12-4 step prism

The 12-4 step prism is a convex isogonal polychoron and member of the step prism family. It has 3 square gyroprisms and 12 phyllic disphenoids as cells, with 4 phyllic disphenoids and 2 square gyroprisms joining at each vertex.

Using the ratio method, the lowest possible ratio between the longest and shortest edges is 1:$$\sqrt{\frac{1+\sqrt3}{2}}$$ ≈ 1:1.16877.

Vertex coordinates
Coordinates for the vertices of a 12-4 step prism inscribed in a dodecagonal duoprism with base lengths a and b are given by: where k is an integer from 0 to 11. If the edge length differences are to be minimized, the ratio of a:b must be equivalent to 1:$$\frac{\sqrt[4]{27}}{3}$$ ≈ 1:0.75984.
 * (a*sin(πk/6), a*cos(πk/6), b*sin(2πk/3), b*cos(2πk/3)),

Isogonal derivatives
Substitution by vertices of these following elements will produce these convex isogonal polychora:
 * Phyllic disphenoid (12): 12-4 step prism
 * Scalene triangle (12): 12-4 step prism
 * Scalene triangle (24): 24-4 step prism
 * Edge (12): 12-4 step prism