# 18-5 step prism

The 18-5 step prism, also known as the 9-2 double step prism, is a convex isogonal polychoron and a member of the step prism family. It has 72 phyllic disphenoids of four kinds as cells, with 16 joining at each vertex. It can also be constructed as the 18-7 step prism, or as the convex hull of two opposite 9-2 step prisms.

18-5 step prism
Rank4
TypeIsogonal
Elements
Cells18+18+18+18 phyllic disphenoids
Faces36+36+36 scalene triangles, 18+18 isosceles triangles
Edges18+18+18+18+18
Vertices18
Vertex figureMetabitriakis snub disphenoid
Measures (circumradius ${\displaystyle {\sqrt {2}}}$, based on a uniform duoprism)
Edge lengths6-valence (18): ${\displaystyle {\sqrt {2}}\approx 1.41421}$
5-valence (18): ${\displaystyle 2\sin {\frac {\pi }{9}}{\sqrt {3+2\cos {\frac {2\pi }{9}}}}\approx 1.45623}$
6-valence (18): ${\displaystyle 2{\sqrt {\sin ^{2}{\frac {\pi }{18}}+\cos ^{2}{\frac {2\pi }{9}}}}\approx 1.57096}$
4-valence (18): ${\displaystyle 2{\sqrt {\sin ^{2}{\frac {\pi }{18}}+\cos ^{2}{\frac {\pi }{9}}}}\approx 1.91120}$
3-valence (18): ${\displaystyle 2{\sqrt {\sin ^{2}{\frac {\pi }{9}}+\cos ^{2}{\frac {\pi }{18}}}}\approx 2.08502}$
Central density1
Related polytopes
Dual18-5 gyrochoron
Abstract & topological properties
Flag count1728
Euler characteristic0
OrientableYes
Properties
SymmetryS2(I2(18)-5), order 36
ConvexYes
NatureTame

Using the ratio method, the lowest possible ratio between the longest and shortest edges is 1:${\displaystyle {\sqrt {2}}}$ ≈ 1:1.41421.

## Vertex coordinates

Coordinates for the vertices of an 18-5 step prism inscribed in an octadecagonal duoprism with base lengths a and b are given by:

• ${\displaystyle \left(a\sin {\frac {k\pi }{9}},\,a\cos {\frac {k\pi }{9}},\,b\sin {\frac {5k\pi }{9}},\,b\cos {\frac {5k\pi }{9}}\right)}$ ,

where k is an integer from 0 to 17. If the edge length differences are to be minimized, the ratio of a:b must be equivalent to 1:${\displaystyle {\sqrt {\frac {\cos {\frac {2\pi }{9}}}{\cos {\frac {\pi }{9}}}}}}$  ≈ 1:0.90289.