Pyritohedral icosahedral alterprism

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 antiprisms, 6 tetragonal disphenoids and 24 sphenoids obtained through the process of alternating the truncated octahedral prism. However, it cannot be made uniform.

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

Vertex coordinates
The vertices of a pyritohedral icosahedral antiprism, assuming that the icosahedra and the tetragonal disphenoids are regular of edge length 1, centered at the origin, are given by the cyclic permutations excluding the last coordinate of:
 * (0, ±1/2, ±(1+$\sqrt{5}$)/4, $\sqrt{2}$/4),
 * (0, ±(1+$\sqrt{5}$)/4, ±1/2, –$\sqrt{2}$/4).

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:
 * (0, ±1/2, ±(3+$\sqrt{17}$)/8, $\sqrt{7+√17}$/8),
 * (0, ±(3+$\sqrt{17}$)/8, ±1/2, –$\sqrt{7+√17}$/8).

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