13-3 step prism

The 13-3 step prism is a convex isogonal polychoron and member of the step prism family. It has 52 phyllic disphenoids of four kinds as cells, with 16 joining at each vertex. It can also be constructed as the 13-4 step prism.

Using the ratio method, the lowest possible ratio between the longest and shortest edges is 1:$$\sqrt{\frac{11+22\sin\frac{\pi}{26}-12\sin\frac{3\pi}{26}-18\cos\frac{\pi}{13}+2\cos\frac{2\pi}{13}}{9+16\sin\frac{\pi}{26}-6\sin\frac{3\pi}{26}-10\cos\frac{\pi}{13}-2\cos\frac{2\pi}{13}}}$$ ≈ 1:1.53455.

Vertex coordinates
Coordinates for the vertices of a 13-3 step prism inscribed in a tridecagonal duoprism with base lengths a and b are given by: where k is an integer from 0 to 12. If the edge length differences are to be minimized, the ratio of a:b must be equivalent to :$$\sqrt{\frac{\cos\frac{2\pi}{13}+\sin\frac{3\pi}{26}}{\cos\frac{2\pi}{13}-\sin\frac{\pi}{26}}}$$ ≈ 1:1.27325.
 * (a*sin(2πk/13), a*cos(2πk/13), b*sin(6πk/13), b*cos(6πk/13)),