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README.md

@@ -144,39 +144,33 @@ $$
   \ge -1 \quad \text{for all } \mathbf{x} \in \nabla \;.
 $$
 
-This relationship is symmetric: the dual of the dual of a polytope is the polytope itself,
-i.e., \\( \nabla^{**} = \nabla \\).
+This relationship is symmetric: the dual of the dual of an IP polytope is the polytope
+itself, i.e., \\( \nabla^{**} = \nabla \\).
 
 ### Weight Systems
 
-Weight systems provide a means to describe simple polytopes known as *simplexes*. More
-broadly, *combined weight systems*, which are collections of individual weight systems,
-can describe any polytope. A combined weight system is a matrix consisting of real
-numbers. The construction process is outlined as follows:
+Weight systems provide a means to describe simple polytopes known as *simplices*. A weight
+system is a tuple of real numbers. The construction process is outlined as follows:
 
-Consider a polytope in \\(\mathbb{R}^n\\) with vertex count \\(k\\), where \\(k\\) is
-bigger than \\(n\\). It is possible to position \\(n\\) of these vertices at arbitrary
-(linearly independent) locations through a linear transformation. The placement of the
-remaining \\(k - n\\) vertices is then determined. Their positions are the defining
-properties of a polytope. To specify these positions independently of the applied linear
-transformation, one can use the following system of equations. If \\(\mathbf{v}_0,
-\mathbf{v}_1, \dots \mathbf{v}_{k-1}\\) are the vertices of the polytope, these relations
-fix \\(k - n\\) vertices in terms of the other \\(n\\):
+Consider an \\(n\\)-dimensional simplex in \\(\mathbb{R}^n\\), i.e., a polytope in
+\\(\mathbb{R}^n\\) with vertex count \\(n + 1\\) and \\(n\\) of its edges extending in
+linearly independent directions. It is possible to position \\(n\\) of its vertices at
+arbitrary (linearly independent) locations through a linear transformation. The placement
+of the remaining vertex is then determined. Its position is the defining property of the
+simplex. To specify the position independently of the applied linear transformation, one
+can use the following equation. If \\(\mathbf{v}_0, \mathbf{v}_1, \dots, \mathbf{v}_n\\)
+are the vertices of the simplex, this relation fixes one vertex in terms of the other
+\\(n\\):
 
-$$
-  \sum_{i=0}^{k-1} q_i^{(j)} \mathbf{v}_i
-  = 0 \quad \text{for } 0 \le j \le k - n - 1 \;,
-$$
-
-where \\(q_i^{(j)}\\) is the matrix of real numbers, the combined weight system. In cases
-where \\(k = n + 1\\), \\(j\\) is limited to the value zero, reducing the matrix to a
-single weight system \\(q_i\\). In this scenario, the polytope is a simplex, and the
-equation simplifies to:
+$$ \sum_{i=0}^n q_i \mathbf{v}_i = 0 \;, $$
 
-$$ \sum_{i=0}^n q_i \mathbf{v}_i = 0 \;. $$
+where \\(q_i\\) is the tuple of real numbers, the weight system.
 
 It is important to note that scaling all weights in a weight system by a common factor
-results in an equivalent weight system that defines the same polytope.
+results in an equivalent weight system that defines the same simplex.
+
+The condition that a simplex is an IP simplex is equivalent to the condition that all
+weights in its weight system are bigger than zero.
 
 For this dataset, the focus is on a specific construction of lattice polytopes described
 in subsequent sections.
@@ -203,16 +197,14 @@ The weights of a lattice polytope are always rational. This characteristic enabl
 rescaling of a weight system so that its weights become integers without any common
 divisor. This rescaling has been performed in this dataset.
 
-Typically, the dual of a lattice polytope defined by a weight system is not a lattice
-polytope. However, our interest lies in a different construction than simply considering
-polytopes defined by (combined) weight systems, as described above. In this construction,
-they are just the starting point. We start with the polytope \\(\nabla\\), arising from a
-weight system as previously described. Then, we define the polytope \\(\Delta\\) as the
-convex hull of the intersection of \\(\nabla^*\\) with the points of the dual lattice. In
-the context of this dataset, the polytope \\(\Delta\\) is referred to as ‘the polytope’.
-Correspondingly, \\(\Delta^{\!*}\\) is referred to as ‘the dual polytope’. The lattice of
-\\(\Delta\\) is taken to be the coarsest lattice possible, such that \\(\nabla\\) is a
-lattice polytope, i.e., the lattice generated by the vertices of \\(\nabla\\). This
+The construction of the lattice polytopes from this dataset works as follows: We start
+with the simplex \\(\nabla\\), arising from a weight system as previously described. Then,
+we define the polytope \\(\Delta\\) as the convex hull of the intersection of
+\\(\nabla^*\\) with the points of the dual lattice. In the context of this dataset, the
+polytope \\(\Delta\\) is referred to as ‘the polytope’. Correspondingly,
+\\(\Delta^{\!*}\\) is referred to as ‘the dual polytope’. The lattice of \\(\nabla\\) and
+\\(\Delta^{\!*}\\) is taken to be the coarsest lattice possible, such that \\(\nabla\\) is
+a lattice polytope, i.e., the lattice generated by the vertices of \\(\nabla\\). This
 construction is exemplified in the following sections.
 
 A weight system is considered an IP weight system if the corresponding \\(\Delta\\) is an
@@ -254,8 +246,11 @@ by dots.
 
 One may notice that a simpler description could be obtained by fixing \\(\mathbf{v}_2 =
 (1, 0)\\) instead of \\(\mathbf{v}_0\\), which would avoid fractional vertex coordinates.
-However, this approach would not illustrate the general case in higher dimensions, where
-this is not possible since there is not always a weight equal to 1.
+However, this approach would not illustrate the construction of the lattice. This is
+because, in this scenario, the lattice points would invariably align with points having
+integer coordinates. In practice, coordinates are often chosen so that lattice points
+correspond to those with integer coordinates. In higher dimensions, this is not trivial,
+as a weight with a value of one is not always present in a weight system.
 
 ### General Dimension
 
@@ -276,3 +271,7 @@ The counts of reflexive single-weight-system polytopes by dimension \\(n\\) are:
 |       3 |                                       95 |
 |       4 |                                  184,026 |
 |       5 |           (this dataset) 185,269,499,015 |
+
+One should note that distinct weight systems may well lead to the same polytope (we have
+not checked how often this occurs). In particular it seems that polytopes with a small
+number of lattice points are generated many times.