute. The regular tetrahedron is a special case of a disphenoid. Other names for the same shape include bisphenoid, isosceles tetrahedron and equifacial tetrahedron.
Orthoschemes
A 3-orthoscheme is a tetrahedron where all four faces are right triangles.[lower-alpha 1] An orthoscheme is an irregular simplex that is the convex hull of a tree in which all edges are mutually perpendicular. In a 3-dimensional orthoscheme, the tree consists of three perpendicular edges connecting all four vertices in a linear path that makes two right-angled turns. The 3-orthoscheme is a tetrahedron having two right angles at each of two vertices, so another name for it is birectangular tetrahedron. It is also called a quadrirectangular tetrahedron because it contains four right angles.[9]
Coxeter also calls quadrirectangular tetrahedra characteristic tetrahedra, because of their integral relationship to the regular polytopes and their symmetry groups.[10] For example, the special case of a 3-orthoscheme with equal-length perpendicular edges is characteristic of the cube, which means that the cube can be subdivided into instances of this orthoscheme. If its three perpendicular edges are of unit length, its remaining edges are two of length √2 and one of length √3, so all its edges are edges or diagonals of the cube. The cube can be dissected into six such 3-orthoschemes four different ways, with all six surrounding the same √3 cube diagonal. The cube can also be dissected into 48 smaller instances of this same characteristic 3-orthoscheme (just one way, by all of its symmetry planes at once).[lower-alpha 2] The characteristic tetrahedron of the cube is an example of a Heronian tetrahedron.
Every regular polytope, including the regular tetrahedron, has its characteristic orthoscheme.[lower-alpha 3] There is a 3-orthoscheme which is the characteristic tetrahedron of the regular tetrahedron. The regular tetrahedron is subdivided into 24 instances of its characteristic tetrahedron by its planes of symmetry.[lower-alpha 4]
Characteristics of the regular tetrahedron[13]
edge arc dihedral
𝒍 $2$ 109°28′16″ $\pi -2{\text{𝜿}}$ 70°31′44″ $\pi -2{\text{𝟁}}$
𝟀 ${\sqrt {\tfrac {4}{3}}}\approx 1.155$ 70°31′44″ $2{\text{𝜿}}$ 60° ${\tfrac {\pi }{3}}$
𝝓 $1$ 54°44′8″ ${\tfrac {\pi }{2}}-{\text{𝜿}}$ 60° ${\tfrac {\pi }{3}}$
𝟁 ${\sqrt {\tfrac {1}{3}}}\approx 0.577$ 54°44′8″ ${\tfrac {\pi }{2}}-{\text{𝜿}}$ 60° ${\tfrac {\pi }{3}}$
$_{0}R/l$ ${\sqrt {\tfrac {3}{2}}}\approx 1.225$
$_{1}R/l$ ${\sqrt {\tfrac {1}{2}}}\approx 0.707$
$_{2}R/l$ ${\sqrt {\tfrac {1}{6}}}\approx 0.408$
${\text{𝜿}}$ 35°15′52″ ${\tfrac {{\text{arc sec }}3}{2}}$
If the regular tetrahedron has edge length 𝒍 = 2, its characteristic tetrahedron's six edges have lengths ${\sqrt {\tfrac {4}{3}}}$, $1$, ${\sqrt {\tfrac {1}{3}}}$ (the exterior right triangle face, the characteristic triangle 𝟀, 𝝓, 𝟁), plus ${\sqrt {\tfrac {3}{2}}}$, ${\sqrt {\tfrac {1}{2}}}$, ${\sqrt {\tfrac {1}{6}}}$ (edges that are the characteristic radii of the regular tetrahedron). The 3-edge path along orthogonal edges of the orthoscheme is $1$, ${\sqrt {\tfrac {1}{3}}}$, ${\sqrt {\tfrac {1}{6}}}$, first from a tetrahedron vertex to an tetrahedron edge center, then turning 90° to an tetrahedron face center, then turning 90° to the tetrahedron center. The orthoscheme has four dissimilar right triangle faces. The exterior face is a 60-90-30 triangle which is one-sixth of a tetrahedron face. The three faces interior to the tetrahedron are: a right triangle with edges $1$, ${\sqrt {\tfrac {3}{2}}}$, ${\sqrt {\tfrac {1}{2}}}$, a right triangle with edges ${\sqrt {\tfrac {1}{3}}}$, ${\sqrt {\tfrac {1}{2}}}$, ${\sqrt {\tfrac {1}{6}}}$, and a right triangle with edges ${\sqrt {\tfrac {4}{3}}}$, ${\sqrt {\tfrac {3}{2}}}$, ${\sqrt {\tfrac {1}{6}}}$.
Space-filling tetrahedra
A space-filling tetrahedron packs with directly congruent or enantiomorphous (mirror image) copies of itself to tile space.[14] The cube can be dissected into six 3-orthoschemes, three left-handed and three right-handed (one of each at each cube face), and cubes can fill space, so the characteristic 3-orthoscheme of the cube is a space-filling tetrahedron in this sense.[lower-alpha 5] A disphenoid can be a space-filling tetrahedron in the directly congruent sense, as in the disphenoid tetrahedral honeycomb. Regular tetrahedra, however, cannot fill space by themselves.[lower-alpha 6]
Fundamental domains
An irregular tetrahedron which is the fundamental domain[15] of a symmetry group is an example of a Goursat tetrahedron. The Goursat tetrahedra generate all the regular polyhedra (and many other uniform polyhedra) by mirror reflections, a process referred to as Wythoff's kaleidoscopic construction.
For polyhedra, Wythoff's construction arranges three mirrors at angles to each other, as in a kaleidoscope. Unlike a cylindrical kaleidoscope, Wythoff's mirrors are located at three faces of a Goursat tetrahedron such that all three mirrors intersect at a single point.[lower-alpha 7]
Among the Goursat tetrahedra which generate 3-dimensional honeycombs we can recognize an orthoscheme (the characteristic tetrahedron of the cube), a double orthoscheme (the characteristic tetrahedron of the cube face-bonded to its mirror image), and the space-filling disphenoid illustrated above.[10] The disphenoid is the double orthoscheme face-bonded to its mirror image (a quadruple orthoscheme). Thus all three of these Goursat tetrahedra, and all the polyhedra they generate by reflections, can be dissected into characteristic tetrahedra of the cube.
Isometries of irregular tetrahedra
The isometries of an irregular (unmarked) tetrahedron depend on the geometry of the tetrahedron, with 7 cases possible. In each case a 3-dimensional point group is formed. Two other isometries (C3, [3]+), and (S4, [2+,4+]) can exist if the face or edge marking are included. Tetrahedral diagrams are included for each type below, with edges colored by isometric equivalence, and are gray colored for unique edges.
Tetrahedron name Edge
equivalence
diagram
Description
Symmetry
Schön. Cox. Orb. Ord.
Regular tetrahedron
Four equilateral triangles
It forms the symmetry group Td, isomorphic to the symmetric group, S4. A regular tetrahedron has Coxeter diagram and Schläfli symbol {3,3}.
Td
T
[3,3]
[3,3]+
*332
332
24
12
Triangular pyramid
An equilateral triangle base and three equal isosceles triangle sides
It gives 6 isometries, corresponding to the 6 isometries of the base. As permutations of the vertices, these 6 isometries are the identity 1, (123), (132), (12), (13) and (23), forming the symmetry group C3v, isomorphic to the symmetric group, S3. A triangular pyramid has Schläfli symbol {3}∨( ).
C3v
C3
[3]
[3]+
*33
33
6
3
Mirrored sphenoid
Two equal scalene triangles with a common base edge
This has two pairs of equal edges (1,3), (1,4) and (2,3), (2,4) and otherwise no edges equal. The only two isometries are 1 and the reflection (34), giving the group Cs, also isomorphic to the cyclic group, Z2.
Cs
=C1h
=C1v
[ ]*2
Irregular tetrahedron
(No symmetry)
Four unequal triangles
Its only isometry is the identity, and the symmetry group is the trivial group. An irregular tetrahedron has Schläfli symbol ( )∨( )∨( )∨( ).
C1[ ]+11
Disphenoids (Four equal triangles)
Tetragonal disphenoid
Four equal isosceles triangles
It has 8 isometries. If edges (1,2) and (3,4) are of different length to the other 4 then the 8 isometries are the identity 1, reflections (12) and (34), and 180° rotations (12)(34), (13)(24), (14)(23) and improper 90° rotations (1234) and (1432) forming the symmetry group D2d. A tetragonal disphenoid has Coxeter diagram and Schläfli symbol s{2,4}.
D2d
S4
[2+,4]
[2+,4+]
2*2
2×
8
4
Rhombic disphenoid
Four equal scalene triangles
It has 4 isometries. The isometries are 1 and the 180° rotations (12)(34), (13)(24), (14)(23). This is the Klein four-group V4 or Z22, present as the point group D2. A rhombic disphenoid has Coxeter diagram and Schläfli symbol sr{2,2}.
D2[2,2]+2224
Generalized disphenoids (2 pairs of equal triangles)
Digonal disphenoid
Two pairs of equal isosceles triangles
This gives two opposite edges (1,2) and (3,4) that are perpendicular but different lengths, and then the 4 isometries are 1, reflections (12) and (34) and the 180° rotation (12)(34). The symmetry group is C2v, isomorphic to the Klein four-group V4. A digonal disphenoid has Schläfli symbol { }∨{ }.
C2v
C2
[2]
[2]+
*22
22
4
2
Phyllic disphenoid
Two pairs of equal scalene or isosceles triangles
This has two pairs of equal edges (1,3), (2,4) and (1,4), (2,3) but otherwise no edges equal. The only two isometries are 1 and the rotation (12)(34), giving the group C2 isomorphic to the cyclic group, Z2.
C2[2]+222
General properties
Volume
The volume of a tetrahedron is given by the pyramid volume formula:
$V={\frac {1}{3}}A_{0}\,h\,$
where A0 is the area of the base and h is the height from the base to the apex. This applies for each of the four choices of the base, so the distances from the apices to the opposite faces are inversely proportional to the areas of these faces.
For a tetrahedron with vertices a = (a1, a2, a3), b = (b1, b2, b3), c = (c1, c2, c3), and d = (d1, d2, d3), the volume is 1/6|det(a − d, b − d, c − d)|, or any other combination of pairs of vertices that form a simply connected graph. This can be rewritten using a dot product and a cross product, yielding
$V={\frac {|(\mathbf {a} -\mathbf {d} )\cdot ((\mathbf {b} -\mathbf {d} )\times (\mathbf {c} -\mathbf {d} ))|}{6}}.$
If the origin of the coordinate system is chosen to coincide with vertex d, then d = 0, so
$V={\frac {|\mathbf {a} \cdot (\mathbf {b} \times \mathbf {c} )|}{6}},$
where a, b, and c represent three edges that meet at one vertex, and a · (b × c) is a scalar triple product. Comparing this formula with that used to compute the volume of a parallelepiped, we conclude that the volume of a tetrahedron is equal to 1/6 of the volume of any parallelepiped that shares three converging edges with it.
The absolute value of the scalar triple product can be represented as the following absolute values of determinants:
$6\cdot V={\begin{Vmatrix}\mathbf {a} &\mathbf {b} &\mathbf {c} \end{Vmatrix}}$ or $6\cdot V={\begin{Vmatrix}\mathbf {a} \\\mathbf {b} \\\mathbf {c} \end{Vmatrix}}$ where ${\begin{cases}\mathbf {a} =(a_{1},a_{2},a_{3}),\\\mathbf {b} =(b_{1},b_{2},b_{3}),\\\mathbf {c} =(c_{1},c_{2},c_{3}),\end{cases}}$ are expressed as row or column vectors.
Hence
$36\cdot V^{2}={\begin{vmatrix}\mathbf {a^{2}} &\mathbf {a} \cdot \mathbf {b} &\mathbf {a} \cdot \mathbf {c} \\\mathbf {a} \cdot \mathbf {b} &\mathbf {b^{2}} &\mathbf {b} \cdot \mathbf {c} \\\mathbf {a} \cdot \mathbf {c} &\mathbf {b} \cdot \mathbf {c} &\mathbf {c^{2}} \end{vmatrix}}$ where ${\begin{cases}\mathbf {a} \cdot \mathbf {b} =ab\cos {\gamma },\\\mathbf {b} \cdot \mathbf {c} =bc\cos {\alpha },\\\mathbf {a} \cdot \mathbf {c} =ac\cos {\beta }.\end{cases}}$
which gives
$V={\frac {abc}{6}}{\sqrt {1+2\cos {\alpha }\cos {\beta }\cos {\gamma }-\cos ^{2}{\alpha }-\cos ^{2}{\beta }-\cos ^{2}{\gamma }}},\,$
where α, β, γ are the plane angles occurring in vertex d. The angle α, is the angle between the two edges connecting the vertex d to the vertices b and c. The angle β, does so for the vertices a and c, while γ, is defined by the position of the vertices a and b.
If we do not require that d = 0 then
$6\cdot V=\left|\det \left({\begin{matrix}a_{1}&b_{1}&c_{1}&d_{1}\\a_{2}&b_{2}&c_{2}&d_{2}\\a_{3}&b_{3}&c_{3}&d_{3}\\1&1&1&1\end{matrix}}\right)\right|\,.$
Given the distances between the vertices of a tetrahedron the volume can be computed using the Cayley–Menger determinant:
$288\cdot V^{2}={\begin{vmatrix}0&1&1&1&1\\1&0&d_{12}^{2}&d_{13}^{2}&d_{14}^{2}\\1&d_{12}^{2}&0&d_{23}^{2}&d_{24}^{2}\\1&d_{13}^{2}&d_{23}^{2}&0&d_{34}^{2}\\1&d_{14}^{2}&d_{24}^{2}&d_{34}^{2}&0\end{vmatrix}}$
where the subscripts i, j ∈ {1, 2, 3, 4} represent the vertices {a, b, c, d} and dij is the pairwise distance between them – i.e., the length of the edge connecting the two vertices. A negative value of the determinant means that a tetrahedron cannot be constructed with the given distances. This formula, sometimes called Tartaglia's formula, is essentially due to the painter Piero della Francesca in the 15th century, as a three dimensional analogue of the 1st century Heron's formula for the area of a triangle.[16]
Let a, b, c be three edges that meet at a point, and x, y, z the opposite edges. Let V be the volume of the tetrahedron; then[17]
$V={\frac {\sqrt {4a^{2}b^{2}c^{2}-a^{2}X^{2}-b^{2}Y^{2}-c^{2}Z^{2}+XYZ}}{12}}$
where
${\begin{aligned}X&=b^{2}+c^{2}-x^{2},\\Y&=a^{2}+c^{2}-y^{2},\\Z&=a^{2}+b^{2}-z^{2}.\end{aligned}}$
The above formula uses six lengths of edges, and the following formula uses three lengths of edges and three angles.
$V={\frac {abc}{6}}{\sqrt {1+2\cos {\alpha }\cos {\beta }\cos {\gamma }-\cos ^{2}{\alpha }-\cos ^{2}{\beta }-\cos ^{2}{\gamma }}}$
Heron-type formula for the volume of a tetrahedron
If U, V, W, u, v, w are lengths of edges of the tetrahedron (first three form a triangle; with u opposite U, v opposite V, w opposite W), then[18]
$V={\frac {\sqrt {\,(-p+q+r+s)\,(p-q+r+s)\,(p+q-r+s)\,(p+q+r-s)}}{192\,u\,v\,w}}$
where
${\begin{aligned}p&={\sqrt {xYZ}},&q&={\sqrt {yZX}},&r&={\sqrt {zXY}},&s&={\sqrt {xyz}},\end{aligned}}$
${\begin{aligned}X&=(w-U+v)\,(U+v+w),&x&=(U-v+w)\,(v-w+U),\\Y&=(u-V+w)\,(V+w+u),&y&=(V-w+u)\,(w-u+V),\\Z&=(v-W+u)\,(W+u+v),&z&=(W-u+v)\,(u-v+W).\end{aligned}}$
Volume divider
Any plane containing a bimedian (connector of opposite edges' midpoints) of a tetrahedron bisects the volume of the tetrahedron.[19]
Non-Euclidean volume
For tetrahedra in hyperbolic space or in three-dimensional elliptic geometry, the dihedral angles of the tetrahedron determine its shape and hence its volume. In these cases, the volume is given by the Murakami–Yano formula.[20] However, in Euclidean space, scaling a tetrahedron changes its volume but not its dihedral angles, so no such formula can exist.
Distance between the edges
Any two opposite edges of a tetrahedron lie on two skew lines, and the distance between the edges is defined as the distance between the two skew lines. Let d be the distance between the skew lines formed by opposite edges a and b − c as calculated here. Then another volume formula is given by
$V={\frac {d|(\mathbf {a} \times \mathbf {(b-c)} )|}{6}}.$
Properties analogous to those of a triangle
The tetrahedron has many properties analogous to those of a triangle, including an insphere, circumsphere, medial tetrahedron, and exspheres. It has respective centers such as incenter, circumcenter, excenters, Spieker center and points such as a centroid. However, there is generally no orthocenter in the sense of intersecting altitudes.[21]
Gaspard Monge found a center that exists in every tetrahedron, now known as the Monge point: the point where the six midplanes of a tetrahedron intersect. A midplane is defined as a plane that is orthogonal to an edge joining any two vertices that also contains the centroid of an opposite edge formed by joining the other two vertices. If the tetrahedron's altitudes do intersect, then the Monge point and the orthocenter coincide to give the class of orthocentric tetrahedron.
An orthogonal line dropped from the Monge point to any face meets that face at the midpoint of the line segment between that face's orthocenter and the foot of the altitude dropped from the opposite vertex.
A line segment joining a vertex of a tetrahedron with the centroid of the opposite face is called a median and a line segment joining the midpoints of two opposite edges is called a bimedian of the tetrahedron. Hence there are four medians and three bimedians in a tetrahedron. These seven line segments are all concurrent at a point called the centroid of the tetrahedron.[22] In addition the four medians are divided in a 3:1 ratio by the centroid (see Commandino's theorem). The centroid of a tetrahedron is the midpoint between its Monge point and circumcenter. These points define the Euler line of the tetrahedron that is analogous to the Euler line of a triangle.
The nine-point circle of the general triangle has an analogue in the circumsphere of a tetrahedron's medial tetrahedron. It is the twelve-point sphere and besides the centroids of the four faces of the reference tetrahedron, it passes through four substitute Euler points, one third of the way from the Monge point toward each of the four vertices. Finally it passes through the four base points of orthogonal lines dropped from each Euler point to the face not containing the vertex that generated the Euler point.[23]
The center T of the twelve-point sphere also lies on the Euler line. Unlike its triangular counterpart, this center lies one third of the way from the Monge point M towards the circumcenter. Also, an orthogonal line through T to a chosen face is coplanar with two other orthogonal lines to the same face. The first is an orthogonal line passing through the corresponding Euler point to the chosen face. The second is an orthogonal line passing through the centroid of the chosen face. This orthogonal line through the twelve-point center lies midway between the Euler point orthogonal line and the centroidal orthogonal line. Furthermore, for any face, the twelve-point center lies at the midpoint of the corresponding Euler point and the orthocenter for that face.
The radius of the twelve-point sphere is one third of the circumradius of the reference tetrahedron.
There is a relation among the angles made by the faces of a general tetrahedron given by[24]
${\begin{vmatrix}-1&\cos {(\alpha _{12})}&\cos {(\alpha _{13})}&\cos {(\alpha _{14})}\\\cos {(\alpha _{12})}&-1&\cos {(\alpha _{23})}&\cos {(\alpha _{24})}\\\cos {(\alpha _{13})}&\cos {(\alpha _{23})}&-1&\cos {(\alpha _{34})}\\\cos {(\alpha _{14})}&\cos {(\alpha _{24})}&\cos {(\alpha _{34})}&-1\\\end{vmatrix}}=0\,$
where αij is the angle between the faces i and j.
The geometric median of the vertex position coordinates of a tetrahedron and its isogonic center are associated, under circumstances analogous to those observed for a triangle. Lorenz Lindelöf found that, corresponding to any given tetrahedron is a point now known as an isogonic center, O, at which the solid angles subtended by the faces are equal, having a common value of π sr, and at which the angles subtended by opposite edges are equal.[25] A solid angle of π sr is one quarter of that subtended by all of space. When all the solid angles at the vertices of a tetrahedron are smaller than π sr, O lies inside the tetrahedron, and because the sum of distances from O to the vertices is a minimum, O coincides with the geometric median, M, of the vertices. In the event that the solid angle at one of the vertices, v, measures exactly π sr, then O and M coincide with v. If however, a tetrahedron has a vertex, v, with solid angle greater than π sr, M still corresponds to v, but O lies outside the tetrahedron.
Geometric relations
A tetrahedron is a 3-simplex. Unlike the case of the other Platonic solids, all the vertices of a regular tetrahedron are equidistant from each other (they are the only possible arrangement of four equidistant points in 3-dimensional space).
A tetrahedron is a triangular pyramid, and the regular tetrahedron is self-dual.
A regular tetrahedron can be embedded inside a cube in two ways such that each vertex is a vertex of the cube, and each edge is a diagonal of one of the cube's faces. For one such embedding, the Cartesian coordinates of the vertices are
(+1, +1, +1);
(−1, −1, +1);
(−1, +1, −1);
(+1, −1, −1).
This yields a tetrahedron with edge-length 2√2, centered at the origin. For the other tetrahedron (which is dual to the first), reverse all the signs. These two tetrahedra's vertices combined are the vertices of a cube, demonstrating that the regular tetrahedron is the 3-demicube.
The volume of this tetrahedron is one-third the volume of the cube. Combining both tetrahedra gives a regular polyhedral compound called the compound of two tetrahedra or stella octangula.
The interior of the stella octangula is an octahedron, and correspondingly, a regular octahedron is the result of cutting off, from a regular tetrahedron, four regular tetrahedra of half the linear size (i.e., rectifying the tetrahedron).
The above embedding divides the cube into five tetrahedra, one of which is regular. In fact, five is the minimum number of tetrahedra required to compose a cube. To see this, starting from a base tetrahedron with 4 vertices, each added tetrahedra adds at most 1 new vertex, so at least 4 more must be added to make a cube, which has 8 vertices.
Inscribing tetrahedra inside the regular compound of five cubes gives two more regular compounds, containing five and ten tetrahedra.
Regular tetrahedra cannot tessellate space by themselves, although this result seems likely enough that Aristotle claimed it was possible. However, two regular tetrahedra can be combined with an octahedron, giving a rhombohedron that can tile space as the tetrahedral-octahedral honeycomb.
However, several irregular tetrahedra are known, of which copies can tile space, for instance the characteristic orthoscheme of the cube and the disphenoid of the disphenoid tetrahedral honeycomb. The complete list remains an open problem.[26]
If one relaxes the requirement that the tetrahedra be all the same shape, one can tile space using only tetrahedra in many different ways. For example, one can divide an octahedron into four identical tetrahedra and combine them again with two regular ones. (As a side-note: these two kinds of tetrahedron have the same volume.)
