Pyritosnub alterprism

The pyritosnub alterprism, also known as the edge-snub octahedral hosochoron or pysna, is a convex isogonal polychoron that consists of 2 pyritosnub cubes, 6 rectangular trapezoprisms, 8 triangular gyroprisms, and 24 skewed wedges. 3 wedges, 1 pyritosnub cube, 1 triangular gyroprism, and 1 rectangular trapezoprism join at each vertex. It can be obtained through the process of alternating one class of edges of the great rhombicuboctahedral prism, such that the octagons turn into rectangles. However, it cannot be made uniform, as it generally has up to 5 edge lengths, which can be minimized to no more than 2 different sizes.

Using the ratio method, the lowest possible ratio between the longest and shortest edges is 1:a ≈ 1:1.30910, where a is the largest real root of 13x4-24x3-6x2+16x+5.

Vertex coordinates
Vertex coordinates for a pyritosnub alterprism created from the vertices of a great rhombicuboctahedral prism of edge length 1, are given by all even permutations of the first three coordinates of:
 * $$\left(±\frac{1+2\sqrt2}{2},\,±\frac{1+\sqrt2}{2},\,±\frac12,\,\frac12\right),$$
 * $$\left(±\frac{1+\sqrt2}{2},\,±\frac{1+2\sqrt2}{2},\,±\frac12,\,-\frac12\right),$$

A variant where the triangular antiprisms being regular octahedra that are a unit distance apart, centered at the origin, is given by the cyclic permutations excluding the last coordinate of:


 * $$\left(±\frac12,\,±\frac{3+\sqrt6}{6},\,±\frac{3+2\sqrt6}{6},\,\frac{\sqrt6}{6}\right),$$
 * $$\left(±\frac12,\,±\frac{3+2\sqrt6}{6},\,±\frac{3+\sqrt6}{6},\,-\frac{\sqrt6}{6}\right).$$

An alternate set of coordinates, assuming that the edge length differences are minimized, centered at the origin, are given by the cyclic permutations excluding the last coordinate of:


 * $$\left(±\frac12,\,±c_1,\,±c_2,\,c_3\right),$$
 * $$\left(±\frac12,\,±c_2,\,±c_1,\,-c_3\right),$$

where


 * $$c_1=\text{root}(208x^4-192x^3-24x^2+32x+5,\, 3) ≈ 0.6545479408664677382876785,$$
 * $$c_2=\text{root}(208x^4-256x^3-24x^2+16x+1,\, 4) ≈ 1.2715982466042828483994902,$$
 * $$c_3=\text{root}(10816x^8-4544x^6+352x^4+32x^2-1,\, 6) ≈ 0.4879113310704798497401873.$$