7-2 step prism

The 7-2 step prism is a convex isogonal polychoron and a member of the step prism family. It has 14 phyllic disphenoids of two different types as cells, with 8 joining at each vertex. It can also be constructed as the 7-3 step prism.

It is the simplest step prism, excluding the pentachoron and the triangular duotegum, which are part of more specific families, as well as the only isogonal polychoron with 7 vertices. It is also the triangular funk tegum.

Using the ratio method, the lowest possible ratio between the longest and shortest edges is 1:$$\sqrt{2\cos\frac\pi7}$$ ≈ 1:1.34236.

Vertex coordinates
Coordinates for the vertices of a 7-2 step prism inscribed in a heptagonal duoprism with base lengths a and b are given by: where k is an integer from 0 to 6. If the edge length differences are to be minimized, the ratio of a:b must be equivalent to 1:$$\sqrt{2\cos\frac\pi7}$$ ≈ 1:1.34236.
 * (a*sin(2πk/7), a*cos(2πk/7), b*sin(4πk/7), b*cos(4πk/7)),

Measures
The hypervolume of a 7-2 step prism inscribed in a heptagonal-heptagonal duoprism with base lengths a and b is given by:
 * $$V=a^2b^2\xi,$$

where ξ ≈ 1.55622 is the largest real root of 884736x3–1613472x–823543, equivalent to 49/(192cos(2π/7)-96)-49/96.

Isogonal derivatives
Substitution by vertices of these following elements will produce these convex isogonal polychora:
 * Phyllic disphenoid (7): 7-2 step prism
 * Scalene triangle (7): 7-2 step prism
 * Scalene triangle (14): 14-2 step prism
 * Edge (7): 7-2 step prism