The tetrahedron is unique among the uniform polyhedra in possessing no parallel faces.
A law of sines for tetrahedra and the space of all shapes of tetrahedra
Main article: Trigonometry of a tetrahedron
A corollary of the usual law of sines is that in a tetrahedron with vertices O, A, B, C, we have
$\sin \angle OAB\cdot \sin \angle OBC\cdot \sin \angle OCA=\sin \angle OAC\cdot \sin \angle OCB\cdot \sin \angle OBA.\,$
One may view the two sides of this identity as corresponding to clockwise and counterclockwise orientations of the surface.
Putting any of the four vertices in the role of O yields four such identities, but at most three of them are independent: If the "clockwise" sides of three of them are multiplied and the product is inferred to be equal to the product of the "counterclockwise" sides of the same three identities, and then common factors are cancelled from both sides, the result is the fourth identity.
Three angles are the angles of some triangle if and only if their sum is 180° (π radians). What condition on 12 angles is necessary and sufficient for them to be the 12 angles of some tetrahedron? Clearly the sum of the angles of any side of the tetrahedron must be 180°. Since there are four such triangles, there are four such constraints on sums of angles, and the number of degrees of freedom is thereby reduced from 12 to 8. The four relations given by this sine law further reduce the number of degrees of freedom, from 8 down to not 4 but 5, since the fourth constraint is not independent of the first three. Thus the space of all shapes of tetrahedra is 5-dimensional.[27]
Law of cosines for tetrahedra
Main article: Trigonometry of a tetrahedron
Let {P1 ,P2, P3, P4} be the points of a tetrahedron. Let Δi be the area of the face opposite vertex Pi and let θij be the dihedral angle between the two faces of the tetrahedron adjacent to the edge PiPj.
The law of cosines for this tetrahedron,[28] which relates the areas of the faces of the tetrahedron to the dihedral angles about a vertex, is given by the following relation:
$\Delta _{i}^{2}=\Delta _{j}^{2}+\Delta _{k}^{2}+\Delta _{l}^{2}-2(\Delta _{j}\Delta _{k}\cos \theta _{il}+\Delta _{j}\Delta _{l}\cos \theta _{ik}+\Delta _{k}\Delta _{l}\cos \theta _{ij})$
Interior point
Let P be any interior point of a tetrahedron of volume V for which the vertices are A, B, C, and D, and for which the areas of the opposite faces are Fa, Fb, Fc, and Fd. Then[29]: p.62, #1609
$PA\cdot F_{\mathrm {a} }+PB\cdot F_{\mathrm {b} }+PC\cdot F_{\mathrm {c} }+PD\cdot F_{\mathrm {d} }\geq 9V.$
For vertices A, B, C, and D, interior point P, and feet J, K, L, and M of the perpendiculars from P to the faces, and suppose the faces have equal areas, then[29]: p.226, #215
$PA+PB+PC+PD\geq 3(PJ+PK+PL+PM).$
Inradius
Denoting the inradius of a tetrahedron as r and the inradii of its triangular faces as ri for i = 1, 2, 3, 4, we have[29]: p.81, #1990
${\frac {1}{r_{1}^{2}}}+{\frac {1}{r_{2}^{2}}}+{\frac {1}{r_{3}^{2}}}+{\frac {1}{r_{4}^{2}}}\leq {\frac {2}{r^{2}}},$
with equality if and only if the tetrahedron is regular.
If A1, A2, A3 and A4 denote the area of each faces, the value of r is given by
$r={\frac {3V}{A_{1}+A_{2}+A_{3}+A_{4}}}$.
This formula is obtained from dividing the tetrahedron into four tetrahedra whose points are the three points of one of the original faces and the incenter. Since the four subtetrahedra fill the volume, we have $V={\frac {1}{3}}A_{1}r+{\frac {1}{3}}A_{2}r+{\frac {1}{3}}A_{3}r+{\frac {1}{3}}A_{4}r$.
Circumradius
Denote the circumradius of a tetrahedron as R. Let a, b, c be the lengths of the three edges that meet at a vertex, and A, B, C the length of the opposite edges. Let V be the volume of the tetrahedron. Then[30][31]
$R={\frac {\sqrt {(aA+bB+cC)(aA+bB-cC)(aA-bB+cC)(-aA+bB+cC)}}{24V}}.$
Circumcenter
The circumcenter of a tetrahedron can be found as intersection of three bisector planes. A bisector plane is defined as the plane centered on, and orthogonal to an edge of the tetrahedron. With this definition, the circumcenter C of a tetrahedron with vertices x0,x1,x2,x3 can be formulated as matrix-vector product:[32]
${\begin{aligned}C&=A^{-1}B&{\text{where}}&\ &A=\left({\begin{matrix}\left[x_{1}-x_{0}\right]^{T}\\\left[x_{2}-x_{0}\right]^{T}\\\left[x_{3}-x_{0}\right]^{T}\end{matrix}}\right)&\ &{\text{and}}&\ &B={\frac {1}{2}}\left({\begin{matrix}\|x_{1}\|^{2}-\|x_{0}\|^{2}\\\|x_{2}\|^{2}-\|x_{0}\|^{2}\\\|x_{3}\|^{2}-\|x_{0}\|^{2}\end{matrix}}\right)\\\end{aligned}}$
In contrast to the centroid, the circumcenter may not always lay on the inside of a tetrahedron. Analogously to an obtuse triangle, the circumcenter is outside of the object for an obtuse tetrahedron.
Centroid
The tetrahedron's center of mass computes as the arithmetic mean of its four vertices, see Centroid.
Faces
The sum of the areas of any three faces is greater than the area of the fourth face.[29]: p.225, #159
Integer tetrahedra
Main article: Heronian tetrahedron
There exist tetrahedra having integer-valued edge lengths, face areas and volume. These are called Heronian tetrahedra. One example has one edge of 896, the opposite edge of 990 and the other four edges of 1073; two faces are isosceles triangles with areas of 436800 and the other two are isosceles with areas of 47120, while the volume is 124185600.[33]
A tetrahedron can have integer volume and consecutive integers as edges, an example being the one with edges 6, 7, 8, 9, 10, and 11 and volume 48.[34]
Related polyhedra and compounds
A regular tetrahedron can be seen as a triangular pyramid.
Regular pyramids
Digonal Triangular Square Pentagonal Hexagonal Heptagonal Octagonal Enneagonal Decagonal...
Improper Regular Equilateral Isosceles
A regular tetrahedron can be seen as a degenerate polyhedron, a uniform digonal antiprism, where base polygons are reduced digons.
Family of uniform n-gonal antiprisms
Antiprism name Digonal antiprism (Trigonal)
Triangular antiprism
(Tetragonal)
Square antiprism
Pentagonal antiprism Hexagonal antiprism Heptagonal antiprism Octagonal antiprism Enneagonal antiprism Decagonal antiprism Hendecagonal antiprism Dodecagonal antiprism ... Apeirogonal antiprism
Polyhedron image ...
Spherical tiling image Plane tiling image
Vertex config. 2.3.3.3 3.3.3.3 4.3.3.3 5.3.3.3 6.3.3.3 7.3.3.3 8.3.3.3 9.3.3.3 10.3.3.3 11.3.3.3 12.3.3.3 ... ∞.3.3.3
A regular tetrahedron can be seen as a degenerate polyhedron, a uniform dual digonal trapezohedron, containing 6 vertices, in two sets of colinear edges.
Family of n-gonal trapezohedra
Trapezohedron name Digonal trapezohedron
(Tetrahedron)
Trigonal trapezohedron Tetragonal trapezohedron Pentagonal trapezohedron Hexagonal trapezohedron Heptagonal trapezohedron Octagonal trapezohedron Decagonal trapezohedron Dodecagonal trapezohedron ... Apeirogonal trapezohedron
Polyhedron image ...
Spherical tiling image Plane tiling image
Face configuration V2.3.3.3 V3.3.3.3 V4.3.3.3 V5.3.3.3 V6.3.3.3 V7.3.3.3 V8.3.3.3 V10.3.3.3 V12.3.3.3 ... V∞.3.3.3
A truncation process applied to the tetrahedron produces a series of uniform polyhedra. Truncating edges down to points produces the octahedron as a rectified tetrahedron. The process completes as a birectification, reducing the original faces down to points, and producing the self-dual tetrahedron once again.
Family of uniform tetrahedral polyhedra
Symmetry: [3,3], (*332) [3,3]+, (332)
{3,3} t{3,3} r{3,3} t{3,3} {3,3} rr{3,3} tr{3,3} sr{3,3}
Duals to uniform polyhedra
V3.3.3 V3.6.6 V3.3.3.3 V3.6.6 V3.3.3 V3.4.3.4 V4.6.6 V3.3.3.3.3
This polyhedron is topologically related as a part of sequence of regular polyhedra with Schläfli symbols {3,n}, continuing into the hyperbolic plane.
*n32 symmetry mutation of regular tilings: {3,n}
Spherical Euclid. Compact hyper. Paraco. Noncompact hyperbolic
3.3 33 34 35 36 37 38 3∞ 312i 39i 36i 33i
The tetrahedron is topologically related to a series of regular polyhedra and tilings with order-3 vertex figures.
*n32 symmetry mutation of regular tilings: {n,3}
Spherical Euclidean Compact hyperb. Paraco. Noncompact hyperbolic
{2,3} {3,3} {4,3} {5,3} {6,3} {7,3} {8,3} {∞,3} {12i,3} {9i,3} {6i,3} {3i,3}
• Compounds of tetrahedra
• Two tetrahedra in a cube
• Compound of five tetrahedra
• Compound of ten tetrahedra
An interesting polyhedron can be constructed from five intersecting tetrahedra. This compound of five tetrahedra has been known for hundreds of years. It comes up regularly in the world of origami. Joining the twenty vertices would form a regular dodecahedron. There are both left-handed and right-handed forms, which are mirror images of each other. Superimposing both forms gives a compound of ten tetrahedra, in which the ten tetrahedra are arranged as five pairs of stellae octangulae. A stella octangula is a compound of two tetrahedra in dual position and its eight vertices define a cube as their convex hull.
The square hosohedron is another polyhedron with four faces, but it does not have triangular faces.
The Szilassi polyhedron and the tetrahedron are the only two known polyhedra in which each face shares an edge with each other face. Furthermore, the Császár polyhedron (itself is the dual of Szilassi polyhedron) and the tetrahedron are the only two known polyhedra in which every diagonal lies on the sides.
Applications
Numerical analysis
In numerical analysis, complicated three-dimensional shapes are commonly broken down into, or approximated by, a polygonal mesh of irregular tetrahedra in the process of setting up the equations for finite element analysis especially in the numerical solution of partial differential equations. These methods have wide applications in practical applications in computational fluid dynamics, aerodynamics, electromagnetic fields, civil engineering, chemical engineering, naval architecture and engineering, and related fields.
Structural engineering
A tetrahedron having stiff edges is inherently rigid. For this reason it is often used to stiffen frame structures such as spaceframes.
Aviation
At some airfields, a large frame in the shape of a tetrahedron with two sides covered with a thin material is mounted on a rotating pivot and always points into the wind. It is built big enough to be seen from the air and is sometimes illuminated. Its purpose is to serve as a reference to pilots indicating wind direction.[35]
Chemistry
The tetrahedron shape is seen in nature in covalently bonded molecules. All sp3-hybridized atoms are surrounded by atoms (or lone electron pairs) at the four corners of a tetrahedron. For instance in a methane molecule (CH
4
) or an ammonium ion (NH+
4
), four hydrogen atoms surround a central carbon or nitrogen atom with tetrahedral symmetry. For this reason, one of the leading journals in organic chemistry is called Tetrahedron. The central angle between any two vertices of a perfect tetrahedron is arccos(−1/3), or approximately 109.47°.[5]
Water, H
2
O
, also has a tetrahedral structure, with two hydrogen atoms and two lone pairs of electrons around the central oxygen atoms. Its tetrahedral symmetry is not perfect, however, because the lone pairs repel more than the single O–H bonds.
Quaternary phase diagrams of mixtures of chemical substances are represented graphically as tetrahedra.
However, quaternary phase diagrams in communication engineering are represented graphically on a two-dimensional plane.
Electricity and electronics
If six equal resistors are soldered together to form a tetrahedron, then the resistance measured between any two vertices is half that of one resistor.[36][37]
Since silicon is the most common semiconductor used in solid-state electronics, and silicon has a valence of four, the tetrahedral shape of the four chemical bonds in silicon is a strong influence on how crystals of silicon form and what shapes they assume.
Color space
Tetrahedra are used in color space conversion algorithms specifically for cases in which the luminance axis diagonally segments the color space (e.g. RGB, CMY).[38]
Games
The Royal Game of Ur, dating from 2600 BC, was played with a set of tetrahedral dice.
Especially in roleplaying, this solid is known as a 4-sided die, one of the more common polyhedral dice, with the number rolled appearing around the bottom or on the top vertex. Some Rubik's Cube-like puzzles are tetrahedral, such as the Pyraminx and Pyramorphix.
Geology
The tetrahedral hypothesis, originally published by William Lowthian Green to explain the formation of the Earth,[39] was popular through the early 20th century.[40][41]
Popular culture
Stanley Kubrick originally intended the monolith in 2001: A Space Odyssey to be a tetrahedron, according to Marvin Minsky, a cognitive scientist and expert on artificial intelligence who advised Kubrick on the HAL 9000 computer and other aspects of the movie. Kubrick scrapped the idea of using the tetrahedron as a visitor who saw footage of it did not recognize what it was and he did not want anything in the movie regular people did not understand.[42]
Tetrahedral graph
Tetrahedral graph
Vertices4
Edges6
Radius1
Diameter1
Girth3
Automorphisms24
Chromatic number4
PropertiesHamiltonian, regular, symmetric, distance-regular, distance-transitive, 3-vertex-connected, planar graph
Table of graphs and parameters
The skeleton of the tetrahedron (comprising the vertices and edges) forms a graph, with 4 vertices, and 6 edges. It is a special case of the complete graph, K4, and wheel graph, W4.[43] It is one of 5 Platonic graphs, each a skeleton of its Platonic solid.
3-fold symmetry
See also
• Boerdijk–Coxeter helix
• Möbius configuration
• Caltrop
• Demihypercube and simplex – n-dimensional analogues
• Pentachoron – 4-dimensional analogue
• Synergetics (Fuller)
• Tetrahedral kite
• Tetrahedral number
• Tetrahedron packing
• Triangular dipyramid – constructed by joining two tetrahedra along one face
• Trirectangular tetrahedron
• Orthoscheme
Notes
1. A 3-orthoscheme is not a disphenoid, because its opposite edges are not of equal length. It is not possible to construct a disphenoid with right triangle or obtuse triangle faces.
2. For a regular k-polytope, the Coxeter-Dynkin diagram of the characteristic k-orthoscheme is the k-polytope's diagram without the generating point ring. The regular k-polytope is subdivided by its symmetry (k-1)-elements into g instances of its characteristic k-orthoscheme that surround its center, where g is the order of the k-polytope's symmetry group.[11]
3. A regular polytope of dimension k has a characteristic k-orthoscheme, and also a characteristic (k-1)-orthoscheme. A regular polyhedron has a characteristic tetrahedron (3-orthoscheme) into which it is subdivided by its planes of symmetry, and also a characteristic triangle (2-orthoscheme) into which its surface is subdivided by its faces' lines of symmetry. After subdividing its surface into characteristic right triangles surrounding each face center, its interior can be subdivided into characteristic tetrahedra by adding radii joining the vertices of the surface right triangles to the polyhedron's center.[12] The interior triangles thus formed will also be right triangles.
4. The 24 characteristic tetrahedra of the regular tetrahedron occur in two mirror-image forms, 12 of each.
5. The characteristic orthoscheme of the cube is one of the Hill tetrahedra, a family of space-filling tetrahedra. All space-filling tetrahedra are scissors-congruent to a cube. Every convex polyhedron is scissors-congruent to an orthoscheme. Every regular convex polyhedron (Platonic solid) can be dissected into some even number of instances of its characteristic orthoscheme.
6. The tetrahedral-octahedral honeycomb fills space with alternating regular tetrahedron cells and regular octahedron cells in a ratio of 2:1.
7. The Coxeter-Dynkin diagram of the generated polyhedron contains three nodes representing the three mirrors. The dihedral angle between each pair of mirrors is encoded in the diagram, as well as the location of a single generating point which is multiplied by mirror reflections into the vertices of the polyhedron. For a regular polyhedron, the Coxeter-Dynkin diagram of the generating characteristic orthoscheme is the generated polyhedron's diagram without the generating point marking.
References
1. Weisstein, Eric W. "Tetrahedron". MathWorld.
2. Ford, Walter Burton; Ammerman, Charles (1913), Plane and Solid Geometry, Macmillan, pp. 294–295
3. Coxeter, Harold Scott MacDonald; Regular Polytopes, Methuen and Co., 1948, Table I(i)
4. Köller, Jürgen, "Tetrahedron", Mathematische Basteleien, 2001
5. Brittin, W. E. (1945). "Valence angle of the tetrahedral carbon atom". Journal of Chemical Education. 22 (3): 145. Bibcode:1945JChEd..22..145B. doi:10.1021/ed022p145.
6. Park, Poo-Sung. "Regular polytope distances", Forum Geometricorum 16, 2016, 227–232. http://forumgeom.fau.edu/FG2016volume16/FG201627.pdf
7. "Sections of a Tetrahedron".
8. Kepler 1619, p. 181.
9. Coxeter, H.S.M. (1989). "Trisecting an Orthoscheme". Computers Math. Applic. 17 (1–3): 59–71. doi:10.1016/0898-1221(89)90148-X.
10. Coxeter 1973, pp. 71–72, §4.7 Characteristic tetrahedra.
11. Coxeter 1973, pp. 130–133, §7.6 The symmetry group of the general regular polytope.
12. Coxeter 1973, p. 130, §7.6; "simplicial subdivision".
13. Coxeter 1973, pp. 292–293, Table I(i); "Tetrahedron, 𝛼3".
14. Coxeter 1973, pp. 33–34, §3.1 Congruent transformations.
15. Coxeter 1973, p. 63, §4.3 Rotation groups in two dimensions; notion of a fundamental region.
16. "Simplex Volumes and the Cayley-Menger Determinant", MathPages.com
17. Kahan, William M. (3 April 2012), What has the Volume of a Tetrahedron to do with Computer Programming Languages? (PDF), p. 11
18. Kahan, William M. (3 April 2012), What has the Volume of a Tetrahedron to do with Computer Programming Languages? (PDF), pp. 16–17
19. Bottema, O. "A Theorem of Bobillier on the Tetrahedron." Elemente der Mathematik 24 (1969): 6–10.
20. Murakami, Jun; Yano, Masakazu (2005), "On the volume of a hyperbolic and spherical tetrahedron", Communications in Analysis and Geometry, 13 (2): 379–400, doi:10.4310/cag.2005.v13.n2.a5, ISSN 1019-8385, MR 2154824
21. Havlicek, Hans; Weiß, Gunter (2003). "Altitudes of a tetrahedron and traceless quadratic forms" (PDF). American Mathematical Monthly. 110 (8): 679–693. arXiv:1304.0179. doi:10.2307/3647851. JSTOR 3647851.
22. Leung, Kam-tim; and Suen, Suk-nam; "Vectors, matrices and geometry", Hong Kong University Press, 1994, pp. 53–54
23. Outudee, Somluck; New, Stephen. The Various Kinds of Centres of Simplices (PDF). Dept of Mathematics, Chulalongkorn University, Bangkok. Archived from the original on 27 February 2009.{{cite book}}: CS1 maint: bot: original URL status unknown (link)
24. Audet, Daniel (May 2011). "Déterminants sphérique et hyperbolique de Cayley-Menger" (PDF). Bulletin AMQ.
25. Lindelof, L. (1867). "Sur les maxima et minima d'une fonction des rayons vecteurs menés d'un point mobile à plusieurs centres fixes". Acta Societatis Scientiarum Fennicae. 8 (Part 1): 189–203.
26. Senechal, Marjorie (1981). "Which tetrahedra fill space?". Mathematics Magazine. Mathematical Association of America. 54 (5): 227–243. doi:10.2307/2689983. JSTOR 2689983.
27. Rassat, André; Fowler, Patrick W. (2004). "Is There a "Most Chiral Tetrahedron"?". Chemistry: A European Journal. 10 (24): 6575–6580. doi:10.1002/chem.200400869. PMID 15558830.
28. Lee, Jung Rye (June 1997). "The Law of Cosines in a Tetrahedron". J. Korea Soc. Math. Educ. Ser. B: Pure Appl. Math.
29. Inequalities proposed in “Crux Mathematicorum”, .
30. Crelle, A. L. (1821). "Einige Bemerkungen über die dreiseitige Pyramide". Sammlung mathematischer Aufsätze u. Bemerkungen 1 (in German). Berlin: Maurer. pp. 105–132. Retrieved 7 August 2018.
31. Todhunter, I. (1886), Spherical Trigonometry: For the Use of Colleges and Schools, p. 129 ( Art. 163 )
32. Lévy, Bruno; Liu, Yang (2010). "Lp Centroidal Voronoi Tessellation and its applications". ACM: 119. {{cite journal}}: Cite journal requires |journal= (help)
33. "Problem 930" (PDF), Solutions, Crux Mathematicorum, 11 (5): 162–166, May 1985
34. Wacław Sierpiński, Pythagorean Triangles, Dover Publications, 2003 (orig. ed. 1962), p. 107. Note however that Sierpiński repeats an erroneous calculation of the volume of the Heronian tetrahedron example above.
35. Federal Aviation Administration (2009), Pilot's Handbook of Aeronautical Knowledge, U. S. Government Printing Office, p. 13-10, ISBN 9780160876110.
36. Klein, Douglas J. (2002). "Resistance-Distance Sum Rules" (PDF). Croatica Chemica Acta. 75 (2): 633–649. Archived from the original (PDF) on 10 June 2007. Retrieved 15 September 2006.
37. Záležák, Tomáš (18 October 2007); "Resistance of a regular tetrahedron" (PDF), retrieved 25 January 2011
38. Vondran, Gary L. (April 1998). "Radial and Pruned Tetrahedral Interpolation Techniques" (PDF). HP Technical Report. HPL-98-95: 1–32. Archived from the original (PDF) on 7 June 2011. Retrieved 11 November 2009.
39. Green, William Lowthian (1875). Vestiges of the Molten Globe, as exhibited in the figure of the earth, volcanic action and physiography. Vol. Part I. London: E. Stanford. Bibcode:1875vmge.book.....G. OCLC 3571917.
40. Holmes, Arthur (1965). Principles of physical geology. Nelson. p. 32. ISBN 9780177612992.
41. Hitchcock, Charles Henry (January 1900). Winchell, Newton Horace (ed.). "William Lowthian Green and his Theory of the Evolution of the Earth's Features". The American Geologist. Vol. XXV. Geological Publishing Company. pp. 1–10.
