Pyritohedral icosahedral antiprism

The pyritohedral icosahedral antiprism or pikap, also known as the alternated truncated octahedral prism or omnisnub tetrahedral antiprism, is a convex isogonal polychoron that consists of 2 pyritohedral icosahedra, 8 triangular gyroprisms, 6 tetragonal disphenoids, and 24 sphenoids. 1 pyritohedral icosahedron, 2 triangular antiprisms, 1 tetragonal disphenoid, and 4 sphenoids join at each vertex. It can be obtained through the process of alternating the truncated octahedral prism. However, it cannot be made uniform, as it generally have 4 edge lengths, which can be minimized to no fewer than 2 different sizes..

Using the ratio method, the lowest possible ratio between the longest and shortest edges is 1:$$\frac{\sqrt{15+\sqrt{17}}}{4}$$ ≈ 1:1.09325.

This polychoron has a kind of pyritohedral antiprismatic symmetry. A variant with chiral tetrahedral antiprism symmetry, called the snub tetrahedral antiprism, also exists.

Vertex coordinates
Vertex coordinates for a pyritohedral icosahedral antiprism, created from the vertices of a truncated octahedral prism of edge length 1, are given by all even permutations in the first 3 coordinates of:
 * $$\left(±\sqrt2,\,±\frac{\sqrt2}{2},\,0,\,\frac12\right),$$
 * $$\left(±\frac{\sqrt2}{2},\,±\sqrt2,\,0,\,-\frac12\right).$$

A variant where the icosahedra and the tetragonal disphenoids are regular of edge length 1, centered at the origin, is given by the cyclic permutations excluding the last coordinate of:
 * $$\left(0,\,±\frac12,\,±\frac{1+\sqrt5}{4},\,\frac{\sqrt2}{4}\right),$$
 * $$\left(0,\,±\frac{1+\sqrt5}{4},\,±\frac12,\,-\frac{\sqrt2}{4}\right).$$

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:
 * $$\left(0,\,±\frac12,\,±\frac{3+\sqrt{17}}{8},\,\frac{\sqrt{7+\sqrt{17}}}{8}\right),$$
 * $$\left(0,\,±\frac{3+\sqrt{17}}{8},\,±\frac12,\,-\frac{\sqrt{7+\sqrt{17}}}{8}\right).$$

Isogonal derivatives
Substitution by vertices of these following elements will produce these convex isogonal polychora:
 * Sphenoid (24): Pyritohedral icosahedral antiprism
 * Triangle (16): Tesseract
 * Isosceles triangle (24): Pyritohedral icosahedral antiprism
 * Scalene triangle (48): Pyritosnub alterprism
 * Edge (12): Octahedral prism
 * Edge (24): Pyritohedral icosahedral antiprism
 * Edge (48): Pyritosnub alterprism