8-2 step prism

The 8-2 step prism is a convex isogonal polychoron, member of the step prism family. It has 4 rhombic disphenoids and 16 phyllic disphenoids of two kinds as cells, with a total of 10 (2 rhombic and 8 phyllic disphenoids) joining at each vertex. It is one of 3 isogonal polychora with 8 vertices, and the only one not to be uniform (the other 2 are the hexadecachoron and tetrahedral prism).

Vertex coordinates
Coordinates for the vertices of an 8-2 step prism inscribed in an octagonal duoprism with base lengths a and b are given by: where k is an integer from 0 to 7. If the edge length differences are to be minimized, the ratio of a:b must be equivalent to 1:$$\frac{\sqrt2}{2}$$ ≈ 1:0.70711.
 * (a*sin(2πk/8), a*cos(2πk/8), b*sin(4πk/8), b*cos(4πk/8)),

This gives the following coordinates for an optimized 8-2 step prism:


 * $$\left(0,\,±\frac12,\,0,\,\frac{\sqrt2}{4}\right),$$
 * $$\left(±\frac12,\,0,\,0,\,-\frac{\sqrt2}{4}\right),$$
 * $$\left(\frac{\sqrt2}{4},\,\frac{\sqrt2}{4},\,\frac{\sqrt2}{4},\,0\right)$$ and all even sign changes of the first 3 coordinates

These coordinates give an 8-2 step prism where the 4 rhombic disphenoids become regular tetrahedra of edge length 1.

Isogonal derivatives
Substitution by vertices of these following elements will produce these convex isogonal polychora:
 * Phyllic disphenoid (8): 8-2 step prism
 * Scalene triangle (8): 8-2 step prism
 * Scalene triangle (16): 16-2 step prism
 * Edge (8): 8-2 step prism