42. "Marvin Minsky: Stanley Kubrick Scraps the Tetrahedron". Web of Stories. Retrieved 20 February 2012.
43. Weisstein, Eric W. "Tetrahedral graph". MathWorld.
Bibliography
• Kepler, Johannes (1619). Harmonices Mundi (The Harmony of the World). Johann Planck.
• Coxeter, H.S.M. (1973). Regular Polytopes (3rd ed.). New York: Dover.
External links
Wikimedia Commons has media related to Tetrahedron.
• Weisstein, Eric W. "Tetrahedron". MathWorld.
• Free paper models of a tetrahedron and many other polyhedra
• An Amazing, Space Filling, Non-regular Tetrahedron that also includes a description of a "rotating ring of tetrahedra", also known as a kaleidocycle.
Polyhedra
Listed by number of faces and type
1–10 faces
• Monohedron
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11–20 faces
• Hendecahedron
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>20 faces
• Icositetrahedron (24)
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• Hexecontahedron (60)
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• Hectotriadiohedron (132)
• Apeirohedron (∞)
elemental things
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convex polyhedron
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non-convex polyhedron
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prismatoids
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Convex polyhedra
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(semiregular or uniform)
• truncated tetrahedron
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• 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.
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
Authority control: National
• Germany
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Wikipedia
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Twiddle factor
A twiddle factor, in fast Fourier transform (FFT) algorithms, is any of the trigonometric constant coefficients that are multiplied by the data in the course of the algorithm. This term was apparently coined by Gentleman & Sande in 1966, and has since become widespread in thousands of papers of the FFT literature.
More specifically, "twiddle factors" originally referred to the root-of-unity complex multiplicative constants in the butterfly operations of the Cooley–Tukey FFT algorithm, used to recursively combine smaller discrete Fourier transforms. This remains the term's most common meaning, but it may also be used for any data-independent multiplicative constant in an FFT.
The prime-factor FFT algorithm is one unusual case in which an FFT can be performed without twiddle factors, albeit only for restricted factorizations of the transform size.
For example, W82 is a twiddle factor used in 8-point radix-2 FFT.
References
• W. M. Gentleman and G. Sande, "Fast Fourier transforms—for fun and profit," Proc. AFIPS 29, 563–578 (1966). doi:10.1145/1464291.1464352
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Wikipedia
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Twin prime
A twin prime is a prime number that is either 2 less or 2 more than another prime number—for example, either member of the twin prime pair (17, 19) or (41, 43). In other words, a twin prime is a prime that has a prime gap of two. Sometimes the term twin prime is used for a pair of twin primes; an alternative name for this is prime twin or prime pair.
Twin primes become increasingly rare as one examines larger ranges, in keeping with the general tendency of gaps between adjacent primes to become larger as the numbers themselves get larger. However, it is unknown whether there are infinitely many twin primes (the so-called twin prime conjecture) or if there is a largest pair. The breakthrough[1] work of Yitang Zhang in 2013, as well as work by James Maynard, Terence Tao and others, has made substantial progress towards proving that there are infinitely many twin primes, but at present this remains unsolved.[2]
Unsolved problem in mathematics:
Are there infinitely many twin primes?
(more unsolved problems in mathematics)
Properties
Usually the pair (2, 3) is not considered to be a pair of twin primes.[3] Since 2 is the only even prime, this pair is the only pair of prime numbers that differ by one; thus twin primes are as closely spaced as possible for any other two primes.
The first several twin prime pairs are
(3, 5), (5, 7), (11, 13), (17, 19), (29, 31), (41, 43), (59, 61), (71, 73), (101, 103), (107, 109), (137, 139), ... OEIS: A077800.
Five is the only prime that belongs to two pairs, as every twin prime pair greater than (3, 5) is of the form $(6n-1,6n+1)$ for some natural number n; that is, the number between the two primes is a multiple of 6.[4] As a result, the sum of any pair of twin primes (other than 3 and 5) is divisible by 12.
Brun's theorem
Main article: Brun's theorem
In 1915, Viggo Brun showed that the sum of reciprocals of the twin primes was convergent.[5] This famous result, called Brun's theorem, was the first use of the Brun sieve and helped initiate the development of modern sieve theory. The modern version of Brun's argument can be used to show that the number of twin primes less than N does not exceed
${\frac {CN}{(\log N)^{2}}}$
for some absolute constant C > 0.[6] In fact, it is bounded above by
${\frac {8C_{2}N}{(\log N)^{2}}}\left[1+\operatorname {\mathcal {O}} \left({\frac {\log \log N}{\log N}}\right)\right],$
where $C_{2}$ is the twin prime constant (slightly less than 2/3), given below.[7]
Twin prime conjecture
The question of whether there exist infinitely many twin primes has been one of the great open questions in number theory for many years. This is the content of the twin prime conjecture, which states that there are infinitely many primes p such that p + 2 is also prime. In 1849, de Polignac made the more general conjecture that for every natural number k, there are infinitely many primes p such that p + 2k is also prime.[8] The case k = 1 of de Polignac's conjecture is the twin prime conjecture.
A stronger form of the twin prime conjecture, the Hardy–Littlewood conjecture (see below), postulates a distribution law for twin primes akin to the prime number theorem.
On 17 April 2013, Yitang Zhang announced a proof that for some integer N that is less than 70 million, there are infinitely many pairs of primes that differ by N.[9] Zhang's paper was accepted in early May 2013.[10] Terence Tao subsequently proposed a Polymath Project collaborative effort to optimize Zhang's bound.[11]
As of 14 April 2014, one year after Zhang's announcement, the bound has been reduced to 246.[12] These improved bounds were discovered using a different approach that was simpler than Zhang's and was discovered independently by James Maynard and Terence Tao. This second approach also gave bounds for the smallest f (m) needed to guarantee that infinitely many intervals of width f (m) contain at least m primes. Moreover (see also the next section) assuming the Elliott–Halberstam conjecture and its generalized form, the Polymath project wiki states that the bound is 12 and 6, respectively.[12]
A strengthening of Goldbach’s conjecture, if proved, would also prove there is an infinite number of twin primes, as would the existence of Siegel zeroes.
Other theorems weaker than the twin prime conjecture
In 1940, Paul Erdős showed that there is a constant c < 1 and infinitely many primes p such that p′ − p < c ln p where p′ denotes the next prime after p. What this means is that we can find infinitely many intervals that contain two primes (p, p′) as long as we let these intervals grow slowly in size as we move to bigger and bigger primes. Here, "grow slowly" means that the length of these intervals can grow logarithmically. This result was successively improved; in 1986 Helmut Maier showed that a constant c < 0.25 can be used. In 2004 Daniel Goldston and Cem Yıldırım showed that the constant could be improved further to c = 0.085786... . In 2005, Goldston, Pintz, and Yıldırım established that c can be chosen to be arbitrarily small,[13][14] i.e.
$\liminf _{n\to \infty }\left({\frac {p_{n+1}-p_{n}}{\log p_{n}}}\right)=0~.$
On the other hand, this result does not rule out that there may not be infinitely many intervals that contain two primes if we only allow the intervals to grow in size as, for example, c ln ln p .
By assuming the Elliott–Halberstam conjecture or a slightly weaker version, they were able to show that there are infinitely many n such that at least two of n, n + 2, n + 6, n + 8, n + 12, n + 18, or n + 20 are prime. Under a stronger hypothesis they showed that for infinitely many n, at least two of n, n + 2, n + 4, and n + 6 are prime.
The result of Yitang Zhang,
$\liminf _{n\to \infty }(p_{n+1}-p_{n})<N~\mathrm {with} ~N=7\times 10^{7},$
is a major improvement on the Goldston–Graham–Pintz–Yıldırım result. The Polymath Project optimization of Zhang's bound and the work of Maynard have reduced the bound: the limit inferior is at most 246.[15][16]
Conjectures
First Hardy–Littlewood conjecture
The Hardy–Littlewood conjecture (named after G. H. Hardy and John Littlewood) is a generalization of the twin prime conjecture. It is concerned with the distribution of prime constellations, including twin primes, in analogy to the prime number theorem. Let $\pi _{2}(x)$ denote the number of primes p ≤ x such that p + 2 is also prime. Define the twin prime constant C2 as[17]
$C_{2}=\prod _ {p\;\mathrm {prime,} \atop p\geq 3}}\left(1-{\frac {1}{(p-1)^{2}}}\right)\approx 0.660161815846869573927812110014\ldots .$
(Here the product extends over all prime numbers p ≥ 3.) Then a special case of the first Hardy-Littlewood conjecture is that
$\pi _{2}(x)\sim 2C_{2}{\frac {x}{(\ln x)^{2}}}\sim 2C_{2}\int _{2}^{x}{\mathrm {d} t \over (\ln t)^{2}}$
in the sense that the quotient of the two expressions tends to 1 as x approaches infinity.[6] (The second ~ is not part of the conjecture and is proven by integration by parts.)
The conjecture can be justified (but not proven) by assuming that ${\tfrac {1}{\ln t}}$ describes the density function of the prime distribution. This assumption, which is suggested by the prime number theorem, implies the twin prime conjecture, as shown in the formula for $\pi _{2}(x)$ above.
The fully general first Hardy–Littlewood conjecture on prime k-tuples (not given here) implies that the second Hardy–Littlewood conjecture is false.
This conjecture has been extended by Dickson's conjecture.
Polignac's conjecture
Polignac's conjecture from 1849 states that for every positive even integer k, there are infinitely many consecutive prime pairs p and p′ such that p′ − p = k (i.e. there are infinitely many prime gaps of size k). The case k = 2 is the twin prime conjecture. The conjecture has not yet been proven or disproven for any specific value of k, but Zhang's result proves that it is true for at least one (currently unknown) value of k. Indeed, if such a k did not exist, then for any positive even natural number N there are at most finitely many n such that $p_{n+1}-p_{n}=m$ for all m < N and so for n large enough we have $p_{n+1}-p_{n}>N,$ which would contradict Zhang's result.[8]
Large twin primes
Beginning in 2007, two distributed computing projects, Twin Prime Search and PrimeGrid, have produced several record-largest twin primes. As of August 2022, the current largest twin prime pair known is 2996863034895 × 21290000 ± 1 ,[18] with 388,342 decimal digits. It was discovered in September 2016.[19]
There are 808,675,888,577,436 twin prime pairs below 1018.[20][21]
An empirical analysis of all prime pairs up to 4.35 × 1015 shows that if the number of such pairs less than x is f (x) ·x /(log x)2 then f (x) is about 1.7 for small x and decreases towards about 1.3 as x tends to infinity. The limiting value of f (x) is conjectured to equal twice the twin prime constant (OEIS: A114907) (not to be confused with Brun's constant), according to the Hardy–Littlewood conjecture.
Other elementary properties
Every third odd number is divisible by 3, and therefore no three successive odd numbers can be prime unless one of them is 3. Five is therefore the only prime that is part of two twin prime pairs. The lower member of a pair is by definition a Chen prime.
It has been proven that the pair (m, m + 2) is a twin prime if and only if
$4((m-1)!+1)\equiv -m{\pmod {m(m+2)}}.$
If m − 4 or m + 6 is also prime then the three primes are called a prime triplet.
For a twin prime pair of the form (6n − 1, 6n + 1) for some natural number n > 1, n must end in the digit 0, 2, 3, 5, 7, or 8 (OEIS: A002822).
Isolated prime
An isolated prime (also known as single prime or non-twin prime) is a prime number p such that neither p − 2 nor p + 2 is prime. In other words, p is not part of a twin prime pair. For example, 23 is an isolated prime, since 21 and 25 are both composite.
The first few isolated primes are
2, 23, 37, 47, 53, 67, 79, 83, 89, 97, ... OEIS: A007510.
It follows from Brun's theorem that almost all primes are isolated in the sense that the ratio of the number of isolated primes less than a given threshold n and the number of all primes less than n tends to 1 as n tends to infinity.
See also
• Cousin prime
• Prime gap
• Prime k-tuple
• Prime quadruplet
• Prime triplet
• Sexy prime
References
1. Thomas, Kelly Devine (Summer 2014). "Yitang Zhang's spectacular mathematical journey". The Institute Letter. Princeton, NJ: Institute for Advanced Study – via ias.edu.
2. Tao, Terry, Ph.D. (presenter) (7 October 2014). Small and large gaps between the primes (video lecture). UCLA Department of Mathematics – via YouTube.
3. "The first 100,000 twin primes (only first member of pair)" (plain text). Lists. The Prime Pages (primes.utm.edu). Martin, TN: U.T. Martin.
4. Caldwell, Chris K. "Are all primes (past 2 and 3) of the forms 6n+1 and 6n−1?". The Prime Pages (primes.utm.edu). Martin, TN: U.T. Martin. Retrieved 2018-09-27.
5. Brun, V. (1915). "Über das Goldbachsche Gesetz und die Anzahl der Primzahlpaare" [On Goldbach's rule and the number of prime number pairs]. Archiv for Mathematik og Naturvidenskab (in German). 34 (8): 3–19. ISSN 0365-4524. JFM 45.0330.16.
6. Bateman, Paul T.; Diamond, Harold G. (2004). Analytic Number Theory. World Scientific. pp. 313 and 334–335. ISBN 981-256-080-7. Zbl 1074.11001.
7. Halberstam, Heini; Richert, Hans-Egon (2010). Sieve Methods. Dover Publications. p. 117.
8. de Polignac, A. (1849). "Recherches nouvelles sur les nombres premiers" [New research on prime numbers]. Comptes rendus (in French). 29: 397–401. [From p. 400] "1er Théorème. Tout nombre pair est égal à la différence de deux nombres premiers consécutifs d'une infinité de manières ..." (1st Theorem. Every even number is equal to the difference of two consecutive prime numbers in an infinite number of ways ...)
9. McKee, Maggie (14 May 2013). "First proof that infinitely many prime numbers come in pairs". Nature. doi:10.1038/nature.2013.12989. ISSN 0028-0836.
10. Zhang, Yitang (2014). "Bounded gaps between primes". Annals of Mathematics. 179 (3): 1121–1174. doi:10.4007/annals.2014.179.3.7. MR 3171761.
11. Tao, Terence (4 June 2013). "Polymath proposal: Bounded gaps between primes".
12. "Bounded gaps between primes". Polymath (michaelnielsen.org). Retrieved 2014-03-27.
13. Goldston, Daniel Alan; Motohashi, Yoichi; Pintz, János; Yıldırım, Cem Yalçın (2006). "Small gaps between primes exist". Japan Academy. Proceedings. Series A. Mathematical Sciences. 82 (4): 61–65. arXiv:math.NT/0505300. doi:10.3792/pjaa.82.61. MR 2222213.
14. Goldston, D.A.; Graham, S.W.; Pintz, J.; Yıldırım, C.Y. (2009). "Small gaps between primes or almost primes". Transactions of the American Mathematical Society. 361 (10): 5285–5330. arXiv:math.NT/0506067. doi:10.1090/S0002-9947-09-04788-6. MR 2515812.
15. Maynard, James (2015). "Small gaps between primes". Annals of Mathematics. Second Series. 181 (1): 383–413. arXiv:1311.4600. doi:10.4007/annals.2015.181.1.7. MR 3272929.
16. Polymath, D.H.J. (2014). "Variants of the Selberg sieve, and bounded intervals containing many primes". Research in the Mathematical Sciences. 1. artc. 12, 83. arXiv:1407.4897. doi:10.1186/s40687-014-0012-7. MR 3373710.
17. Sloane, N. J. A. (ed.). "Sequence A005597 (Decimal expansion of the twin prime constant)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2019-11-01.
18. Caldwell, Chris K. " 2996863034895 × 21290000 − 1 ". The Prime Database. Martin, TN: UT Martin.
19. "World record twin primes found!". primegrid.com. 20 September 2016.
20. Sloane, N. J. A. (ed.). "Sequence A007508 (Number of twin prime pairs below 10n)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2019-11-01.
21. Oliveira e Silva, Tomás (7 April 2008). "Tables of values of π(x) and of π2(x)". Aveiro University. Retrieved 7 January 2011.
Further reading
• Sloane, Neil; Plouffe, Simon (1995). The Encyclopedia of Integer Sequences. San Diego, CA: Academic Press. ISBN 0-12-558630-2.
External links
• "Twins", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Top-20 Twin Primes at Chris Caldwell's Prime Pages
• Xavier Gourdon, Pascal Sebah: Introduction to Twin Primes and Brun's Constant
• "Official press release" of 58711-digit twin prime record
• Weisstein, Eric W. "Twin Primes". MathWorld.
• The 20 000 first twin primes
• Polymath: Bounded gaps between primes
• Sudden Progress on Prime Number Problem Has Mathematicians Buzzing
Prime number classes
By formula
• Fermat (22n + 1)
• Mersenne (2p − 1)
• Double Mersenne (22p−1 − 1)
• Wagstaff (2p + 1)/3
• Proth (k·2n + 1)
• Factorial (n! ± 1)
• Primorial (pn# ± 1)
• Euclid (pn# + 1)
• Pythagorean (4n + 1)
• Pierpont (2m·3n + 1)
• Quartan (x4 + y4)
• Solinas (2m ± 2n ± 1)
• Cullen (n·2n + 1)
• Woodall (n·2n − 1)
• Cuban (x3 − y3)/(x − y)
• Leyland (xy + yx)
• Thabit (3·2n − 1)
• Williams ((b−1)·bn − 1)
• Mills (⌊A3n⌋)
By integer sequence
• Fibonacci
• Lucas
• Pell
• Newman–Shanks–Williams
• Perrin
• Partitions
• Bell
• Motzkin
By property
• Wieferich (pair)
• Wall–Sun–Sun
• Wolstenholme
• Wilson
• Lucky
• Fortunate
• Ramanujan
• Pillai
• Regular
• Strong
• Stern
• Supersingular (elliptic curve)
• Supersingular (moonshine theory)
• Good
• Super
• Higgs
• Highly cototient
• Unique
Base-dependent
• Palindromic
• Emirp
• Repunit (10n − 1)/9
• Permutable
• Circular
• Truncatable
• Minimal
• Delicate
• Primeval
• Full reptend
• Unique
• Happy
• Self
• Smarandache–Wellin
• Strobogrammatic
• Dihedral
• Tetradic
Patterns
• Twin (p, p + 2)
• Bi-twin chain (n ± 1, 2n ± 1, 4n ± 1, …)
• Triplet (p, p + 2 or p + 4, p + 6)
• Quadruplet (p, p + 2, p + 6, p + 8)
• k-tuple
• Cousin (p, p + 4)
• Sexy (p, p + 6)
• Chen
• Sophie Germain/Safe (p, 2p + 1)
• Cunningham (p, 2p ± 1, 4p ± 3, 8p ± 7, ...)
• Arithmetic progression (p + a·n, n = 0, 1, 2, 3, ...)
• Balanced (consecutive p − n, p, p + n)
By size
• Mega (1,000,000+ digits)
• Largest known
• list
Complex numbers
• Eisenstein prime
• Gaussian prime
Composite numbers
• Pseudoprime
• Catalan
• Elliptic
• Euler
• Euler–Jacobi
• Fermat
• Frobenius
• Lucas
• Somer–Lucas
• Strong
• Carmichael number
• Almost prime
• Semiprime
• Sphenic number
• Interprime
• Pernicious
Related topics
• Probable prime
• Industrial-grade prime
• Illegal prime
• Formula for primes
• Prime gap
First 60 primes
• 2
• 3
• 5
• 7
• 11
• 13
• 17
• 19
• 23
• 29
• 31
• 37
• 41
• 43
• 47
• 53
• 59
• 61
• 67
• 71
• 73
• 79
• 83
• 89
• 97
• 101
• 103
• 107
• 109
• 113
• 127
• 131
• 137
• 139
• 149
• 151
• 157
• 163
• 167
• 173
• 179
• 181
• 191
• 193
• 197
• 199
• 211
• 223
• 227
• 229
• 233
• 239
• 241
• 251
• 257
• 263
• 269
• 271
• 277
• 281
List of prime numbers
Prime number conjectures
• Hardy–Littlewood
• 1st
• 2nd
• Agoh–Giuga
• Andrica's
• Artin's
• Bateman–Horn
• Brocard's
• Bunyakovsky
• Chinese hypothesis
• Cramér's
• Dickson's
• Elliott–Halberstam
• Firoozbakht's
• Gilbreath's
• Grimm's
• Landau's problems
• Goldbach's
• weak
• Legendre's
• Twin prime
• Legendre's constant
• Lemoine's
• Mersenne
• Oppermann's
• Polignac's
• Pólya
• Schinzel's hypothesis H
• Waring's prime number
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Wikipedia
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Twin-width
The twin-width of an undirected graph is a natural number associated with the graph, used to study the parameterized complexity of graph algorithms. Intuitively, it measures how similar the graph is to a cograph, a type of graph that can be reduced to a single vertex by repeatedly merging together twins, vertices that have the same neighbors. The twin-width is defined from a sequence of repeated mergers where the vertices are not required to be twins, but have nearly equal sets of neighbors.
Definition
Twin-width is defined for finite simple undirected graphs. These have a finite set of vertices, and a set of edges that are unordered pairs of vertices. The open neighborhood of any vertex is the set of other vertices that it is paired with in edges of the graph; the closed neighborhood is formed from the open neighborhood by including the vertex itself. Two vertices are true twins when they have the same closed neighborhood, and false twins when they have the same open neighborhood; more generally, both true twins and false twins can be called twins, without qualification.[1]
The cographs have many equivalent definitions,[2] but one of them is that these are the graphs that can be reduced to a single vertex by a process of repeatedly finding any two twin vertices and merging them into a single vertex. For a cograph, this reduction process will always succeed, no matter which choice of twins to merge is made at each step. For a graph that is not a cograph, it will always get stuck in a subgraph with more than two vertices that has no twins.[1]
The definition of twin-width mimics this reduction process. A contraction sequence, in this context, is a sequence of steps, beginning with the given graph, in which each step replaces a pair of vertices by a single vertex. This produces a sequence of graphs, with edges colored red and black; in the given graph, all edges are assumed to be black. When two vertices are replaced by a single vertex, the neighborhood of the new vertex is the union of the neighborhoods of the replaced vertices. In this new neighborhood, an edge that comes from black edges in the neighborhoods of both vertices remains black; all other edges are colored red.[1]
A contraction sequence is called a $d$-sequence if, throughout the sequence, every vertex touches at most $d$ red edges. The twin-width of a graph is the smallest value of $d$ for which it has a $d$-sequence.[1]
A dense graph may still have bounded twin-width; for instance, the cographs include all complete graphs. A variation of twin-width, sparse twin-width, applies to families of graphs rather than to individual graphs. For a family of graphs that is closed under taking induced subgraphs and has bounded twin-width, the following properties are equivalent:[3]
• The graphs in the family are sparse, meaning that they have a number of edges bounded by a linear function of their number of vertices.
• The family does not include all complete bipartite graphs.
• The family of all subgraphs of graphs in the given family has bounded twin-width.
