18-4 step prism

The 18-4 step prism is a convex isogonal polychoron and a member of the step prism family. It has 9 rhombic disphenoids and 72 phyllic disphenoids of four kinds as cells, with 18 (2 rhombic and 16 phyllic disphenoids) joining at each vertex.

Using the ratio method, the lowest possible ratio between the longest and shortest edges is 1:$$\sqrt{\frac{12+8\cos\frac{\pi}{9}-4\cos\frac{2\pi}{9}}{9-2\cos\frac{\pi}{9}-2\sin\frac{\pi}{18}}}$$ ≈ 1:1.55857.

Vertex coordinates
Coordinates for the vertices of an 18-4 step prism inscribed in an octadecagonal duoprism with base lengths a and b are given by: where k is an integer from 0 to 17. If the edge length differences are to be minimized, the ratio of a:b must be equivalent to 1:$$\frac{2\cos\frac{\pi}{18}}{\sqrt3}$$ ≈ 1:1.13716.
 * (a*sin(πk/9), a*cos(πk/9), b*sin(4πk/9), b*cos(4πk/9)),