12-3 step prism

The 12-3 step prism is a convex isogonal polychoron and a member of the step prism family. It has 4 triangular gyroprisms and 24 phyllic disphenoids of two kinds as cells, with 8 disphenoids and 2 gyroprisms joining at each vertex.

Using the ratio method, the lowest possible ratio between the longest and shortest edges is 1:$$\frac{\sqrt{9+6\sqrt3}}{3}$$ ≈ 1:1.46789.

Vertex coordinates
Coordinates for the vertices of a 12-3 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:$$\sqrt{\frac{1+\sqrt3}{2}}$$ ≈ 1:1.16877.
 * (a*sin(πk/6), a*cos(πk/6), b*sin(πk/2), b*cos(πk/2)),