• The family has bounded expansion, meaning that all its shallow minors are sparse.
Such a family is said to have bounded sparse twin-width.[3]
The concept of twin-width can be generalized from graphs to various totally ordered structures (including graphs equipped with a total ordering on their vertices), and is in many ways simpler for ordered structures than for unordered graphs.[4] It is also possible to formulate equivalent definitions for other notions of graph width using contraction sequences with different requirements than having bounded degree.[5]
Graphs of bounded twin-width
Cographs have twin-width zero. In the reduction process for cographs, there will be no red edges: when two vertices are merged, their neighborhoods are equal, so there are no edges coming from only one of the two neighborhoods to be colored red. In any other graph, any contraction sequence will produce some red edges, and the twin-width will be greater than zero.[1]
The path graphs with at most three vertices are cographs, but every larger path graph has twin-width one. For a contraction sequence that repeatedly merges the last two edges of the path, only the edge incident to the single merged vertex will be red, so this is a 1-sequence. Trees have twin-width at most two, and for some trees this is tight. A 2-contraction sequence for any tree may be found by choosing a root, and then repeatedly merging two leaves that have the same parent or, if this is not possible, merging the deepest leaf into its parent. The only red edges connect leaves to their parents, and when there are two at the same parent they can be merged, keeping the red degree at most two.[1]
More generally, the following classes of graphs have bounded twin-width, and a contraction sequence of bounded width can be found for them in polynomial time:
• Every graph of bounded clique-width, or of bounded rank-width, also has bounded twin-width. The twin-width is at most exponential in the clique-width, and at most doubly exponential in the rank-width.[1] These graphs include, for instance, the distance-hereditary graphs, the k-leaf powers for bounded values of k, and the graphs of bounded treewidth.
• Indifference graphs (equivalently, unit interval graphs or proper interval graphs) have twin-width at most two.[6]
• Unit disk graphs defined from sets of unit disks that cover each point of the plane a bounded number of times have bounded twin-width. The same is true for unit ball graphs in higher dimensions.[1]
• The permutation graphs coming from permutations with a forbidden permutation pattern have bounded twin-width. This allows twin-width to be applied to algorithmic problems on permutations with forbidden patterns.[1]
• Every family of graphs defined by forbidden minors has bounded twin-width. For instance, by Wagner's theorem, the forbidden minors for planar graphs are the two graphs $K_{5}$ and $K_{3,3}$, so the planar graphs have bounded twin-width.[1]
• Every graph of bounded stack number or bounded queue number also has bounded twin-width.[3] There exist families of graphs of bounded sparse twin-width that do not have bounded stack number, but the corresponding question for queue number remains open.[7]
• The strong product of any two graphs of bounded twin-width, one of which has bounded degree, again has bounded twin-width. This can be used to prove the bounded twin-width of classes of graphs that have decompositions into strong products of paths and bounded-treewidth graphs, such as the k-planar graphs.[3] For the lexicographic product of graphs, the twin-width is exactly the maximum of the widths of the two factor graphs.[8] Twin-width also behaves well under several other standard graph products, but not the modular product of graphs.[9]
In every hereditary family of graphs of bounded twin-width, it is possible to find a family of total orders for the vertices of its graphs so that the inherited ordering on an induced subgraph is also an ordering in the family, and so that the family is small with respect to these orders. This means that, for a total order on $n$ vertices, the number of graphs in the family consistent with that order is at most singly exponential in $n$. Conversely, every hereditary family of ordered graphs that is small in this sense has bounded twin-width.[4] It was originally conjectured that every hereditary family of labeled graphs that is small, in the sense that the number of graphs is at most a singly exponential factor times $n!$, has bounded twin-width.[3] However, this conjecture was disproved using a family of induced subgraphs of an infinite Cayley graph that are small as labeled graphs but do not have bounded twin-width.[10]
There exist graphs of unbounded twin-width within the following families of graphs:
• Graphs of bounded degree.[1]
• Interval graphs.[1]
• Unit disk graphs.[1]
In each of these cases, the result follows by a counting argument: there are more graphs of the given type than there can be graphs of bounded twin-width.[1]
Properties
If a graph has bounded twin-width, then it is possible to find a versatile tree of contractions. This is a large family of contraction sequences, all of some (larger) bounded width, so that at each step in each sequence there are linearly many disjoint pairs of vertices each of which could be contracted at the next step in the sequence. It follows from this that the number of graphs of bounded twin-width on any set of $n$ given vertices is larger than $n!$ by only a singly exponential factor, that the graphs of bounded twin-width have an adjacency labelling scheme with only a logarithmic number of bits per vertex, and that they have universal graphs of polynomial size in which each $n$-vertex graph of bounded twin-width can be found as an induced subgraph.[3]
Algorithms
The graphs of twin-width at most one can be recognized in polynomial time.[8] However, it is NP-complete to determine whether a given graph has twin-width at most four, and NP-hard to approximate the twin-width with an approximation ratio better than 5/4. Under the exponential time hypothesis, computing the twin-width requires time at least exponential in $n/\log n$, on $n$-vertex graphs.[11] In practice, it is possible to compute the twin-width of graphs of moderate size using SAT solvers.[12] For most of the known families of graphs of bounded twin-width, it is possible to construct a contraction sequence of bounded width in polynomial time.[1]
Once a contraction sequence has been given or constructed, many different algorithmic problems can be solved using it, in many cases more efficiently than is possible for graphs that do not have bounded twin-width. As detailed below, these include exact parameterized algorithms and approximation algorithms for NP-hard problems, as well as some problems that have classical polynomial time algorithms but can nevertheless be sped up using the assumption of bounded twin-width.
Parameterized algorithms
An algorithmic problem on graphs having an associated parameter is called fixed-parameter tractable if it has an algorithm that, on graphs with $n$ vertices and parameter value $k$, runs in time $O(n^{c}\,f(k))$ for some constant $c$ and computable function $f$. For instance, a running time of $O(n2^{k})$ would be fixed-parameter tractable in this sense. This style of analysis is generally applied to problems that do not have a known polynomial-time algorithm, because otherwise fixed-parameter tractability would be trivial. Many such problems have been shown to be fixed-parameter tractable with twin-width as a parameter, when a contraction sequence of bounded width is given as part of the input. This applies, in particular, to the graph families of bounded twin-width listed above, for which a contraction sequence can be constructed efficiently. However, it is not known how to find a good contraction sequence for an arbitrary graph of low twin-width, when no other structure in the graph is known.
The fixed-parameter tractable problems for graphs of bounded twin-width with given contraction sequences include:
• Testing whether the given graph models any given property in the first-order logic of graphs. Here, both the twin-width and the description length of the property are parameters of the analysis. Problems of this type include subgraph isomorphism for subgraphs of bounded size, and the vertex cover and dominating set problems for covers or dominating sets of bounded size.[1] The dependence of these general methods on the length of the logical formula describing the property is tetrational, but for independent set, dominating set, and related problems it can be reduced to exponential in the size of the independent or dominating set, and for subgraph isomorphism it can be reduced to factorial in the number of vertices of the subgraph. For instance, the time to find a $k$-vertex independent set, for an $n$-vertex graph with a given $d$-sequence, is $O(k^{2}d^{2k}n)$, by a dynamic programming algorithm that considers small connected subgraphs of the red graphs in the forward direction of the contraction sequence. These time bounds are optimal, up to logarithmic factors in the exponent, under the exponential time hypothesis.[6] For an extension of the first-order logic of graphs to graphs with totally ordered vertices, and logical predicates that can test this ordering, model checking is still fixed-parameter tractable for hereditary graph families of bounded twin-width, but not (under standard complexity-theoretic assumptions) for hereditary families of unbounded twin-width.[13]
• Coloring graphs of bounded twin-width, using a number of colors that is bounded by a function of their twin-width and of the size of their largest clique. For instance, triangle-free graphs of twin-width $d$ can be $(d+2)$-colored by a greedy coloring algorithm that colors vertices in the reverse of the order they were contracted away.[6] This result shows that the graphs of bounded twin-width are χ-bounded.[6][14] For graph families of bounded sparse twin-width, the generalized coloring numbers are bounded. Here, the generalized coloring number $\operatorname {col} _{r}(G)$ is at most $k$ if the vertices can be linearly ordered in such a way that each vertex can reach at most $k-1$ earlier vertices in the ordering, through paths of length $r$ through later vertices in the ordering.[15]
Speedups of classical algorithms
In graphs of bounded twin-width, it is possible to perform a breadth-first search, on a graph with $n$ vertices, in time $O(n\log n)$, even when the graph is dense and has more edges than this time bound.[6]
Approximation algorithms
Twin-width has also been applied in approximation algorithms. In particular, in the graphs of bounded twin-width, it is possible to find an approximation to the minimum dominating set with bounded approximation ratio. This is in contrast to more general graphs, for which it is NP-hard to obtain an approximation ratio that is better than logarithmic.[6]
The maximum independent set and graph coloring problems can be approximated to within an approximation ratio of $n^{\varepsilon }$, for every $\varepsilon >0$, in polynomial time on graphs of bounded twin-width. In contrast, without the assumption of bounded twin-width, it is NP-hard to achieve any approximation ratio of this form with $\varepsilon <1$.[16]
References
1. Bonnet, Édouard; Kim, Eun Jung; Thomassé, Stéphan; Watrigant, Rémi (2022), "Twin-width I: Tractable FO model checking", Journal of the ACM, 69 (1): A3:1–A3:46, arXiv:2004.14789, doi:10.1145/3486655, MR 4402362
2. "Cograph graphs", Information System on Graph Class Inclusions
3. Bonnet, Édouard; Geniet, Colin; Kim, Eun Jung; Thomassé, Stéphan; Watrigant, Rémi (2022), "Twin-width II: small classes", Combinatorial Theory, 2 (2): P10:1–P10:42, arXiv:2006.09877, doi:10.5070/C62257876, MR 4449818
4. Bonnet, Édouard; Giocanti, Ugo; de Mendez, Patrice Ossona; Simon, Pierre; Thomassé, Stéphan; Torunczyk, Szymon (2022), "Twin-width IV: ordered graphs and matrices", in Leonardi, Stefano; Gupta, Anupam (eds.), STOC '22: 54th Annual ACM SIGACT Symposium on Theory of Computing, Rome, Italy, June 20–24, 2022, Association for Computing Machinery, pp. 924–937, arXiv:2102.03117, doi:10.1145/3519935.3520037, S2CID 235743245
5. Bonnet, Édouard; Kim, Eun Jung; Reinald, Amadeus; Thomassé, Stéphan (2022), "Twin-width VI: the lens of contraction sequences", in Naor, Joseph (Seffi); Buchbinder, Niv (eds.), Proceedings of the 2022 ACM-SIAM Symposium on Discrete Algorithms, SODA 2022, Virtual Conference / Alexandria, VA, USA, January 9–12, 2022, Society for Industrial and Applied Mathematics, pp. 1036–1056, arXiv:2111.00282, doi:10.1137/1.9781611977073.45, S2CID 240354166
6. Bonnet, Édouard; Geniet, Colin; Kim, Eun Jung; Thomassé, Stéphan; Watrigant, Rémi (2021), "Twin-width III: Max independent set, min dominating set, and coloring", in Bansal, Nikhil; Merelli, Emanuela; Worrell, James (eds.), 48th International Colloquium on Automata, Languages, and Programming, ICALP 2021, July 12–16, 2021, Glasgow, Scotland (Virtual Conference), LIPIcs, vol. 198, Schloss Dagstuhl – Leibniz-Zentrum für Informatik, pp. 35:1–35:20, arXiv:2007.14161, doi:10.4230/LIPIcs.ICALP.2021.35, ISBN 9783959771955, S2CID 235736798
7. Dujmović, Vida; Eppstein, David; Hickingbotham, Robert; Morin, Pat; Wood, David R. (August 2021), "Stack-Number is not bounded by queue-number", Combinatorica, 42 (2): 151–164, arXiv:2011.04195, doi:10.1007/s00493-021-4585-7, S2CID 226281691
8. Bonnet, Édouard; Kim, Eun Jung; Reinald, Amadeus; Thomassé, Stéphan; Watrigant, Rémi (2022), "Twin-width and polynomial kernels", Algorithmica, 84 (11): 3300–3337, doi:10.1007/s00453-022-00965-5, MR 4500778
9. Pettersson, William; Sylvester, John (2022), "Bounds on the twin-width of product graphs", arXiv:2202.11556 [math.CO]
10. Bonnet, Édouard; Geniet, Colin; Tessera, Romain; Thomassé, Stéphan (2022), "Twin-width VII: groups", arXiv:2204.12330 [math.GR]
11. Bergé, Pierre; Bonnet, Édouard; Déprés, Hugues (2022), "Deciding twin-width at most 4 Is NP-complete", in Bojanczyk, Mikolaj; Merelli, Emanuela; Woodruff, David P. (eds.), 49th International Colloquium on Automata, Languages, and Programming, ICALP 2022, July 4–8, 2022, Paris, France, LIPIcs, vol. 229, Schloss Dagstuhl – Leibniz-Zentrum für Informatik, pp. 18:1–18:20, arXiv:2112.08953, doi:10.4230/LIPIcs.ICALP.2022.18, ISBN 9783959772358, S2CID 245218775
12. Schidler, André; Szeider, Stefan (2022), "A SAT approach to twin-width", in Phillips, Cynthia A.; Speckmann, Bettina (eds.), Proceedings of the Symposium on Algorithm Engineering and Experiments, ALENEX 2022, Alexandria, VA, USA, January 9–10, 2022, Society for Industrial and Applied Mathematics, pp. 67–77, arXiv:2110.06146, doi:10.1137/1.9781611977042.6, S2CID 238634418
13. Simon, Pierre; Toruńczyk, Szymon (2021), Ordered graphs of bounded twin-width, arXiv:2102.06881
14. Pilipczuk, Marek; Sokołowski, Michał (2023), "Graphs of bounded twin-width are quasi-polynomially $\chi $-bounded", Journal of Combinatorial Theory, Series B, 161: 382–406, arXiv:2202.07608, doi:10.1016/j.jctb.2023.02.006, MR 4568111
15. Dreier, Jan; Gajarský, Jakub; Jiang, Yiting; Ossona de Mendez, Patrice; Raymond, Jean-Florent (2022), "Twin-width and generalized coloring numbers", Discrete Mathematics, 345 (3), Paper 112746, arXiv:1602.09052, doi:10.1016/j.disc.2021.112746, MR 4349879, S2CID 233296212
16. Bergé, Pierre; Bonnet, Édouard; Déprés, Hugues; Watrigant, Rémi (2023), "Approximating highly inapproximable problems on graphs of bounded twin-width", in Berenbrink, Petra; Bouyer, Patricia; Dawar, Anuj; Kanté, Mamadou Moustapha (eds.), 40th International Symposium on Theoretical Aspects of Computer Science, STACS 2023, March 7-9, 2023, Hamburg, Germany, LIPIcs, vol. 254, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, pp. 10:1–10:15, arXiv:2207.07708, doi:10.4230/LIPIcs.STACS.2023.10
Further reading
• Ahn, Jungho; Hendrey, Kevin; Kim, Donggyu; Oum, Sang-Il (2022), "Bounds for the twin-width of graphs", SIAM Journal on Discrete Mathematics, 36 (3): 2352–2366, arXiv:2110.03957, doi:10.1137/21M1452834, MR 4487903
• Balabán, Jakub; Hlinený, Petr (2021), "Twin-width is linear in the poset width", in Golovach, Petr A.; Zehavi, Meirav (eds.), 16th International Symposium on Parameterized and Exact Computation, IPEC 2021, September 8–10, 2021, Lisbon, Portugal, LIPIcs, vol. 214, Schloss Dagstuhl – Leibniz-Zentrum für Informatik, pp. 6:1–6:13, arXiv:2106.15337, doi:10.4230/LIPIcs.IPEC.2021.6, ISBN 9783959772167, S2CID 235669802
• Balabán, Jakub; Hlinený, Petr; Jedelský, Jan (2022), "Twin-width and transductions of proper k-mixed-thin graphs", in Bekos, Michael A.; Kaufmann, Michael (eds.), Graph-Theoretic Concepts in Computer Science - 48th International Workshop, WG 2022, Tübingen, Germany, June 22–24, 2022, Revised Selected Papers, Lecture Notes in Computer Science, vol. 13453, Springer, pp. 43–55, arXiv:2202.12536, doi:10.1007/978-3-031-15914-5_4
• Bonnet, Édouard; Chakraborty, Dibyayan; Kim, Eun Jung; Köhler, Noleen; Lopes, Raul; Thomassé, Stéphan (2022), "Twin-Width VIII: Delineation and win-wins", in Dell, Holger; Nederlof, Jesper (eds.), 17th International Symposium on Parameterized and Exact Computation, IPEC 2022, September 7–9, 2022, Potsdam, Germany, LIPIcs, vol. 249, Schloss Dagstuhl – Leibniz-Zentrum für Informatik, pp. 9:1–9:18, arXiv:2204.00722, doi:10.4230/LIPIcs.IPEC.2022.9
• Bonnet, Édouard; Déprés, Hugues (2023), "Twin-width can be exponential in treewidth", Journal of Combinatorial Theory, Series B, 161: 1–14, arXiv:2204.07670, doi:10.1016/j.jctb.2023.01.003, MR 4543125
• Bonnet, Édouard; Giocanti, Ugo; de Mendez, Patrice Ossona; Thomassé, Stéphan (2023), "Twin-width V: linear minors, modular counting, and matrix multiplication", in Berenbrink, Petra; Bouyer, Patricia; Dawar, Anuj; Kanté, Mamadou Moustapha (eds.), 40th International Symposium on Theoretical Aspects of Computer Science, STACS 2023, March 7–9, 2023, Hamburg, Germany, LIPIcs, vol. 254, Schloss Dagstuhl – Leibniz-Zentrum für Informatik, pp. 15:1–15:16, doi:10.4230/LIPIcs.STACS.2023.15
• Bonnet, Édouard; Kwon, O-joung; Wood, David R. (2022), "Reduced bandwidth: a qualitative strengthening of twin-width in minor-closed classes (and beyond)", arXiv:2202.11858 [math.CO]
• Bonnet, Édouard; Nešetřil, Jaroslav; de Mendez, Patrice Ossona; Siebertz, Sebastian; Thomassé, Stéphan (2021), Twin-width and permutations, arXiv:2102.06880
• Gajarský, Jakub; Pilipczuk, Michal; Przybyszewski, Wojciech; Toruńczyk, Szymon (2022), "Twin-width and types", in Bojanczyk, Mikolaj; Merelli, Emanuela; Woodruff, David P. (eds.), 49th International Colloquium on Automata, Languages, and Programming, ICALP 2022, July 4–8, 2022, Paris, France, LIPIcs, vol. 229, Schloss Dagstuhl – Leibniz-Zentrum für Informatik, pp. 123:1–123:21, arXiv:2206.08248, doi:10.4230/LIPIcs.ICALP.2022.123, ISBN 9783959772358
• Gajarský, Jakub; Pilipczuk, Michal; Toruńczyk, Szymon (2022), "Stable graphs of bounded twin-width", in Baier, Christel; Fisman, Dana (eds.), LICS '22: 37th Annual ACM/IEEE Symposium on Logic in Computer Science, Haifa, Israel, August 2–5, 2022, Association for Computing Machinery, pp. 39:1–39:12, arXiv:2107.03711, doi:10.1145/3531130.3533356
• Geniet, Colin; Thomassé, Stéphan (2022), "First order logic and twin-width in tournaments and dense oriented graphs", arXiv:2207.07683 [cs.LO]
• Jacob, Hugo; Pilipczuk, Marcin (2022), "Bounding twin-width for bounded-treewidth graphs, planar graphs, and bipartite graphs", in Bekos, Michael A.; Kaufmann, Michael (eds.), Graph-Theoretic Concepts in Computer Science – 48th International Workshop, WG 2022, Tübingen, Germany, June 22–24, 2022, Revised Selected Papers, Lecture Notes in Computer Science, vol. 13453, Springer, pp. 287–299, arXiv:2201.09749, doi:10.1007/978-3-031-15914-5_21
• Pilipczuk, Michal; Sokolowski, Marek; Zych-Pawlewicz, Anna (2022), "Compact representation for matrices of bounded twin-width", in Berenbrink, Petra; Monmege, Benjamin (eds.), 39th International Symposium on Theoretical Aspects of Computer Science, STACS 2022, March 15–18, 2022, Marseille, France (Virtual Conference), LIPIcs, vol. 219, Schloss Dagstuhl – Leibniz-Zentrum für Informatik, pp. 52:1–52:14, arXiv:2110.08106, doi:10.4230/LIPIcs.STACS.2022.52, ISBN 9783959772228, S2CID 239009596
• Przybyszewski, Wojciech (2022), VC-density and abstract cell decomposition for edge relation in graphs of bounded twin-width, arXiv:2202.04006
• Thomassé, Stéphan (2022), "A brief tour in twin-width (invited talk)", in Bojanczyk, Mikolaj; Merelli, Emanuela; Woodruff, David P. (eds.), 49th International Colloquium on Automata, Languages, and Programming, ICALP 2022, July 4–8, 2022, Paris, France, LIPIcs, vol. 229, Schloss Dagstuhl – Leibniz-Zentrum für Informatik, pp. 6:1–6:5, doi:10.4230/LIPIcs.ICALP.2022.6, ISBN 9783959772358
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Wikipedia
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Dragon curve
A dragon curve is any member of a family of self-similar fractal curves, which can be approximated by recursive methods such as Lindenmayer systems. The dragon curve is probably most commonly thought of as the shape that is generated from repeatedly folding a strip of paper in half, although there are other curves that are called dragon curves that are generated differently.
"Dragon fractal" redirects here. For the filled Julia sets, see Douady rabbit.
Heighway dragon
The Heighway dragon (also known as the Harter–Heighway dragon or the Jurassic Park dragon) was first investigated by NASA physicists John Heighway, Bruce Banks, and William Harter. It was described by Martin Gardner in his Scientific American column Mathematical Games in 1967. Many of its properties were first published by Chandler Davis and Donald Knuth. It appeared on the section title pages of the Michael Crichton novel Jurassic Park.[1]
Construction
The Heighway dragon can be constructed from a base line segment by repeatedly replacing each segment by two segments with a right angle and with a rotation of 45° alternatively to the right and to the left:[2]
The Heighway dragon is also the limit set of the following iterated function system in the complex plane:
$f_{1}(z)={\frac {(1+i)z}{2}}$
$f_{2}(z)=1-{\frac {(1-i)z}{2}}$
with the initial set of points $S_{0}=\{0,1\}$.
