# Isogonal polytope

An isogonal polytope or vertex-transitive polytope is a polytope whose vertices are identical under its highest symmetry group. In an isogonal polytope, there is a singular vertex figure, and all of the vertices lie on a hypersphere. The dual of an isogonal polytope is an isotopic polytope, which are made out of one facet type. All regular, uniform and scaliform polytopes are isogonal. However, not all isogonal polytopes can be made equilateral, such as the snub decachoron.

If an isogonal polytope is also isotopic, it is called a noble polytope. Self-dual isogonal polytopes are also noble.

## Optimization

Most isogonal polytopes can be optimized in the sense of minimizing the edge length variations. However, some isogonal polytopes cannot be optimized in any meaningful way, such as the antiditetragoltriates, because their optimized forms are not topologically identical to these shapes. If an isogonal polytope has no variations in its highest symmetry, such as the triangular duoantiprism, it is considered to be optimized.

There are two methods of optimization: the absolute-value method and the ratio method, which may yield different solutions. Both methods rely on a set variable, defined by a constant. All regular, uniform and scaliform polytopes are optimized polytopes in both of the two methods.

### Absolute-value method

This method uses all possible sums of absolute values of differences between two edge length types, and then computing the minimum value of the resulting function. The number of sums required for an isogonal polytope with n edge types is n(n-1)/2. The feasibility of this method depends on the number of variables and the number of edge lengths.

As an example, a truncated octahedron in BC3 symmetry can be defined by all permutations and sign changes of (0, a, b), where a and b are nonzero and distinct from each other. The variable a can be set to a constant value such as 1, yielding (0, 1, b). The two edge lengths are given by:

• d1 = 2 (distance between (0, 1, b) and (1, 0, b))
• d2 = 2|b-1| (distance between (0, 1, b) and (0, b, 1))

Since there are only two edge lengths, only one absolute value sum is needed: f(x) = |2-2|b-1||. The minimum of this function is attained when b is equal to 2, which evaluates to 0. Therefore, any optimized truncated octahedron in this sense is equilateral and hence uniform.

### Ratio method

This method is dependent on the polytope's largest and smallest edge lengths. It involves dividing the largest edge length by the smallest edge length, giving a value equal to or greater than 1, and then finding the lowest possible value. Due to the unpredictability of edge lengths, a convenient method is to enumerate all divisions between two edge types and their reciprocals, and then finding the lowest possible value above (or equal to) 1 that is not within the "area" of the functions. For an isogonal polytope with n edge lengths, n(n-1) divisions are required. For topologically similar isogonal polytopes, the largest and smallest edge lengths may vary between edge length types depending on the variable used.

We can use the truncated octahedron example from earlier. In this case the two edge lengths can be interpreted as functions. For b > 2, 2|b-1| is larger than 2, and for 1 < b < 2, 2|b-1| is smaller than 2. Dividing the appropriate edge lengths results in a minimum ratio of 1:1 at b = 2, implying that any optimized truncated octahedron in this sense is equilateral and hence uniform.

It is more useful in most cases than the absolute-value method, especially when an isogonal polytope has no uniform realization. The optimized form in this case is the closest to what a "uniform" variant would look like, whereas the result obtained through the absolute-value method has a bigger ratio in some cases. Notable examples are the step prisms, which are more reliably optimized generically than the absolute-value method.

## Types of convex isogonal polytopes

### Polygons

• Regular polygons (infinite, half symmetry variants exist for even-sided polygons with two alternating edge lengths)