9-3 step prism

The 9-3 step prism is a convex isogonal polychoron and a member of the step prism family. It has 3 triangular gyroprisms and 9 phyllic disphenoids as cells, with 4 disphenoids and 2 gyroprisms joining at each vertex..

It is one of three isogonal polychora with 9 vertices (the others are the 9-2 step prism and triangular duoprism), as well as the simplest step prism to have cells other than tetrahedra.

Using the ratio method, the lowest possible ratio between the longest and shortest edges is 1:$$\sqrt{\frac{3+2\cos\frac{2\pi}{9}-2\sin\frac{\pi}{18}}{3}}$$ ≈ 1:1.18107.

Vertex coordinates
Coordinates for the vertices of a 9-3 step prism inscribed in an enneagonal duoprism with base lengths a and b are given by: where k is an integer from 0 to 8. If the edge length differences are to be minimized, the ratio of a:b must be equivalent to 1:$$\sqrt{\frac{1+2\cos\frac{2\pi}{9}}{3}}$$ ≈ 1:0.91871.
 * (a*sin(2πk/9), a*cos(2πk/9), 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 (9): 9-3 step prism
 * Scalene triangle (9): 9-3 step prism
 * Scalene triangle (18): 18-3 step prism
 * Edge (9): 9-3 step prism