Using pairs of real numbers instead, this is the same as the two functions consisting of
$f_{1}(x,y)={\frac {1}{\sqrt {2}}}{\begin{pmatrix}\cos 45^{\circ }&-\sin 45^{\circ }\\\sin 45^{\circ }&\cos 45^{\circ }\end{pmatrix}}{\begin{pmatrix}x\\y\end{pmatrix}}$
$f_{2}(x,y)={\frac {1}{\sqrt {2}}}{\begin{pmatrix}\cos 135^{\circ }&-\sin 135^{\circ }\\\sin 135^{\circ }&\cos 135^{\circ }\end{pmatrix}}{\begin{pmatrix}x\\y\end{pmatrix}}+{\begin{pmatrix}1\\0\end{pmatrix}}$
Folding the dragon
The Heighway dragon curve can be constructed by folding a strip of paper, which is how it was originally discovered.[1] Take a strip of paper and fold it in half to the right. Fold it in half again to the right. If the strip was opened out now, unbending each fold to become a 90-degree turn, the turn sequence would be RRL, i.e. the second iteration of the Heighway dragon. Fold the strip in half again to the right, and the turn sequence of the unfolded strip is now RRLRRLL – the third iteration of the Heighway dragon. Continuing folding the strip in half to the right to create further iterations of the Heighway dragon (in practice, the strip becomes too thick to fold sharply after four or five iterations).
The folding patterns of this sequence of paper strips, as sequences of right (R) and left (L) folds, are:
• 1st iteration: R
• 2nd iteration: R R L
• 3rd iteration: R R L R R L L
• 4th iteration: R R L R R L L R R R L L R L L.
Each iteration can be found by copying the previous iteration, then an R, then a second copy of the previous iteration in reverse order with the L and R letters swapped.[1]
Properties
• Many self-similarities can be seen in the Heighway dragon curve. The most obvious is the repetition of the same pattern tilted by 45° and with a reduction ratio of $\textstyle {\sqrt {2}}$. Based on these self-similarities, many of its lengths are simple rational numbers.
Lengths
Self-similarities
• The dragon curve can tile the plane. One possible tiling replaces each edge of a square tiling with a dragon curve, using the recursive definition of the dragon starting from a line segment. The initial direction to expand each segment can be determined from a checkerboard coloring of a square tiling, expanding vertical segments into black tiles and out of white tiles, and expanding horizontal segments into white tiles and out of black ones.[3]
• As a non-self-crossing space-filling curve, the dragon curve has fractal dimension exactly 2. For a dragon curve with initial segment length 1, its area is 1/2, as can be seen from its tilings of the plane.[1]
• The boundary of the set covered by the dragon curve has infinite length, with fractal dimension
$2\log _{2}\lambda \approx 1.523627086202492,$
where
$\lambda ={\frac {1+(28-3{\sqrt {87}})^{1/3}+(28+3{\sqrt {87}})^{1/3}}{3}}\approx 1.69562076956$
is the real solution of the equation $\lambda ^{3}-\lambda ^{2}-2=0.$[4]
Twindragon
See also: Complex-base system § Base −1 ± i
The twindragon (also known as the Davis–Knuth dragon) can be constructed by placing two Heighway dragon curves back to back. It is also the limit set of the following iterated function system:
$f_{1}(z)={\frac {(1+i)z}{2}}$
$f_{2}(z)=1-{\frac {(1+i)z}{2}}$
where the initial shape is defined by the following set $S_{0}=\{0,1,1-i\}$.
It can be also written as a Lindenmayer system – it only needs adding another section in initial string:
• angle 90°
• initial string FX+FX+
• string rewriting rules
• X ↦ X+YF
• Y ↦ FX−Y.
It is also the locus of points in the complex of plane with the same integer part when written in base $(-1\pm i)$.[5]
Terdragon
The terdragon can be written as a Lindenmayer system:
• angle 120°
• initial string F
• string rewriting rules
• F ↦ F+F−F.
It is the limit set of the following iterated function system:
$f_{1}(z)=\lambda z$
$f_{2}(z)={\frac {i}{\sqrt {3}}}z+\lambda $
$f_{3}(z)=\lambda z+\lambda ^{*}$
${\mbox{where }}\lambda ={\frac {1}{2}}-{\frac {i}{2{\sqrt {3}}}}{\text{ and }}\lambda ^{*}={\frac {1}{2}}+{\frac {i}{2{\sqrt {3}}}}.$
Lévy dragon
The Lévy C curve is sometimes known as the Lévy dragon.[6]
Variants
The dragon curve belongs to a basic set of iteration functions consisting of two lines with four possible orientations at perpendicular angles:
Curve Creators and Creation Year of Dragon Family Members
Lévy Curve Ernesto Cesàro (1906), Georg Faber (1910), Paul Lévy (1914)
Dragon curve John Heighway (1966), Bruce Banks (1966), William Harter (1966)
Davis Diamond Chandler Davis (1970), Donald J. Knuth (1970)
Knuth Wedge Chandler Davis (1970), Donald J. Knuth (1970)
Unicorn Curve Peter van Roy (1989)
Lion Curve Bernt Rainer Wahl (1989)
It is possible to change the turn angle from 90° to other angles. Changing to 120° yields a structure of triangles, while 60° gives the following curve:
A discrete dragon curve can be converted to a dragon polyomino as shown. Like discrete dragon curves, dragon polyominoes approach the fractal dragon curve as a limit.
Occurrences of the dragon curve in solution sets
Having obtained the set of solutions to a linear differential equation, any linear combination of the solutions will, because of the superposition principle, also obey the original equation. In other words, new solutions are obtained by applying a function to the set of existing solutions. This is similar to how an iterated function system produces new points in a set, though not all IFS are linear functions. In a conceptually similar vein, a set of Littlewood polynomials can be arrived at by such iterated applications of a set of functions.
A Littlewood polynomial is a polynomial: $p(x)=\sum _{i=0}^{n}a_{i}x^{i}\,$ where all $a_{i}=\pm 1$.
For some $|w|<1$ we define the following functions:
$f_{+}(z)=1+wz$
$f_{-}(z)=1-wz$
Starting at z=0 we can generate all Littlewood polynomials of degree d using these functions iteratively d+1 times.[7] For instance: $f_{+}(f_{-}(f_{-}(0)))=1+(1-w)w=1+1w-1w^{2}$
It can be seen that for $w=(1+i)/2$, the above pair of functions is equivalent to the IFS formulation of the Heighway dragon. That is, the Heighway dragon, iterated to a certain iteration, describe the set of all Littlewood polynomials up to a certain degree, evaluated at the point $w=(1+i)/2$. Indeed, when plotting a sufficiently high number of roots of the Littlewood polynomials, structures similar to the dragon curve appear at points close to these coordinates.[7][8][9]
See also
• List of fractals by Hausdorff dimension
• Complex-base system
References
1. Tabachnikov, Sergei (2014), "Dragon curves revisited", The Mathematical Intelligencer, 36 (1): 13–17, doi:10.1007/s00283-013-9428-y, MR 3166985, S2CID 14420269
2. Edgar, Gerald (2008), "Heighway's Dragon", in Edgar, Gerald (ed.), Measure, Topology, and Fractal Geometry, Undergraduate Texts in Mathematics (2nd ed.), New York: Springer, pp. 20–22, doi:10.1007/978-0-387-74749-1, ISBN 978-0-387-74748-4, MR 2356043
3. Edgar (2008), "Heighway’s Dragon Tiles the Plane", pp. 74–75.
4. Edgar (2008), "Heighway Dragon Boundary", pp. 194–195.
5. Knuth, Donald (1998). "Positional Number Systems". The art of computer programming. Vol. 2 (3rd ed.). Boston: Addison-Wesley. p. 206. ISBN 0-201-89684-2. OCLC 48246681.
6. Bailey, Scott; Kim, Theodore; Strichartz, Robert S. (2002), "Inside the Lévy dragon", The American Mathematical Monthly, 109 (8): 689–703, doi:10.2307/3072395, JSTOR 3072395, MR 1927621.
7. "The n-Category Café".
8. "Week285".
9. "The Beauty of Roots". 2011-12-11.
External links
Wikimedia Commons has media related to Dragon curve.
• Weisstein, Eric W., "Dragon Curve", MathWorld
• Knuth on the Dragon Curve
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• Fractal art
• List of fractals by Hausdorff dimension
• The Fractal Geometry of Nature (1982 book)
• The Beauty of Fractals (1986 book)
• Chaos: Making a New Science (1987 book)
• Kaleidoscope
• Chaos theory
Mathematics of paper folding
Flat folding
• Big-little-big lemma
• Crease pattern
• Huzita–Hatori axioms
• Kawasaki's theorem
• Maekawa's theorem
• Map folding
• Napkin folding problem
• Pureland origami
• Yoshizawa–Randlett system
Strip folding
• Dragon curve
• Flexagon
• Möbius strip
• Regular paperfolding sequence
3d structures
• Miura fold
• Modular origami
• Paper bag problem
• Rigid origami
• Schwarz lantern
• Sonobe
• Yoshimura buckling
Polyhedra
• Alexandrov's uniqueness theorem
• Blooming
• Flexible polyhedron (Bricard octahedron, Steffen's polyhedron)
• Net
• Source unfolding
• Star unfolding
Miscellaneous
• Fold-and-cut theorem
• Lill's method
Publications
• Geometric Exercises in Paper Folding
• Geometric Folding Algorithms
• Geometric Origami
• A History of Folding in Mathematics
• Origami Polyhedra Design
• Origamics
People
• Roger C. Alperin
• Margherita Piazzola Beloch
• Robert Connelly
• Erik Demaine
• Martin Demaine
• Rona Gurkewitz
• David A. Huffman
• Tom Hull
• Kôdi Husimi
• Humiaki Huzita
• Toshikazu Kawasaki
• Robert J. Lang
• Anna Lubiw
• Jun Maekawa
• Kōryō Miura
• Joseph O'Rourke
• Tomohiro Tachi
• Eve Torrence
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Wikipedia
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Peter Twinn
Peter Frank George Twinn CBE (9 January 1916 – 29 October 2004[1]) was a British mathematician, Second World War codebreaker and entomologist. The first professional mathematician to be recruited to GC&CS.[2] Head of ISK from 1943, the unit responsible for decrypting over 100,000 Abwehr communications.[3]
Peter Frank George Twinn
Born
Peter Frank George Twinn
(1916-01-09)9 January 1916
Streatham, South London
Died29 October 2004(2004-10-29) (aged 88)
NationalityEnglish
CitizenshipBritish
Education
• Manchester Grammar School
• Dulwich College
Alma materBrasenose College, Oxford
Occupations
• Codebreaker
• civil servant
Employer
• GC&CS
Early life and education
Born in Streatham, South London, Twinn was the son of a senior General Post Office official.[1] After attending Manchester Grammar School and Dulwich College, he graduated in mathematics at Brasenose College, Oxford.[1] He won a scholarship to pursue postgraduate studies in physics.[4]
Cryptography
Twinn was the first professional mathematician to join GC&CS.[2] In early 1939, he applied after seeing an advertisement, working first in London before moving to Bletchley Park. He worked with Dilly Knox and Alan Turing on German Enigma ciphers. In early 1942, he became the head of the Abwehr Enigma section.
Recruitment to GC&CS
He was in the middle of a postgraduate scholarship studying Physics when he saw an advertisement for a job with the government. "I was a bit unsettled," he remembered. "I'd finished my university degree and I didn't quite know what to do." The advertisement indicated that they were looking for mathematicians, but was unclear about what else was involved.
In that unsettled period after the Munich Agreement, international relations between the major European powers were tense and getting tenser.
" They offered me this job at the princely salary of, I think, £275 a year," he said, "which sounded all right to me, and I was taken along on the first day to be introduced to Dilly Knox." He began as an assistant to Alfred Dilwyn ("Dilly") Knox, who headed a team of codebreakers at GC&CS.
An eccentric but brilliant character, Dilly Knox was the first British codebreaker to work on the Enigma cipher. Like most GC&CS experts, he was a classicist. But, as war loomed, GC&CS began employing mathematicians, as well as chess players and crossword experts. Twinn was in fact the first mathematician to join the team.
Knox believed in throwing his new recruits in at the deep end. He gave Twinn a mere five minutes' training before telling him to go and get on with it.
On the eve of war
Twinn was the first British cryptographer to read a German military Enigma message, having obtained vital information from Polish cryptanalysts in July 1939. Twinn said that "It was a trifling exercise, but I repeat for the umpteenth time, no credit to me."
In July 1939 GC&CS moved from London to Bletchley Park. The mansion in the park was used by the staff, but many other buildings had to be constructed to accommodate the large number of people who worked for GC&CS during the war. These temporary buildings were known as the "huts".
Enigma
The Enigma machine dated back to 1919, when Hugo Alexander Koch, a Dutchman, patented an invention that he called a secret writing machine. Soon Arthur Scherbius, an engineer, was experimenting with this and similar machines and became enthusiastic about encryption machines that used rotors. He recommended them to Siegfried Turkel, the director of the Institute of Criminology in Vienna, who also became interested in them.
In the meantime, Koch had set up a company with the hope of selling his encryption machine for commercial use; one disadvantage was that numbers had to be spelt out in words. Industry was not interested, but in 1926 the German Navy looked at the Koch machine. Senior officers were impressed with it and ordered a large number. The purchase of the device – called Enigma – was kept strictly secret.
The Enigma machine was complicated, with a keyboard, like the ones used on a typewriter, containing all the letters of the alphabet. Each of the 26 letters was connected electrically to one of three rotors, each provided with a ring. Each ring also held the 26 letters of the alphabet. Further electrical connections led from the rotors to 26 illuminated letters.
When an operator, enciphering a message, pressed a key, an electric current passed through the machine and the rotors turned mechanically, but not in unison. Every time a key was pushed, the first rotor would rotate one letter. This happened 26 times until the first rotor had made a complete revolution. Then the second rotor would start to rotate. And so on.
When a key was pressed, a light came on behind the cipher text letter, always different from the original letter in the plain text. The illuminated letters made up the coded message.
The system worked in reverse. The person decoding a cipher message would use an Enigma with identical settings. When he pressed the cipher text letter, the letter in the original plain text message lit up. The illuminated letters made up the original message.
To make the codes more difficult to break, each of the rotors could be taken out and replaced in a different order. Also, the rings on the rotors could be put in a different order each day – for example, on one day the first rotor could be set at B, the next day at F, and so on. The military version of Enigma was provided with a plug board, like an old telephone switchboard. This allowed an extra switching of the letters, both before they entered the rotors and after leaving them. The plug board had 26 holes. Connections were made with wires and plugs. With three rotors and, say, six pairs of letters connected with the plug board, there would be 105,456 different combinations of the alphabet.
In December 1938 the Germans added additional rotors (up to six) and the number of combinations increased dramatically. The Germans believed that messages sent on their most sophisticated Enigma machines were so well coded that they could not be decoded. But Twinn and his colleagues proved them wrong.
About 10,000 people worked at Bletchley. The core group was the small number of cryptanalysts trying to crack the Enigma machine; at the beginning, this group consisted of no more than ten people, with Knox and Twinn in charge.
The British codebreakers had been working on the commercial version of Enigma, the easier of the two to break, during the 1920s and 1930s, and they had made much progress in breaking the military version. But Twinn and his colleagues were stymied because they could not work out the order in which the Enigma keys were wired up.
In July 1939, a month or so before the war started, Knox and some others travelled to Poland. Polish cryptologists, some of whom were brilliant, handed over to their British colleagues key information about Enigma, including replica machines.
The British discovered that Enigma machines were wired alphabetically: A to the first contact, B to the second, and so on. This was the order given in the diagram attached to the patent application. But Twinn and his colleagues thought it such an obvious thing to do that nobody considered it worth trying.
In early 1940 Twinn made the first break into Enigma. This could have been done much earlier if only they had tried the alphabetical system detailed in the patent application.
The ability to read German encoded military messages was of inestimable help to the Allies in winning the war. It was achieved largely because of the efforts of Twinn, Knox, Alan Turing (who later became the father of artificial intelligence) and others at Bletchley Park. Turing, a brilliant mathematician, developed a machine called the “bombe”, which speeded up the deciphering process by trial and error — a crucial development for the codebreakers.
German Naval Enigma
Twinn worked with Turing on breaking the German Naval Enigma. Their success helped allied convoys to avoid German U-boats.
Intelligence Services Knox
In October 1941, Dilly Knox solved the Abwehr Enigma.[3] Intelligence Services Knox (ISK) was established to decrypt Abwehr communications.[3] In early 1942, with Knox seriously ill, Twinn took change of running ISK[5] and was appointed head after Knox's death.[3] By the end of the war, ISK had decrypted and disseminated 140,800 messages.[3]
Intelligence gained from these Abwehr decrypts played an important part in ensuring the success of Double-Cross operations by MI5 and M16, and in Operation Fortitude, the Allied campaign to deceive the Germans about D-Day.[5]
Post-war career
Twinn's carried on government work after the war in a number of departments, including, in the late 1960s, as Director of Hovercraft in the Ministry for Technology. Later he became Secretary of the Royal Aircraft Establishment in Farnborough. In the early 1970s, he was the second secretary of the Natural Environment Research Council. He was appointed CBE in the 1980 Birthday Honours.
Twinn became interested in entomology, gaining his doctorate from the University of London in the jumping mechanism of click beetles. He co-authored A Provisional Atlas of the Longhorn Beetle (Coleoptera Cerambycidae) (1999), a study of the distribution of a number of beetle species.
Twinn had an interest in music and played the clarinet and viola. Twinn married Rosamund Case, whom he had met at Bletchley Park through his interest in music, in 1944; they had a son and three daughters.
Publications
• Peter F. G. Twinn and P. T. Harding, "Provisional atlas of the longhorn beetles (Coleoptera, Cerambycidae) of Britain", Huntingdon: Biological Records Centre, 1999. ISBN 1-870393-43-0
Notes
1. Dan van der Vat, "Obituary: Peter Twinn", The Guardian, 20 November 2004
2. Batey 2009
3. Batey 2009, p. xi
4. "Peter Twinn – Obituary", The Times, 24 November 2004
5. "Peter Twinn", The Daily Telegraph, 17 November 2004
References
• Batey, Mavis (2009). Dilly: The Man Who Broke Enigmas. Dialogue. ISBN 978-1-906447-01-4.
• "Peter Twinn", The Telegraph, London, 17 November 2004, archived from the original on 2 March 2007, retrieved 31 July 2013
External links
• Telegraph obituary
Authority control
International
• ISNI
• VIAF
National
• United States
• Netherlands
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Wikipedia
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Hinged dissection
In geometry, a hinged dissection, also known as a swing-hinged dissection or Dudeney dissection,[1] is a kind of geometric dissection in which all of the pieces are connected into a chain by "hinged" points, such that the rearrangement from one figure to another can be carried out by swinging the chain continuously, without severing any of the connections.[2] Typically, it is assumed that the pieces are allowed to overlap in the folding and unfolding process;[3] this is sometimes called the "wobbly-hinged" model of hinged dissection.[4]
History
The concept of hinged dissections was popularised by the author of mathematical puzzles, Henry Dudeney. He introduced the famous hinged dissection of a square into a triangle (pictured) in his 1907 book The Canterbury Puzzles.[5] The Wallace–Bolyai–Gerwien theorem, first proven in 1807, states that any two equal-area polygons must have a common dissection. However, the question of whether two such polygons must also share a hinged dissection remained open until 2007, when Erik Demaine et al. proved that there must always exist such a hinged dissection, and provided a constructive algorithm to produce them.[4][6][7] This proof holds even under the assumption that the pieces may not overlap while swinging, and can be generalised to any pair of three-dimensional figures which have a common dissection (see Hilbert's third problem).[6][8] In three dimensions, however, the pieces are not guaranteed to swing without overlap.[9]
Other hinges
Other types of "hinges" have been considered in the context of dissections. A twist-hinge dissection is one which use a three-dimensional "hinge" which is placed on the edges of pieces rather than their vertices, allowing them to be "flipped" three-dimensionally.[10][11] As of 2002, the question of whether any two polygons must have a common twist-hinged dissection remains unsolved.[12]
References
1. Akiyama, Jin; Nakamura, Gisaku (2000). "Dudeney Dissection of Polygons". Discrete and Computational Geometry. Lecture Notes in Computer Science. Vol. 1763. pp. 14–29. doi:10.1007/978-3-540-46515-7_2. ISBN 978-3-540-67181-7.
2. Pitici, Mircea (September 2008). "Hinged Dissections". Math Explorers Club. Cornell University. Retrieved 19 December 2013.
3. O'Rourke, Joseph (2003). "Computational Geometry Column 44". arXiv:cs/0304025v1.
4. "Problem 47: Hinged Dissections". The Open Problems Project. Smith College. 8 December 2012. Retrieved 19 December 2013.
5. Frederickson 2002, p.1
6. Abbot, Timothy G.; Abel, Zachary; Charlton, David; Demaine, Erik D.; Demaine, Martin L.; Kominers, Scott D. (2008). "Hinged Dissections Exist". Proceedings of the twenty-fourth annual symposium on Computational geometry - SCG '08. p. 110. arXiv:0712.2094. doi:10.1145/1377676.1377695. ISBN 9781605580715. S2CID 3264789.
7. Bellos, Alex (30 May 2008). "The science of fun". The Guardian. Retrieved 20 December 2013.
8. Phillips, Tony (November 2008). "Tony Phillips' Take on Math in the Media". Math in the Media. Retrieved 20 December 2013.
9. O'Rourke, Joseph (March 2008). "Computational Geometry Column 50" (PDF). ACM SIGACT News. 39 (1). Retrieved 20 December 2013.
10. Frederickson 2002, p.6
11. Frederickson, Greg N. (2007). Symmetry and Structure in Twist-Hinged Dissections of Polygonal Rings and Polygonal Anti-Rings (PDF). Bridges 2007. The Bridges Organization. Retrieved 20 December 2013.
12. Frederickson 2002, p. 7
Bibliography
• Frederickson, Greg N. (26 August 2002). Hinged Dissections: Swinging and Twisting. Cambridge University Press. ISBN 978-0521811927. Retrieved 19 December 2013.
External links
• An applet demonstrating Dudeney's hinged square-triangle dissection
• A gallery of hinged dissections
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Wikipedia
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Twist knot
In knot theory, a branch of mathematics, a twist knot is a knot obtained by repeatedly twisting a closed loop and then linking the ends together. (That is, a twist knot is any Whitehead double of an unknot.) The twist knots are an infinite family of knots, and are considered the simplest type of knots after the torus knots.
Construction
A twist knot is obtained by linking together the two ends of a twisted loop. Any number of half-twists may be introduced into the loop before linking, resulting in an infinite family of possibilities. The following figures show the first few twist knots:
• One half-twist
(trefoil knot, 31)
• Two half-twists
(figure-eight knot, 41)
• Three half-twists
(52 knot)
• Four half-twists
(stevedore knot, 61)
• Five half-twists
(72 knot)
• Six half-twists
(81 knot)
Properties
All twist knots have unknotting number one, since the knot can be untied by unlinking the two ends. Every twist knot is also a 2-bridge knot.[1] Of the twist knots, only the unknot and the stevedore knot are slice knots.[2] A twist knot with $n$ half-twists has crossing number $n+2$. All twist knots are invertible, but the only amphichiral twist knots are the unknot and the figure-eight knot.
