# 9-2 step prism

9-2 step prism
Rank4
TypeIsogonal
Elements
Cells9+9+9 phyllic disphenoids
Faces18+18 scalene triangles, 9+9 isosceles triangles
Edges9+9+9+9
Vertices9
Vertex figureRidge-triakis bi-apiculated tetrahedron
Measures (circumradius ${\displaystyle {\sqrt {2}}}$, based on a uniform duoprism)
Edge lengths7-valence (9): ${\displaystyle 2\sin {\frac {\pi }{9}}{\sqrt {3+2\cos {\frac {2\pi }{9}}}}\approx 1.45623}$
4-valence (9): ${\displaystyle 2{\sqrt {\sin ^{2}{\frac {\pi }{9}}+\cos ^{2}{\frac {\pi }{18}}}}\approx 2.08502}$
3-valence (9): ${\displaystyle {\sqrt {4+2\cos {\frac {\pi }{9}}-2\sin {\frac {\pi }{18}}}}\approx 2.35204}$
4-valence (9): ${\displaystyle {\sqrt {6}}\approx 2.44949}$
Central density1
Related polytopes
Dual9-2 gyrochoron
Abstract & topological properties
Flag count648
Euler characteristic0
OrientableYes
Properties
SymmetryS2(I2(9)-2), order 18
Flag orbits36
ConvexYes
NatureTame

The 9-2 step prism is a convex isogonal polychoron and a member of the step prism family. It has 27 phyllic disphenoids of three kinds as cells, with 12 joining at each vertex. It can also be constructed as the 9-4 step prism.

It is one of 3 isogonal polychora with 9 vertices, the others are the uniform triangular duoprism and the 9-3 step prism.

Using the ratio method, the lowest possible ratio between the longest and shortest edges is 1:${\displaystyle {\sqrt {\frac {2\cos {\frac {\pi }{9}}}{\cos {\frac {2\pi }{9}}}}}}$ ≈ 1:1.56632.

## Vertex coordinates

Coordinates for the vertices of a 9-2 step prism inscribed in an enneagonal duoprism with base lengths a and b are given by:

• (a*sin(2πk/9), a*cos(2πk/9), b*sin(4πk/9), b*cos(4πk/9)),

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:${\displaystyle {\sqrt {1+2\cos {\frac {\pi }{9}}}}}$ ≈ 1:1.69688.

## Isogonal derivatives

Substitution by vertices of these following elements will produce these convex iisogonal polychora:

• Phyllic disphenoid (9): 9-2 step prism
• Scalene triangle (9): 9-2 step prism
• Scalene triangle (19): 18-2 step prism
• Edge (9): 9-2 step prism