Invariants
The invariants of a twist knot depend on the number $n$ of half-twists. The Alexander polynomial of a twist knot is given by the formula
$\Delta (t)={\begin{cases}{\frac {n+1}{2}}t-n+{\frac {n+1}{2}}t^{-1}&{\text{if }}n{\text{ is odd}}\\-{\frac {n}{2}}t+(n+1)-{\frac {n}{2}}t^{-1}&{\text{if }}n{\text{ is even,}}\\\end{cases}}$
and the Conway polynomial is
$\nabla (z)={\begin{cases}{\frac {n+1}{2}}z^{2}+1&{\text{if }}n{\text{ is odd}}\\1-{\frac {n}{2}}z^{2}&{\text{if }}n{\text{ is even.}}\\\end{cases}}$
When $n$ is odd, the Jones polynomial is
$V(q)={\frac {1+q^{-2}+q^{-n}-q^{-n-3}}{q+1}},$
and when $n$ is even, it is
$V(q)={\frac {q^{3}+q-q^{3-n}+q^{-n}}{q+1}}.$
References
1. Rolfsen, Dale (2003). Knots and links. Providence, R.I: AMS Chelsea Pub. pp. 114. ISBN 0-8218-3436-3.
2. Weisstein, Eric W. "Twist 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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Wikipedia
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Twisted K-theory
In mathematics, twisted K-theory (also called K-theory with local coefficients[1]) is a variation on K-theory, a mathematical theory from the 1950s that spans algebraic topology, abstract algebra and operator theory.
More specifically, twisted K-theory with twist H is a particular variant of K-theory, in which the twist is given by an integral 3-dimensional cohomology class. It is special among the various twists that K-theory admits for two reasons. First, it admits a geometric formulation. This was provided in two steps; the first one was done in 1970 (Publ. Math. de l'IHÉS) by Peter Donovan and Max Karoubi; the second one in 1988 by Jonathan Rosenberg in Continuous-Trace Algebras from the Bundle Theoretic Point of View.
In physics, it has been conjectured to classify D-branes, Ramond-Ramond field strengths and in some cases even spinors in type II string theory. For more information on twisted K-theory in string theory, see K-theory (physics).
In the broader context of K-theory, in each subject it has numerous isomorphic formulations and, in many cases, isomorphisms relating definitions in various subjects have been proven. It also has numerous deformations, for example, in abstract algebra K-theory may be twisted by any integral cohomology class.
Definition
To motivate Rosenberg's geometric formulation of twisted K-theory, start from the Atiyah–Jänich theorem, stating that
$Fred({\mathcal {H}}),$
the Fredholm operators on Hilbert space ${\mathcal {H}}$, is a classifying space for ordinary, untwisted K-theory. This means that the K-theory of the space $M$ consists of the homotopy classes of maps
$[M\rightarrow Fred({\mathcal {H}})]$
from $M$ to $Fred({\mathcal {H}}).$
A slightly more complicated way of saying the same thing is as follows. Consider the trivial bundle of $Fred({\mathcal {H}})$ over $M$, that is, the Cartesian product of $M$ and $Fred({\mathcal {H}})$. Then the K-theory of $M$ consists of the homotopy classes of sections of this bundle.
We can make this yet more complicated by introducing a trivial
$PU({\mathcal {H}})$
bundle $P$ over $M$, where $PU({\mathcal {H}})$ is the group of projective unitary operators on the Hilbert space ${\mathcal {H}}$. Then the group of maps
$[P\rightarrow Fred({\mathcal {H}})]_{PU({\mathcal {H}})}$
from $P$ to $Fred({\mathcal {H}})$ which are equivariant under an action of $PU({\mathcal {H}})$ is equivalent to the original groups of maps
$[M\rightarrow Fred({\mathcal {H}})].$
This more complicated construction of ordinary K-theory is naturally generalized to the twisted case. To see this, note that $PU({\mathcal {H}})$ bundles on $M$ are classified by elements $H$ of the third integral cohomology group of $M$. This is a consequence of the fact that $PU({\mathcal {H}})$ topologically is a representative Eilenberg–MacLane space
$K(\mathbf {Z} ,3)$.
The generalization is then straightforward. Rosenberg has defined
$K_{H}(M)$,
the twisted K-theory of $M$ with twist given by the 3-class $H$, to be the space of homotopy classes of sections of the trivial $Fred({\mathcal {H}})$ bundle over $M$ that are covariant with respect to a $PU({\mathcal {H}})$ bundle $P_{H}$ fibered over $M$ with 3-class $H$, that is
$K_{H}(M)=[P_{H}\rightarrow Fred({\mathcal {H}})]_{PU({\mathcal {H}})}.$
Equivalently, it is the space of homotopy classes of sections of the $Fred({\mathcal {H}})$ bundles associated to a $PU({\mathcal {H}})$ bundle with class $H$.
Relation to K-theory
When $H$ is the trivial class, twisted K-theory is just untwisted K-theory, which is a ring. However, when $H$ is nontrivial this theory is no longer a ring. It has an addition, but it is no longer closed under multiplication.
However, the direct sum of the twisted K-theories of $M$ with all possible twists is a ring. In particular, the product of an element of K-theory with twist $H$ with an element of K-theory with twist $H'$ is an element of K-theory twisted by $H+H'$. This element can be constructed directly from the above definition by using adjoints of Fredholm operators and construct a specific 2 x 2 matrix out of them (see the reference 1, where a more natural and general Z/2-graded version is also presented). In particular twisted K-theory is a module over classical K-theory.
Calculations
Physicist typically want to calculate twisted K-theory using the Atiyah–Hirzebruch spectral sequence.[2] The idea is that one begins with all of the even or all of the odd integral cohomology, depending on whether one wishes to calculate the twisted $K_{0}$ or the twisted $K^{0}$, and then one takes the cohomology with respect to a series of differential operators. The first operator, $d_{3}$, for example, is the sum of the three-class $H$, which in string theory corresponds to the Neveu-Schwarz 3-form, and the third Steenrod square,[3] so
$d_{3}^{p,q}=Sq^{3}+H$
No elementary form for the next operator, $d_{5}$, has been found, although several conjectured forms exist. Higher operators do not contribute to the $K$-theory of a 10-manifold, which is the dimension of interest in critical superstring theory. Over the rationals Michael Atiyah and Graeme Segal have shown that all of the differentials reduce to Massey products of $M$.[4]
After taking the cohomology with respect to the full series of differentials one obtains twisted $K$-theory as a set, but to obtain the full group structure one in general needs to solve an extension problem.
Example: the three-sphere
The three-sphere, $S^{3}$, has trivial cohomology except for $H^{0}(S^{3})$ and $H^{3}(S^{3})$ which are both isomorphic to the integers. Thus the even and odd cohomologies are both isomorphic to the integers. Because the three-sphere is of dimension three, which is less than five, the third Steenrod square is trivial on its cohomology and so the first nontrivial differential is just $d_{5}=H$. The later differentials increase the degree of a cohomology class by more than three and so are again trivial; thus the twisted $K$-theory is just the cohomology of the operator $d_{3}$ which acts on a class by cupping it with the 3-class $H$.
Imagine that $H$ is the trivial class, zero. Then $d_{3}$ is also trivial. Thus its entire domain is its kernel, and nothing is in its image. Thus $K_{H}^{0}(S^{3})$ is the kernel of $d_{3}$ in the even cohomology, which is the full even cohomology, which consists of the integers. Similarly $K_{H}^{1}(S^{3})$ consists of the odd cohomology quotiented by the image of $d_{3}$, in other words quotiented by the trivial group. This leaves the original odd cohomology, which is again the integers. In conclusion, $K^{0}$ and $K^{1}$ of the three-sphere with trivial twist are both isomorphic to the integers. As expected, this agrees with the untwisted $K$-theory.
Now consider the case in which $H$ is nontrivial. $H$ is defined to be an element of the third integral cohomology, which is isomorphic to the integers. Thus $H$ corresponds to a number, which we will call $n$. $d_{3}$ now takes an element $m$ of $H^{0}$ and yields the element $nm$ of $H^{3}$. As $n$ is not equal to zero by assumption, the only element of the kernel of $d_{3}$ is the zero element, and so $K_{H=n}^{0}(S^{3})=0$. The image of $d_{3}$ consists of all elements of the integers that are multiples of $n$. Therefore, the odd cohomology, $\mathbb {Z} $, quotiented by the image of $d_{3}$, $n\mathbb {Z} $, is the cyclic group of order $n$, $\mathbb {Z} /n$. In conclusion
$K_{H=n}^{1}(S^{3})=\mathbb {Z} /n$
In string theory this result reproduces the classification of D-branes on the 3-sphere with $n$ units of $H$-flux, which corresponds to the set of symmetric boundary conditions in the supersymmetric $SU(2)$ WZW model at level $n-2$.
There is an extension of this calculation to the group manifold of SU(3).[5] In this case the Steenrod square term in $d_{3}$, the operator $d_{5}$, and the extension problem are nontrivial.
See also
• K-theory (physics)
• Wess–Zumino–Witten model
• Bundle gerbe
Notes
1. Donavan, Peter; Karoubi, Max (1970). "Graded Brauer groups and $K$-theory with local coefficients". Publications Mathématiques de l'IHÉS. 38: 5–25.
2. A guide to such calculations in the case of twisted K-theory can be found in E8 Gauge Theory, and a Derivation of K-Theory from M-Theory by Emanuel Diaconescu, Gregory Moore and Edward Witten (DMW).
3. (DMW) also provide a crash course in Steenrod squares for physicists.
4. In Twisted K-theory and cohomology.
5. In D-Brane Instantons and K-Theory Charges by Juan Maldacena, Gregory Moore and Nathan Seiberg.
References
• "Graded Brauer groups and K-theory with local coefficients", by Peter Donovan and Max Karoubi. Publ. Math. IHÉS Nr. 38, pp. 5–25 (1970).
• D-Brane Instantons and K-Theory Charges by Juan Maldacena, Gregory Moore and Nathan Seiberg
• Twisted K-theory and Cohomology by Michael Atiyah and Graeme Segal
• Twisted K-theory and the K-theory of Bundle Gerbes by Peter Bouwknegt, Alan Carey, Varghese Mathai, Michael Murray and Danny Stevenson.
• Twisted K-theory, old and new
External links
• Strings 2002, Michael Atiyah lecture, "Twisted K-theory and physics"
• The Verlinde algebra is twisted equivariant K-theory (PDF)
• Riemann–Roch and index formulae in twisted K-theory (PDF)
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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Twisted Poincaré duality
In mathematics, the twisted Poincaré duality is a theorem removing the restriction on Poincaré duality to oriented manifolds. The existence of a global orientation is replaced by carrying along local information, by means of a local coefficient system.
Twisted Poincaré duality for de Rham cohomology
Another version of the theorem with real coefficients features de Rham cohomology with values in the orientation bundle. This is the flat real line bundle denoted $o(M)$, that is trivialized by coordinate charts of the manifold $M$, with transition maps the sign of the Jacobian determinant of the charts transition maps. As a flat line bundle, it has a de Rham cohomology, denoted by
$H^{*}(M;\mathbb {R} ^{w})$ or $H^{*}(M;o(M))$.
For M a compact manifold, the top degree cohomology is equipped with a so-called trace morphism
$\theta \colon H^{d}(M;o(M))\to \mathbb {R} $,
that is to be interpreted as integration on M, i.e., evaluating against the fundamental class.
Poincaré duality for differential forms is then the conjunction, for M connected, of the following two statements:
• The trace morphism is a linear isomorphism.
• The cup product, or exterior product of differential forms
$\cup \colon H^{*}(M;\mathbb {R} )\otimes H^{d-*}(M,o(M))\to H^{d}(M,o(M))\simeq \mathbb {R} $
is non-degenerate.
The oriented Poincaré duality is contained in this statement, as understood from the fact that the orientation bundle o(M) is trivial if the manifold is oriented, an orientation being a global trivialization, i.e., a nowhere vanishing parallel section.
See also
• Local system
• Dualizing sheaf
• Verdier duality
References
• Some references are provided in the answers to this thread on MathOverflow.
• The online book Algebraic and geometric surgery by Andrew Ranicki.
• Bott, Raoul; Tu, Loring W. (1982). Differential forms in algebraic topology. Graduate Texts in Mathematics. Vol. 82. New York-Berlin: Springer-Verlag. doi:10.1007/978-1-4757-3951-0. ISBN 0-387-90613-4. MR 0658304.
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Twisted geometries
Twisted geometries are discrete geometries that play a role in loop quantum gravity and spin foam models, where they appear in the semiclassical limit of spin networks.[1][2][3] A twisted geometry can be visualized as collections of polyhedra dual to the nodes of the spin network's graph.[4] Intrinsic and extrinsic curvatures are defined in a manner similar to Regge calculus, but with the generalisation of including a certain type of metric discontinuities: the face shared by two adjacent polyhedra has a unique area, but its shape can be different. This is a consequence of the quantum geometry of spin networks: ordinary Regge calculus is "too rigid" to account for all the geometric degrees of freedom described by the semiclassical limit of a spin network.
The name twisted geometry captures the relation between these additional degrees of freedom and the off-shell presence of torsion in the theory, but also the fact that this classical description can be derived from Twistor theory, by assigning a pair of twistors to each link of the graph, and suitably constraining their helicities and incidence relations.[5][6]
References
1. L. Freidel and S. Speziale (2010). "Twisted geometries: A geometric parametrisation of SU(2) phase space". Phys. Rev. D. 82 (8): 084040. arXiv:1001.2748. Bibcode:2010PhRvD..82h4040F. doi:10.1103/PhysRevD.82.084040. S2CID 119110824.
2. C. Rovelli and S. Speziale (2010). "On the geometry of loop quantum gravity on a graph". Phys. Rev. D. 82 (4): 044018. arXiv:1005.2927. Bibcode:2010PhRvD..82d4018R. doi:10.1103/PhysRevD.82.044018. S2CID 118396168.
3. E. R. Livine and J. Tambornino (2012). "Spinor Representation for Loop Quantum Gravity". J. Math. Phys. 53 (1): 012503. arXiv:1105.3385. Bibcode:2012JMP....53a2503L. doi:10.1063/1.3675465. S2CID 119607941.
4. E. Bianchi, P. Dona and S. Speziale (2011). "Polyhedra in loop quantum gravity". Phys. Rev. D. 83 (4): 044035. arXiv:1009.3402. Bibcode:2011PhRvD..83d4035B. doi:10.1103/PhysRevD.83.044035. S2CID 14414561.
5. L. Freidel and S. Speziale (2010). "From twistors to twisted geometries". Phys. Rev. D. 82 (8): 084041. arXiv:1006.0199. Bibcode:2010PhRvD..82h4041F. doi:10.1103/PhysRevD.82.084041. S2CID 119292655.
6. S. Speziale and Wolfgang M. Wieland (2012). "The twistorial structure of loop-gravity transition amplitudes". Phys. Rev. D. 86 (12): 124023. arXiv:1207.6348. Bibcode:2012PhRvD..86l4023S. doi:10.1103/PhysRevD.86.124023. S2CID 59406729.
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Twisted sheaf
In mathematics, a twisted sheaf is a variant of a coherent sheaf. Precisely, it is specified by: an open covering in the étale topology Ui, coherent sheaves Fi over Ui, a Čech 2-cocycle θ on the covering Ui as well as the isomorphisms
$g_{ij}:F_{j}|_{U_{ij}}{\overset {\sim }{\to }}F_{i}|_{U_{ij}}$
satisfying
• $g_{ii}=\operatorname {id} _{F_{i}}$,
• $g_{ij}=g_{ji}^{-1},$
• $g_{ij}\circ g_{jk}\circ g_{ki}=\theta _{ijk}\operatorname {id} _{F_{i}}.$
The notion of twisted sheaves was introduced by Jean Giraud. The above definition due to Căldăraru is down-to-earth but is equivalent to a more sophisticated definition in terms of gerbe; see § 2.1.3 of (Lieblich 2007).
See also
• Reflexive sheaf
• Torsion sheaf
References
• Căldăraru, Andrei (2002). "Derived categories of twisted sheaves on elliptic threefolds". Journal für die reine und angewandte Mathematik (Crelle's Journal). 2002 (544): 161–179. arXiv:math/0012083. doi:10.1515/CRLL.2002.022. S2CID 119117575.
• Lieblich, Max (2007). "Moduli of twisted sheaves". Duke Mathematical Journal. 138. doi:10.1215/S0012-7094-07-13812-2. S2CID 14067307.
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Twisting properties
Twisting properties in general terms are associated with the properties of samples that identify with statistics that are suitable for exchange.
Description
Starting with a sample $\{x_{1},\ldots ,x_{m}\}$ observed from a random variable X having a given distribution law with a non-set parameter, a parametric inference problem consists of computing suitable values – call them estimates – of this parameter precisely on the basis of the sample. An estimate is suitable if replacing it with the unknown parameter does not cause major damage in next computations. In algorithmic inference, suitability of an estimate reads in terms of compatibility with the observed sample.
In turn, parameter compatibility is a probability measure that we derive from the probability distribution of the random variable to which the parameter refers. In this way we identify a random parameter Θ compatible with an observed sample. Given a sampling mechanism $M_{X}=(g_{\theta },Z)$, the rationale of this operation lies in using the Z seed distribution law to determine both the X distribution law for the given θ, and the Θ distribution law given an X sample. Hence, we may derive the latter distribution directly from the former if we are able to relate domains of the sample space to subsets of Θ support. In more abstract terms, we speak about twisting properties of samples with properties of parameters and identify the former with statistics that are suitable for this exchange, so denoting a well behavior w.r.t. the unknown parameters. The operational goal is to write the analytic expression of the cumulative distribution function $F_{\Theta }(\theta )$, in light of the observed value s of a statistic S, as a function of the S distribution law when the X parameter is exactly θ.
Method
Given a sampling mechanism $M_{X}=(g_{\theta },Z)$ for the random variable X, we model ${\boldsymbol {X}}=\{X_{1},\ldots ,X_{m}\}$ to be equal to $\{g_{\theta }(Z_{1}),\ldots ,g_{\theta }(Z_{m})\}$. Focusing on a relevant statistic $S=h_{1}(X_{1},\ldots ,X_{m})$ for the parameter θ, the master equation reads
$s=h(g_{\theta }(z_{1}),\ldots ,g_{\theta }(z_{m}))=\rho (\theta ;z_{1},\ldots ,z_{m}).$
When s is a well-behaved statistic w.r.t the parameter, we are sure that a monotone relation exists for each ${\boldsymbol {z}}=\{z_{1},\ldots ,z_{m}\}$ between s and θ. We are also assured that Θ, as a function of ${\boldsymbol {Z}}$ for given s, is a random variable since the master equation provides solutions that are feasible and independent of other (hidden) parameters.[1]
The direction of the monotony determines for any ${\boldsymbol {z}}$ a relation between events of the type $s\geq s'\leftrightarrow \theta \geq \theta '$ or vice versa $s\geq s'\leftrightarrow \theta \leq \theta '$, where $s'$ is computed by the master equation with $\theta '$. In the case that s assumes discrete values the first relation changes into $s\geq s'\rightarrow \theta \geq \theta '\rightarrow s\geq s'+\ell $ where $\ell >0$ is the size of the s discretization grain, idem with the opposite monotony trend. Resuming these relations on all seeds, for s continuous we have either
$F_{\Theta \mid S=s}(\theta )=F_{S\mid \Theta =\theta }(s)$
or
$F_{\Theta \mid S=s}(\theta )=1-F_{S\mid \Theta =\theta }(s)$
For s discrete we have an interval where $F_{\Theta \mid S=s}(\theta )$ lies, because of $\ell >0$. The whole logical contrivance is called a twisting argument. A procedure implementing it is as follows.
Algorithm
Generating a parameter distribution law through a twisting argument
Given a sample $\{x_{1},\ldots ,x_{m}\}$ from a random variable with parameter θ unknown,
1. Identify a well behaving statistic S for the parameter θ and its discretization grain $\ell $ (if any);
2. decide the monotony versus;
3. compute $F_{\Theta }(\theta )\in \left(q_{1}(F_{S|\Theta =\theta }(s)),q_{2}(F_{S|\Theta =\theta }(s))\right)$ where:
• if S is continuous $q_{1}=q_{2}$
• if S is discrete
1. $q_{2}(F_{S}(s))=q_{1}(F_{S}(s-\ell )$ if s does not decrease with θ
2. $q_{1}(F_{S}(s))=q_{2}(F_{S}(s-\ell )$ if s does not increase with θ and
3. $q_{i}(F_{S})=1-F_{S}$ if s does not decrease with θ and $q_{i}(F_{S})=F_{S}$ if s does not increase with θ for $i=1,2$.
Remark
The rationale behind twisting arguments does not change when parameters are vectors, though some complication arises from the management of joint inequalities. Instead, the difficulty of dealing with a vector of parameters proved to be the Achilles heel of Fisher's approach to the fiducial distribution of parameters.[2] Also Fraser’s constructive probabilities[3] devised for the same purpose do not treat this point completely.
Example
For ${\boldsymbol {x}}$ drawn from a gamma distribution, whose specification requires values for the parameters λ and k, a twisting argument may be stated by following the below procedure. Given the meaning of these parameters we know that
$(k\leq k')\leftrightarrow (s_{k}\leq s_{k'}){\text{ for fixed }}\lambda ,$
$(\lambda \leq \lambda ')\leftrightarrow (s_{\lambda '}\leq s_{\lambda }){\text{ for fixed }}k,$
where $s_{k}=\prod _{i=1}^{m}x_{i}$ and $s_{\lambda }=\sum _{i=1}^{m}x_{i}$. This leads to a joint cumulative distribution function
$F_{\Lambda ,K}(\lambda ,k)=F_{\Lambda \,\mid \,K=k}(\lambda )F_{K}(k)=F_{K\,\mid \,\Lambda =\lambda }(k)F_{\Lambda }(\lambda ).$
Using the first factorization and replacing $s_{k}$ with $r_{k}={\frac {s_{k}}{s_{\lambda }^{m}}}$ in order to have a distribution of $K$ that is independent of $\Lambda $, we have
$F_{\Lambda \,\mid \,K=k}(\lambda )=1-{\frac {\Gamma (km,\lambda s_{\Lambda })}{\Gamma (km)}}$
$F_{K}(k)=1-F_{R_{k}}(r_{K})$
with m denoting the sample size, $s_{\Lambda }$ and $r_{K}$ are the observed statistics (hence with indices denoted by capital letters), $\Gamma (a,b)$ the incomplete gamma function and $F_{R_{k}}(r_{K})$ the Fox's H function that can be approximated with a gamma distribution again with proper parameters (for instance estimated through the method of moments) as a function of k and m.
With a sample size $m=30,s_{\Lambda }=72.82$ and $r_{K}=$ $4.5\times 10^{-46}$, you may find the joint p.d.f. of the Gamma parameters K and $\Lambda $ on the left. The marginal distribution of K is reported in the picture on the right.
Notes
1. By default, capital letters (such as U, X) will denote random variables and small letters (u, x) their corresponding realizations.
2. Fisher 1935.
3. Fraser 1966.
References
• Fisher, M.A. (1935). "The fiducial argument in statistical inference". Annals of Eugenics. 6 (4): 391–398. doi:10.1111/j.1469-1809.1935.tb02120.x. hdl:2440/15222.
• Fraser, D. A. S. (1966). "Structural probability and generalization". Biometrika. 53 (1/2): 1–9. doi:10.2307/2334048. JSTOR 2334048.
• Apolloni, B; Malchiodi, D.; Gaito, S. (2006). Algorithmic Inference in Machine Learning. International Series on Advanced Intelligence. Vol. 5 (2nd ed.). Adelaide: Magill. Advanced Knowledge International
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Twistor correspondence
In mathematical physics, the twistor correspondence or Penrose–Ward correspondence is a bijection between instantons on complexified Minkowski space and holomorphic vector bundles on twistor space, which as a complex manifold is $\mathbb {P} ^{3}$, or complex projective 3-space. Twistor space was introduced by Roger Penrose, while Richard Ward formulated the correspondence between instantons and vector bundles on twistor space.
Statement
There is a bijection between
1. Gauge equivalence classes of anti-self dual Yang–Mills (ASDYM) connections on complexified Minkowski space $M_{\mathbb {C} }\cong \mathbb {C} ^{4}$ with gauge group $\mathrm {GL} (n,\mathbb {C} )$ (the complex general linear group)
2. Holomorphic rank n vector bundles $E$ over projective twistor space ${\mathcal {PT}}\cong \mathbb {P} ^{3}-\mathbb {P} ^{1}$ which are trivial on each degree one section of ${\mathcal {PT}}\rightarrow \mathbb {P} ^{1}$.[1][2]
where $\mathbb {P} ^{n}$ is the complex projective space of dimension $n$.
Applications
ADHM construction
On the anti-self dual Yang–Mills side, the solutions, known as instantons, extend to solutions on compactified Euclidean 4-space. On the twistor side, the vector bundles extend from ${\mathcal {PT}}$ to $\mathbb {P} ^{3}$, and the reality condition on the ASDYM side corresponds to a reality structure on the algebraic bundles on the twistor side. Holomorphic vector bundles over $\mathbb {P} ^{3}$ have been extensively studied in the field of algebraic geometry, and all relevant bundles can be generated by the monad construction[3] also known as the ADHM construction, hence giving a classification of instantons.
References
1. Dunajski, Maciej (2010). Solitons, instantons, and twistors. Oxford: Oxford University Press. ISBN 9780198570622.
2. Ward, R.S. (April 1977). "On self-dual gauge fields". Physics Letters A. 61 (2): 81–82. doi:10.1016/0375-9601(77)90842-8.
3. Atiyah, M.F.; Hitchin, N.J.; Drinfeld, V.G.; Manin, Yu.I. (March 1978). "Construction of instantons". Physics Letters A. 65 (3): 185–187. doi:10.1016/0375-9601(78)90141-X.
Topics of twistor theory
Objectives
Principles
• Background independence
Final objective
• Quantum gravity
• Theory of everything
Mathematical concepts
Twistors
• Penrose transform
• Twistor space
Physical concepts
• Twistor string theory
• Twistor correspondence
• Twistor theory
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Twistor space
In mathematics and theoretical physics (especially twistor theory), twistor space is the complex vector space of solutions of the twistor equation $\nabla _{A'}^{(A}\Omega _{^{}}^{B)}=0$. It was described in the 1960s by Roger Penrose and Malcolm MacCallum.[1] According to Andrew Hodges, twistor space is useful for conceptualizing the way photons travel through space, using four complex numbers. He also posits that twistor space may aid in understanding the asymmetry of the weak nuclear force.[2]
Informal motivation
In the (translated) words of Jacques Hadamard: "the shortest path between two truths in the real domain passes through the complex domain." Therefore when studying four-dimensional space $\mathbb {R} ^{4}$ it might be valuable to identify it with $\mathbb {C} ^{2}.$ However, since there is no canonical way of doing so, instead all isomorphisms respecting orientation and metric between the two are considered. It turns out that complex projective 3-space $\mathbb {CP} ^{3}$ parametrizes such isomorphisms together with complex coordinates. Thus one complex coordinate describes the identification and the other two describe a point in $\mathbb {R} ^{4}$. It turns out that vector bundles with self-dual connections on $\mathbb {R} ^{4}$(instantons) correspond bijectively to holomorphic vector bundles on complex projective 3-space $\mathbb {CP} ^{3}$.
Formal definition
For Minkowski space, denoted $\mathbb {M} $, the solutions to the twistor equation are of the form
$\Omega ^{A}(x)=\omega ^{A}-ix^{AA'}\pi _{A'}$
where $\omega ^{A}$ and $\pi _{A'}$ are two constant Weyl spinors and $x^{AA'}=\sigma _{\mu }^{AA'}x^{\mu }$ is a point in Minkowski space. The $\sigma _{\mu }=(I,{\vec {\sigma }})$ are the Pauli matrices, with $A,A^{\prime }=1,2$ the indexes on the matrices. This twistor space is a four-dimensional complex vector space, whose points are denoted by $Z^{\alpha }=(\omega ^{A},\pi _{A'})$, and with a hermitian form
$\Sigma (Z)=\omega ^{A}{\bar {\pi }}_{A}+{\bar {\omega }}^{A'}\pi _{A'}$
which is invariant under the group SU(2,2) which is a quadruple cover of the conformal group C(1,3) of compactified Minkowski spacetime.
Points in Minkowski space are related to subspaces of twistor space through the incidence relation
$\omega ^{A}=ix^{AA'}\pi _{A'}.$
This incidence relation is preserved under an overall re-scaling of the twistor, so usually one works in projective twistor space, denoted $\mathbb {PT} $, which is isomorphic as a complex manifold to $\mathbb {CP} ^{3}$.
Given a point $x\in M$ it is related to a line in projective twistor space where we can see the incidence relation as giving the linear embedding of a $\mathbb {CP} ^{1}$ parametrized by $\pi _{A'}$.
The geometric relation between projective twistor space and complexified compactified Minkowski space is the same as the relation between lines and two-planes in twistor space; more precisely, twistor space is
$\mathbb {T} :=\mathbb {C} ^{4}.$ :=\mathbb {C} ^{4}.}
It has associated to it the double fibration of flag manifolds $\mathbb {P} \xleftarrow {\mu } \mathbb {F} \xrightarrow {\nu } \mathbb {M} $ where $\mathbb {P} $ is the projective twistor space
$\mathbb {P} =F_{1}(\mathbb {T} )=\mathbb {CP} ^{3}=\mathbf {P} (\mathbb {C} ^{4})$
and $\mathbb {M} $ is the compactified complexified Minkowski space
$\mathbb {M} =F_{2}(\mathbb {T} )=\operatorname {Gr} _{2}(\mathbb {C} ^{4})=\operatorname {Gr} _{2,4}(\mathbb {C} )$
and the correspondence space between $\mathbb {P} $ and $\mathbb {M} $ is
$\mathbb {F} =F_{1,2}(\mathbb {T} )$
In the above, $\mathbf {P} $ stands for projective space, $\operatorname {Gr} $ a Grassmannian, and $F$ a flag manifold. The double fibration gives rise to two correspondences (see also Penrose transform), $c=\nu \circ \mu ^{-1}$ and $c^{-1}=\mu \circ \nu ^{-1}.$
The compactified complexified Minkowski space $\mathbb {M} $ is embedded in $\mathbf {P} _{5}\cong \mathbf {P} (\wedge ^{2}\mathbb {T} )$ by the Plücker embedding; the image is the Klein quadric.
References
1. Penrose, R.; MacCallum, M.A.H. (February 1973). "Twistor theory: An approach to the quantisation of fields and space-time". Physics Reports. 6 (4): 241–315. doi:10.1016/0370-1573(73)90008-2.
2. Hodges, Andrew (2010). One to Nine: The Inner Life of Numbers. Doubleday Canada. p. 142. ISBN 978-0-385-67266-5.
• Ward, R.S.; Wells, R.O. (1991). Twistor Geometry and Field Theory. Cambridge University Press. ISBN 0-521-42268-X.
• Huggett, S.A.; Tod, K.P. (1994). An introduction to twistor theory. Cambridge University Press. ISBN 978-0-521-45689-0.
Topics of twistor theory
Objectives
Principles
• Background independence
Final objective
• Quantum gravity
• Theory of everything
Mathematical concepts
Twistors
• Penrose transform
• Twistor space
Physical concepts
• Twistor string theory
• Twistor correspondence
• Twistor theory
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Twistor theory
In theoretical physics, twistor theory was proposed by Roger Penrose in 1967[1] as a possible path[2] to quantum gravity and has evolved into a widely studied branch of theoretical and mathematical physics. Penrose's idea was that twistor space should be the basic arena for physics from which space-time itself should emerge. It has led to powerful mathematical tools that have applications to differential and integral geometry, nonlinear differential equations and representation theory, and in physics to general relativity, quantum field theory, and the theory of scattering amplitudes. Twistor theory arose in the context of the rapidly expanding mathematical developments in Einstein's theory of general relativity in the late 1950s and in the 1960s and carries a number of influences from that period. In particular, Roger Penrose has credited Ivor Robinson as an important early influence in the development of twistor theory, through his construction of so-called Robinson congruences.[3]
Overview
Mathematically, projective twistor space $\mathbb {PT} $ is a 3-dimensional complex manifold, complex projective 3-space $\mathbb {CP} ^{3}$. It has the physical interpretation of the space of massless particles with spin. It is the projectivisation of a 4-dimensional complex vector space, non-projective twistor space $\mathbb {T} $ with a Hermitian form of signature (2,2) and a holomorphic volume form. This can be most naturally understood as the space of chiral (Weyl) spinors for the conformal group $SO(4,2)/\mathbb {Z} _{2}$ of Minkowski space; it is the fundamental representation of the spin group $SU(2,2)$ of the conformal group. This definition can be extended to arbitrary dimensions except that beyond dimension four, one defines projective twistor space to be the space of projective pure spinors for the conformal group.[4][5]
In its original form, twistor theory encodes physical fields on Minkowski space into complex analytic objects (mathematical objects that can be studied using complex analysis) on twistor space via the Penrose transform. This is especially natural for massless fields of arbitrary spin. In the first instance these are obtained via contour integral formulae in terms of free holomorphic functions on regions in twistor space. The holomorphic twistor functions that give rise to solutions to the massless field equations can be more deeply understood as Čech representatives of analytic cohomology classes on regions in $\mathbb {PT} $. These correspondences have been extended to certain nonlinear fields, including self-dual gravity in Penrose's nonlinear graviton construction[6] and self-dual Yang–Mills fields in the so-called Ward construction;[7] the former gives rise to deformations of the underlying complex structure of regions in $\mathbb {PT} $, and the latter to certain holomorphic vector bundles over regions in $\mathbb {PT} $. These constructions have had wide applications, including inter alia the theory of integrable systems.[8][9][10]
The self-duality condition is a major limitation for incorporating the full nonlinearities of physical theories, although it does suffice for Yang–Mills–Higgs monopoles and instantons (see ADHM construction).[11] An early attempt to overcome this restriction was the introduction of ambitwistors by Edward Witten[12] and by Isenberg, Yasskin & Green.[13] Ambitwistor space is the space of complexified light rays or massless particles and can be regarded as a complexification or cotangent bundle of the original twistor description. These apply to general fields but the field equations are no longer so simply expressed.
Twistorial formulae for interactions beyond the self-dual sector first arose from Witten's twistor string theory.[14] This is a quantum theory of holomorphic maps of a Riemann surface into twistor space. It gave rise to the remarkably compact RSV (Roiban, Spradlin & Volovich) formulae for tree-level S-matrices of Yang–Mills theories,[15] but its gravity degrees of freedom gave rise to a version of conformal supergravity limiting its applicability; conformal gravity is an unphysical theory containing ghosts, but its interactions are combined with those of Yang–Mills theory in loop amplitudes calculated via twistor string theory.[16]
Despite its shortcomings, twistor string theory led to rapid developments in the study of scattering amplitudes. One was the so-called MHV formalism[17] loosely based on disconnected strings, but was given a more basic foundation in terms of a twistor action for full Yang–Mills theory in twistor space.[18] Another key development was the introduction of BCFW recursion.[19] This has a natural formulation in twistor space[20][21] that in turn led to remarkable formulations of scattering amplitudes in terms of Grassmann integral formulae[22][23] and polytopes.[24] These ideas have evolved more recently into the positive Grassmannian[25] and amplituhedron.
Twistor string theory was extended first by generalising the RSV Yang–Mills amplitude formula, and then by finding the underlying string theory. The extension to gravity was given by Cachazo & Skinner,[26] and formulated as a twistor string theory for maximal supergravity by David Skinner.[27] Analogous formulae were then found in all dimensions by Cachazo, He & Yuan for Yang–Mills theory and gravity[28] and subsequently for a variety of other theories.[29] They were then understood as string theories in ambitwistor space by Mason & Skinner[30] in a general framework that includes the original twistor string and extends to give a number of new models and formulae.[31][32][33] As string theories they have the same critical dimensions as conventional string theory; for example the type II supersymmetric versions are critical in ten dimensions and are equivalent to the full field theory of type II supergravities in ten dimensions (this is distinct from conventional string theories that also have a further infinite hierarchy of massive higher spin states that provide an ultraviolet completion). They extend to give formulae for loop amplitudes[34][35] and can be defined on curved backgrounds.[36]
The twistor correspondence
Denote Minkowski space by $M$, with coordinates $x^{a}=(t,x,y,z)$ and Lorentzian metric $\eta _{ab}$ signature $(1,3)$. Introduce 2-component spinor indices $A=0,1;\;A'=0',1',$ and set
$x^{AA'}={\frac {1}{\sqrt {2}}}{\begin{pmatrix}t-z&x+iy\\x-iy&t+z\end{pmatrix}}.$
Non-projective twistor space $\mathbb {T} $ is a four-dimensional complex vector space with coordinates denoted by $Z^{\alpha }=\left(\omega ^{A},\,\pi _{A'}\right)$ where $\omega ^{A}$ and $\pi _{A'}$ are two constant Weyl spinors. The hermitian form can be expressed by defining a complex conjugation from $\mathbb {T} $ to its dual $\mathbb {T} ^{*}$ by ${\bar {Z}}_{\alpha }=\left({\bar {\pi }}_{A},\,{\bar {\omega }}^{A'}\right)$ so that the Hermitian form can be expressed as
$Z^{\alpha }{\bar {Z}}_{\alpha }=\omega ^{A}{\bar {\pi }}_{A}+{\bar {\omega }}^{A'}\pi _{A'}.$
This together with the holomorphic volume form, $\varepsilon _{\alpha \beta \gamma \delta }Z^{\alpha }dZ^{\beta }\wedge dZ^{\gamma }\wedge dZ^{\delta }$ is invariant under the group SU(2,2), a quadruple cover of the conformal group C(1,3) of compactified Minkowski spacetime.
Points in Minkowski space are related to subspaces of twistor space through the incidence relation
$\omega ^{A}=ix^{AA'}\pi _{A'}.$
The incidence relation is preserved under an overall re-scaling of the twistor, so usually one works in projective twistor space $\mathbb {PT} ,$ which is isomorphic as a complex manifold to $\mathbb {CP} ^{3}$. A point $x\in M$ thereby determines a line $\mathbb {CP} ^{1}$ in $\mathbb {PT} $ parametrised by $\pi _{A'}.$ A twistor $Z^{\alpha }$ is easiest understood in space-time for complex values of the coordinates where it defines a totally null two-plane that is self-dual. Take $x$ to be real, then if $Z^{\alpha }{\bar {Z}}_{\alpha }$ vanishes, then $x$ lies on a light ray, whereas if $Z^{\alpha }{\bar {Z}}_{\alpha }$ is non-vanishing, there are no solutions, and indeed then $Z^{\alpha }$ corresponds to a massless particle with spin that are not localised in real space-time.
Variations
Supertwistors
Supertwistors are a supersymmetric extension of twistors introduced by Alan Ferber in 1978.[37] Non-projective twistor space is extended by fermionic coordinates where ${\mathcal {N}}$ is the number of supersymmetries so that a twistor is now given by $\left(\omega ^{A},\,\pi _{A'},\,\eta ^{i}\right),i=1,\ldots ,{\mathcal {N}}$ with $\eta ^{i}$ anticommuting. The super conformal group $SU(2,2|{\mathcal {N}})$ naturally acts on this space and a supersymmetric version of the Penrose transform takes cohomology classes on supertwistor space to massless supersymmetric multiplets on super Minkowski space. The ${\mathcal {N}}=4$ case provides the target for Penrose's original twistor string and the ${\mathcal {N}}=8$ case is that for Skinner's supergravity generalisation.
Hyperkähler manifolds
Hyperkähler manifolds of dimension $4k$ also admit a twistor correspondence with a twistor space of complex dimension $2k+1$.[38]
Palatial twistor theory
The nonlinear graviton construction encodes only anti-self-dual, i.e., left-handed fields.[6] A first step towards the problem of modifying twistor space so as to encode a general gravitational field is the encoding of right-handed fields. Infinitesimally, these are encoded in twistor functions or cohomology classes of homogeneity −6. The task of using such twistor functions in a fully nonlinear way so as to obtain a right-handed nonlinear graviton has been referred to as the (gravitational) googly problem (the word "googly" is a term used in the game of cricket for a ball bowled with right-handed helicity using the apparent action that would normally give rise to left-handed helicity).[39] The most recent proposal in this direction by Penrose in 2015 was based on noncommutative geometry on twistor space and referred to as palatial twistor theory.[40] The theory is named after Buckingham Palace, where Michael Atiyah suggested to Penrose the use of a type of "noncommutative algebra", an important component of the theory (the underlying twistor structure in palatial twistor theory was modeled not on the twistor space but on the non-commutative holomorphic twistor quantum algebra).[41]
See also
• Background independence
• Complex spacetime
• History of loop quantum gravity
• Robinson congruences
• Spin network
Notes
1. Penrose, R. (1967). "Twistor Algebra". Journal of Mathematical Physics. 8 (2): 345–366. Bibcode:1967JMP.....8..345P. doi:10.1063/1.1705200.
2. Penrose, R.; MacCallum, M.A.H. (1973). "Twistor theory: An approach to the quantisation of fields and space-time". Physics Reports. 6 (4): 241–315. Bibcode:1973PhR.....6..241P. doi:10.1016/0370-1573(73)90008-2.
3. Penrose, Roger (1987). "On the Origins of Twistor Theory". In Rindler, Wolfgang; Trautman, Andrzej (eds.). Gravitation and Geometry, a Volume in Honour of Ivor Robinson. Bibliopolis. ISBN 88-7088-142-3.
4. Penrose, Roger; Rindler, Wolfgang (1986). Spinors and Space-Time. Cambridge University Press. pp. Appendix. doi:10.1017/cbo9780511524486. ISBN 9780521252676.
5. Hughston, L. P.; Mason, L. J. (1988). "A generalised Kerr-Robinson theorem". Classical and Quantum Gravity. 5 (2): 275. Bibcode:1988CQGra...5..275H. doi:10.1088/0264-9381/5/2/007. ISSN 0264-9381. S2CID 250783071.
6. Penrose, R (1976). "Non-linear gravitons and curved twistor theory". Gen. Rel. Grav. 7 (1): 31–52. Bibcode:1976GReGr...7...31P. doi:10.1007/BF00762011. S2CID 123258136.
7. Ward, R. S. (1977). "On self-dual gauge fields". Physics Letters A. 61 (2): 81–82. Bibcode:1977PhLA...61...81W. doi:10.1016/0375-9601(77)90842-8.
8. Ward, R. S. (1990). Twistor geometry and field theory. Wells, R. O. (Raymond O'Neil), 1940-. Cambridge [England]: Cambridge University Press. ISBN 978-0521422680. OCLC 17260289.
9. Mason, Lionel J; Woodhouse, Nicholas M J (1996). Integrability, self-duality, and twistor theory. Oxford: Clarendon Press. ISBN 9780198534983. OCLC 34545252.
10. Dunajski, Maciej (2010). Solitons, instantons, and twistors. Oxford: Oxford University Press. ISBN 9780198570622. OCLC 507435856.
11. Atiyah, M.F.; Hitchin, N.J.; Drinfeld, V.G.; Manin, Yu.I. (1978). "Construction of instantons". Physics Letters A. 65 (3): 185–187. Bibcode:1978PhLA...65..185A. doi:10.1016/0375-9601(78)90141-x.
12. Witten, Edward (1978). "An interpretation of classical Yang–Mills theory". Physics Letters B. 77 (4–5): 394–398. Bibcode:1978PhLB...77..394W. doi:10.1016/0370-2693(78)90585-3.
13. Isenberg, James; Yasskin, Philip B.; Green, Paul S. (1978). "Non-self-dual gauge fields". Physics Letters B. 78 (4): 462–464. Bibcode:1978PhLB...78..462I. doi:10.1016/0370-2693(78)90486-0.
14. Witten, Edward (6 October 2004). "Perturbative Gauge Theory as a String Theory in Twistor Space". Communications in Mathematical Physics. 252 (1–3): 189–258. arXiv:hep-th/0312171. Bibcode:2004CMaPh.252..189W. doi:10.1007/s00220-004-1187-3. S2CID 14300396.
15. Roiban, Radu; Spradlin, Marcus; Volovich, Anastasia (2004-07-30). "Tree-level S matrix of Yang–Mills theory". Physical Review D. 70 (2): 026009. arXiv:hep-th/0403190. Bibcode:2004PhRvD..70b6009R. doi:10.1103/PhysRevD.70.026009. S2CID 10561912.
16. Berkovits, Nathan; Witten, Edward (2004). "Conformal supergravity in twistor-string theory". Journal of High Energy Physics. 2004 (8): 009. arXiv:hep-th/0406051. Bibcode:2004JHEP...08..009B. doi:10.1088/1126-6708/2004/08/009. ISSN 1126-6708. S2CID 119073647.
17. Cachazo, Freddy; Svrcek, Peter; Witten, Edward (2004). "MHV vertices and tree amplitudes in gauge theory". Journal of High Energy Physics. 2004 (9): 006. arXiv:hep-th/0403047. Bibcode:2004JHEP...09..006C. doi:10.1088/1126-6708/2004/09/006. ISSN 1126-6708. S2CID 16328643.
18. Adamo, Tim; Bullimore, Mathew; Mason, Lionel; Skinner, David (2011). "Scattering amplitudes and Wilson loops in twistor space". Journal of Physics A: Mathematical and Theoretical. 44 (45): 454008. arXiv:1104.2890. Bibcode:2011JPhA...44S4008A. doi:10.1088/1751-8113/44/45/454008. S2CID 59150535.
19. Britto, Ruth; Cachazo, Freddy; Feng, Bo; Witten, Edward (2005-05-10). "Direct Proof of the Tree-Level Scattering Amplitude Recursion Relation in Yang–Mills Theory". Physical Review Letters. 94 (18): 181602. arXiv:hep-th/0501052. Bibcode:2005PhRvL..94r1602B. doi:10.1103/PhysRevLett.94.181602. PMID 15904356. S2CID 10180346.
20. Mason, Lionel; Skinner, David (2010-01-01). "Scattering amplitudes and BCFW recursion in twistor space". Journal of High Energy Physics. 2010 (1): 64. arXiv:0903.2083. Bibcode:2010JHEP...01..064M. doi:10.1007/JHEP01(2010)064. ISSN 1029-8479. S2CID 8543696.
21. Arkani-Hamed, N.; Cachazo, F.; Cheung, C.; Kaplan, J. (2010-03-01). "The S-matrix in twistor space". Journal of High Energy Physics. 2010 (3): 110. arXiv:0903.2110. Bibcode:2010JHEP...03..110A. doi:10.1007/JHEP03(2010)110. ISSN 1029-8479. S2CID 15898218.
22. Arkani-Hamed, N.; Cachazo, F.; Cheung, C.; Kaplan, J. (2010-03-01). "A duality for the S matrix". Journal of High Energy Physics. 2010 (3): 20. arXiv:0907.5418. Bibcode:2010JHEP...03..020A. doi:10.1007/JHEP03(2010)020. ISSN 1029-8479. S2CID 5771375.
23. Mason, Lionel; Skinner, David (2009). "Dual superconformal invariance, momentum twistors and Grassmannians". Journal of High Energy Physics. 2009 (11): 045. arXiv:0909.0250. Bibcode:2009JHEP...11..045M. doi:10.1088/1126-6708/2009/11/045. ISSN 1126-6708. S2CID 8375814.
24. Hodges, Andrew (2013-05-01). "Eliminating spurious poles from gauge-theoretic amplitudes". Journal of High Energy Physics. 2013 (5): 135. arXiv:0905.1473. Bibcode:2013JHEP...05..135H. doi:10.1007/JHEP05(2013)135. ISSN 1029-8479. S2CID 18360641.
25. Arkani-Hamed, Nima; Bourjaily, Jacob L.; Cachazo, Freddy; Goncharov, Alexander B.; Postnikov, Alexander; Trnka, Jaroslav (2012-12-21). "Scattering Amplitudes and the Positive Grassmannian". arXiv:1212.5605 [hep-th].
26. Cachazo, Freddy; Skinner, David (2013-04-16). "Gravity from Rational Curves in Twistor Space". Physical Review Letters. 110 (16): 161301. arXiv:1207.0741. Bibcode:2013PhRvL.110p1301C. doi:10.1103/PhysRevLett.110.161301. PMID 23679592. S2CID 7452729.
27. Skinner, David (2013-01-04). "Twistor Strings for N=8 Supergravity". arXiv:1301.0868 [hep-th].
28. Cachazo, Freddy; He, Song; Yuan, Ellis Ye (2014-07-01). "Scattering of massless particles: scalars, gluons and gravitons". Journal of High Energy Physics. 2014 (7): 33. arXiv:1309.0885. Bibcode:2014JHEP...07..033C. doi:10.1007/JHEP07(2014)033. ISSN 1029-8479. S2CID 53685436.
29. Cachazo, Freddy; He, Song; Yuan, Ellis Ye (2015-07-01). "Scattering equations and matrices: from Einstein to Yang–Mills, DBI and NLSM". Journal of High Energy Physics. 2015 (7): 149. arXiv:1412.3479. Bibcode:2015JHEP...07..149C. doi:10.1007/JHEP07(2015)149. ISSN 1029-8479. S2CID 54062406.
30. Mason, Lionel; Skinner, David (2014-07-01). "Ambitwistor strings and the scattering equations". Journal of High Energy Physics. 2014 (7): 48. arXiv:1311.2564. Bibcode:2014JHEP...07..048M. doi:10.1007/JHEP07(2014)048. ISSN 1029-8479. S2CID 53666173.
31. Berkovits, Nathan (2014-03-01). "Infinite tension limit of the pure spinor superstring". Journal of High Energy Physics. 2014 (3): 17. arXiv:1311.4156. Bibcode:2014JHEP...03..017B. doi:10.1007/JHEP03(2014)017. ISSN 1029-8479. S2CID 28346354.
32. Geyer, Yvonne; Lipstein, Arthur E.; Mason, Lionel (2014-08-19). "Ambitwistor Strings in Four Dimensions". Physical Review Letters. 113 (8): 081602. arXiv:1404.6219. Bibcode:2014PhRvL.113h1602G. doi:10.1103/PhysRevLett.113.081602. PMID 25192087. S2CID 40855791.
33. Casali, Eduardo; Geyer, Yvonne; Mason, Lionel; Monteiro, Ricardo; Roehrig, Kai A. (2015-11-01). "New ambitwistor string theories". Journal of High Energy Physics. 2015 (11): 38. arXiv:1506.08771. Bibcode:2015JHEP...11..038C. doi:10.1007/JHEP11(2015)038. ISSN 1029-8479. S2CID 118801547.
34. Adamo, Tim; Casali, Eduardo; Skinner, David (2014-04-01). "Ambitwistor strings and the scattering equations at one loop". Journal of High Energy Physics. 2014 (4): 104. arXiv:1312.3828. Bibcode:2014JHEP...04..104A. doi:10.1007/JHEP04(2014)104. ISSN 1029-8479. S2CID 119194796.
35. Geyer, Yvonne; Mason, Lionel; Monteiro, Ricardo; Tourkine, Piotr (2015-09-16). "Loop Integrands for Scattering Amplitudes from the Riemann Sphere". Physical Review Letters. 115 (12): 121603. arXiv:1507.00321. Bibcode:2015PhRvL.115l1603G. doi:10.1103/PhysRevLett.115.121603. PMID 26430983. S2CID 36625491.
36. Adamo, Tim; Casali, Eduardo; Skinner, David (2015-02-01). "A worldsheet theory for supergravity". Journal of High Energy Physics. 2015 (2): 116. arXiv:1409.5656. Bibcode:2015JHEP...02..116A. doi:10.1007/JHEP02(2015)116. ISSN 1029-8479. S2CID 119234027.
37. Ferber, A. (1978), "Supertwistors and conformal supersymmetry", Nuclear Physics B, 132 (1): 55–64, Bibcode:1978NuPhB.132...55F, doi:10.1016/0550-3213(78)90257-2.
38. Hitchin, N. J.; Karlhede, A.; Lindström, U.; Roček, M. (1987). "Hyper-Kähler metrics and supersymmetry". Communications in Mathematical Physics. 108 (4): 535–589. Bibcode:1987CMaPh.108..535H. doi:10.1007/BF01214418. ISSN 0010-3616. MR 0877637. S2CID 120041594.
39. Penrose 2004, p. 1000.
40. Penrose, Roger (2015). "Palatial twistor theory and the twistor googly problem". Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences. 373 (2047): 20140237. Bibcode:2015RSPTA.37340237P. doi:10.1098/rsta.2014.0237. PMID 26124255. S2CID 13038470.
41. "Michael Atiyah's Imaginative State of Mind" – Quanta Magazine.
References
• Roger Penrose (2004), The Road to Reality, Alfred A. Knopf, ch. 33, pp. 958–1009.
• Roger Penrose and Wolfgang Rindler (1984), Spinors and Space-Time; vol. 1, Two-Spinor Calculus and Relativitic Fields, Cambridge University Press, Cambridge.
• Roger Penrose and Wolfgang Rindler (1986), Spinors and Space-Time; vol. 2, Spinor and Twistor Methods in Space-Time Geometry, Cambridge University Press, Cambridge.
Further reading
• Atiyah, Michael; Dunajski, Maciej; Mason, Lionel J. (2017). "Twistor theory at fifty: from contour integrals to twistor strings". Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences. 473 (2206): 20170530. arXiv:1704.07464. Bibcode:2017RSPSA.47370530A. doi:10.1098/rspa.2017.0530. PMC 5666237. PMID 29118667. S2CID 5735524.
• Baird, P., "An Introduction to Twistors."
• Huggett, S. and Tod, K. P. (1994). An Introduction to Twistor Theory, second edition. Cambridge University Press. ISBN 9780521456890. OCLC 831625586.
• Hughston, L. P. (1979) Twistors and Particles. Springer Lecture Notes in Physics 97, Springer-Verlag. ISBN 978-3-540-09244-5.
• Hughston, L. P. and Ward, R. S., eds (1979) Advances in Twistor Theory. Pitman. ISBN 0-273-08448-8.
• Mason, L. J. and Hughston, L. P., eds (1990) Further Advances in Twistor Theory, Volume I: The Penrose Transform and its Applications. Pitman Research Notes in Mathematics Series 231, Longman Scientific and Technical. ISBN 0-582-00466-7.
• Mason, L. J., Hughston, L. P., and Kobak, P. K., eds (1995) Further Advances in Twistor Theory, Volume II: Integrable Systems, Conformal Geometry, and Gravitation. Pitman Research Notes in Mathematics Series 232, Longman Scientific and Technical. ISBN 0-582-00465-9.
• Mason, L. J., Hughston, L. P., Kobak, P. K., and Pulverer, K., eds (2001) Further Advances in Twistor Theory, Volume III: Curved Twistor Spaces. Research Notes in Mathematics 424, Chapman and Hall/CRC. ISBN 1-58488-047-3.
• Penrose, Roger (1967), "Twistor Algebra", Journal of Mathematical Physics, 8 (2): 345–366, Bibcode:1967JMP.....8..345P, doi:10.1063/1.1705200, MR 0216828, archived from the original on 2013-01-12
• Penrose, Roger (1968), "Twistor Quantisation and Curved Space-time", International Journal of Theoretical Physics, 1 (1): 61–99, Bibcode:1968IJTP....1...61P, doi:10.1007/BF00668831, S2CID 123628735
• Penrose, Roger (1969), "Solutions of the Zero‐Rest‐Mass Equations", Journal of Mathematical Physics, 10 (1): 38–39, Bibcode:1969JMP....10...38P, doi:10.1063/1.1664756, archived from the original on 2013-01-12
• Penrose, Roger (1977), "The Twistor Programme", Reports on Mathematical Physics, 12 (1): 65–76, Bibcode:1977RpMP...12...65P, doi:10.1016/0034-4877(77)90047-7, MR 0465032
• Penrose, Roger (1999). "The Central Programme of Twistor Theory". Chaos, Solitons and Fractals. 10 (2–3): 581–611. Bibcode:1999CSF....10..581P. doi:10.1016/S0960-0779(98)00333-6.
• Witten, Edward (2004), "Perturbative Gauge Theory as a String Theory in Twistor Space", Communications in Mathematical Physics, 252 (1–3): 189–258, arXiv:hep-th/0312171, Bibcode:2004CMaPh.252..189W, doi:10.1007/s00220-004-1187-3, S2CID 14300396
External links
• Penrose, Roger (1999), "Einstein's Equation and Twistor Theory: Recent Developments"
• Penrose, Roger; Hadrovich, Fedja. "Twistor Theory."
• Hadrovich, Fedja, "Twistor Primer."
• Penrose, Roger. "On the Origins of Twistor Theory."
• Jozsa, Richard (1976), "Applications of Sheaf Cohomology in Twistor Theory."
• Dunajski, Maciej (2009). "Twistor Theory and Differential Equations". J. Phys. A: Math. Theor. 42 (40): 404004. arXiv:0902.0274. Bibcode:2009JPhA...42N4004D. doi:10.1088/1751-8113/42/40/404004. S2CID 62774126.
• Andrew Hodges, Summary of recent developments.
• Huggett, Stephen (2005), "The Elements of Twistor Theory."
• Mason, L. J., "The twistor programme and twistor strings: From twistor strings to quantum gravity?"
• Sämann, Christian (2006). Aspects of Twistor Geometry and Supersymmetric Field Theories within Superstring Theory (PhD). Universit ̈at Hannover. arXiv:hep-th/0603098.
• Sparling, George (1999), "On Time Asymmetry."
• Spradlin, Marcus (2012). "Progress and Prospects in Twistor String Theory" (PDF). hdl:11299/130081.
• MathWorld: Twistors.
• Universe Review: "Twistor Theory."
• Twistor newsletter archives.
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Twists of elliptic curves
In the mathematical field of algebraic geometry, an elliptic curve E over a field K has an associated quadratic twist, that is another elliptic curve which is isomorphic to E over an algebraic closure of K. In particular, an isomorphism between elliptic curves is an isogeny of degree 1, that is an invertible isogeny. Some curves have higher order twists such as cubic and quartic twists. The curve and its twists have the same j-invariant.
Applications of twists include cryptography,[1] the solution of Diophantine equations,[2][3] and when generalized to hyperelliptic curves, the study of the Sato–Tate conjecture.[4]
Quadratic twist
First assume $K$ is a field of characteristic different from 2. Let $E$ be an elliptic curve over $K$ of the form:
$y^{2}=x^{3}+a_{2}x^{2}+a_{4}x+a_{6}.\,$
Given $d\neq 0$ not a square in $K$, the quadratic twist of $E$ is the curve $E^{d}$, defined by the equation:
$dy^{2}=x^{3}+a_{2}x^{2}+a_{4}x+a_{6}.\,$
or equivalently
$y^{2}=x^{3}+da_{2}x^{2}+d^{2}a_{4}x+d^{3}a_{6}.\,$
The two elliptic curves $E$ and $E^{d}$ are not isomorphic over $K$, but rather over the field extension $K({\sqrt {d}})$. Qualitatively speaking, the arithmetic of a curve and its quadratic twist can look very different in the field $K$, while the complex analysis of the curves is the same; and so a family of curves related by twisting becomes a useful setting in which to study the arithmetic properties of elliptic curves.[5]
Twists can also be defined when the base field $K$ is of characteristic 2. Let $E$ be an elliptic curve over $K$ of the form:
$y^{2}+a_{1}xy+a_{3}y=x^{3}+a_{2}x^{2}+a_{4}x+a_{6}.\,$
Given $d\in K$ such that $X^{2}+X+d$ is an irreducible polynomial over $K$, the quadratic twist of $E$ is the curve $E^{d}$, defined by the equation:
$y^{2}+a_{1}xy+a_{3}y=x^{3}+(a_{2}+da_{1}^{2})x^{2}+a_{4}x+a_{6}+da_{3}^{2}.\,$
The two elliptic curves $E$ and $E^{d}$ are not isomorphic over $K$, but over the field extension $K[X]/(X^{2}+X+d)$.
Quadratic twist over finite fields
If $K$ is a finite field with $q$ elements, then for all $x$ there exist a $y$ such that the point $(x,y)$ belongs to either $E$ or $E^{d}$. In fact, if $(x,y)$ is on just one of the curves, there is exactly one other $y'$ on that same curve (which can happen if the characteristic is not $2$).
As a consequence, $|E(K)|+|E^{d}(K)|=2q+2$ or equivalently $t_{E^{d}}=-t_{E}$, where $t_{E}$ is the trace of the Frobenius endomorphism of the curve.
Quartic twist
It is possible to "twist" elliptic curves with j-invariant equal to 1728 by quartic characters;[6] twisting a curve $E$ by a quartic twist, one obtains precisely four curves: one is isomorphic to $E$, one is its quadratic twist, and only the other two are really new. Also in this case, twisted curves are isomorphic over the field extension given by the twist degree.
Cubic twist
Analogously to the quartic twist case, an elliptic curve over $K$ with j-invariant equal to zero can be twisted by cubic characters. The curves obtained are isomorphic to the starting curve over the field extension given by the twist degree.
Generalization
Twists can be defined for other smooth projective curves as well. Let $K$ be a field and $C$ be curve over that field, i.e., a projective variety of dimension 1 over $K$ that is irreducible and geometrically connected. Then a twist $C'$ of $C$ is another smooth projective curve for which there exists a ${\bar {K}}$-isomorphism between $C'$ and $C$, where the field ${\bar {K}}$ is the algebraic closure of $K$.[4]
Examples
• Twisted Hessian curves
• Twisted Edwards curve
• Twisted tripling-oriented Doche–Icart–Kohel curve
References
1. Bos, Joppe W.; Halderman, J. Alex; Heninger, Nadia; Moore, Jonathan; Naehrig, Michael; Wustrow, Eric (2014). "Elliptic Curve Cryptography in Practice". In Christin, Nicolas; Safavi-Naini, Reihaneh (eds.). Financial Cryptography and Data Security. Vol. 8437. Berlin, Heidelberg: Springer. pp. 157–175. doi:10.1007/978-3-662-45472-5_11. ISBN 978-3-662-45471-8. Retrieved 2022-04-10.
2. Mazur, B.; Rubin, K. (September 2010). "Ranks of twists of elliptic curves and Hilbert's tenth problem". Inventiones Mathematicae. 181 (3): 541–575. arXiv:0904.3709. Bibcode:2010InMat.181..541M. doi:10.1007/s00222-010-0252-0. ISSN 0020-9910. S2CID 3394387.
3. Poonen, Bjorn; Schaefer, Edward F.; Stoll, Michael (2007-03-15). "Twists of X(7) and primitive solutions to x2+y3=z7". Duke Mathematical Journal. 137 (1). arXiv:math/0508174. doi:10.1215/S0012-7094-07-13714-1. ISSN 0012-7094. S2CID 2326034.
4. Lombardo, Davide; Lorenzo García, Elisa (February 2019). "Computing twists of hyperelliptic curves". Journal of Algebra. 519: 474–490. arXiv:1611.04856. Bibcode:2016arXiv161104856L. doi:10.1016/j.jalgebra.2018.08.035. S2CID 119143097.
5. Rubin, Karl; Silverberg, Alice (2002-07-08). "Ranks of elliptic curves". Bulletin of the American Mathematical Society. 39 (4): 455–474. doi:10.1090/S0273-0979-02-00952-7. ISSN 0273-0979. MR 1920278.
6. Gouvêa, F.; Mazur, B. (1991). "The square-free sieve and the rank of elliptic curves" (PDF). Journal of the American Mathematical Society. 4 (1): 1–23. doi:10.1090/S0894-0347-1991-1080648-7. JSTOR 2939253.
• P. Stevenhagen (2008). Elliptic Curves (PDF). Universiteit Leiden.
• C. L. Stewart and J. Top (October 1995). "On Ranks of Twists of Elliptic Curves and Power-Free Values of Binary Forms". Journal of the American Mathematical Society. 8 (4): 943–973. doi:10.1090/S0894-0347-1995-1290234-5. JSTOR 2152834.
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Two-Sided Matching
Two-Sided Matching: A Study in Game-Theoretic Modeling and Analysis is a book on matching markets in economics and game theory, particularly concentrating on the stable marriage problem. It was written by Alvin E. Roth and Marilda Sotomayor, with a preface by Robert Aumann,[1][2] and published in 1990 by the Cambridge University Press as volume 18 in their series of Econometric Society monographs.[3] For this work, Roth and Sotomayor won the 1990 Frederick W. Lanchester Prize of the Institute for Operations Research and the Management Sciences.[4]
Two-Sided Matching: A Study in Game-Theoretic Modeling and Analysis
Author
• Alvin E. Roth
• Marilda Sotomayor
SeriesEconometric Society monographs
SubjectMatching markets
PublisherCambridge University Press
Publication date
1990
Topics
The book's introduction discusses the National Resident Matching Program and its use of stable marriage to assign medical students to hospital positions, and collects the problems in economics that the theory of matching markets is positioned to solve. Following this, it has three main sections.[2][4][5]
The first of these sections discusses the stable matching problem in its simplest form, in which two equal-sized groups of agents are to be matched one-to-one. It discusses the stability of solutions (the property that no pair of agents both prefer being matched to each other to their assigned matches), the lattice of stable matchings, the Gale–Shapley algorithm for finding stable solutions, and two key properties of this algorithm: that among all stable solutions it chooses the one that gives one group of agents their most-preferred stable match, and that it is an honest mechanism that incentivizes this group of agents to report their preferences truthfully.[4][5]
The second part of the book, which reviewer Ulrich Kamecke describes as its most central, concerns extensions of these results to the many-one matching needed for the National Resident Matching Program, and to the specific economic factors that made that program successful compared to comparable programs elsewhere, and that have impeded its success. One example concerns the two-body problem of married couples who would both prefer to be assigned to the same place, a constraint that adds considerable complexity to the matching problem and may prevent a stable solution from existing.[1][4]
The third part of the book concerns a different direction in which these ideas have been extended, to matching markets such as those for real estate in which indivisible goods are traded, with money used to transfer utility. It includes results in auction theory, linear and nonlinear utility functions, and the assignment game of Lloyd Shapley and Martin Shubik.[4][5][6]
Audience and reception
Two-Sided Matching presents known material on its topics, rather than introducing new research, but it is not a textbook. Instead, its aim is to provide a survey of this area aimed at economic practitioners, with arguments for the importance of its material based on its pragmatic significance rather than its mathematical beauty. Nevertheless, it also has material of interest to researchers, including an extensive bibliography and a concluding list of open problems for future research.[4] Compared to other books on stable matching, including Marriages Stables by Donald Knuth and The Stable Marriage Problem: Structure and Algorithms by Dan Gusfield and Robert W. Irving, Two-Sided Matching focuses much more on the economic, application-specific, and strategic issues of stable matching, and much less on its algorithmic issues.[2]
Alan Kirman calls the book a "clear and elegant account" of its material, writing that its focus on practical applications makes it "of particular interest".[7] Theodore Bergstrom writes that it will also "delight economists who want to think beautiful thoughts about important practical problems".[1] Benny Moldovanu predicts that it "will become the standard source of reference" for its material.[8] And Uriel Rothblum calls it the kind of once-a-generation book that can "change the way in which an entire field of study is viewed."[2]
References
1. Bergstrom, Theodore C. (June 1992), "Review of Two-Sided Matching", Journal of Economic Literature, 30 (2): 896–898, JSTOR 2727713
2. Rothblum, Uriel G. (January 1992), "Review of Two-Sided Matching", Games and Economic Behavior, 4 (1): 161–165, doi:10.1016/0899-8256(92)90011-g
3. Wieczorek, A., "Review of Two-Sided Matching", zbMATH, Zbl 0726.90003
4. Kamecke, Ulrich (November 1992), "Review of Two-Sided Matching", Economica, New Series, 59 (236): 487–489, doi:10.2307/2554894, JSTOR 2554894
5. Potters, Jos (1993), "Review of Two-Sided Matching", Mathematical Reviews, MR 1119308
6. Winters, Jan Kees (October 1992), "Review of Two-Sided Matching", European Journal of Political Economy, 8 (3): 510–514, doi:10.1016/0176-2680(92)90017-b
7. Kirman, Alan P. (July 1992), "Review of Two-Sided Matching", The Economic Journal, 102 (413): 975–976, doi:10.2307/2234601, JSTOR 2234601
8. Moldovanu, B. (January 1992), "Review of Two-Sided Matching", Journal of Economics, 55: 116–117, ProQuest 1299512649
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