# 18-3 step prism

18-3 step prism
Rank4
TypeIsogonal
SpaceSpherical
Info
SymmetryI2(18)+×2×I, order 36
Elements
Vertex figureBase-tritriakis tetragonal antiwedge
Cells18+18+18+18 phyllic disphenoids, 6 triangular antiprisms
Faces6 triangles, 18+36+36+36+36 scalene triangles
Edges18+18+18+18+18+18
Vertices18
Measures (circumradi $\sqrt2$ , based on a uniform duoprism)
Edge lengths10-valence (18): $\sqrt{1+4\sin^2\frac{\pi}{18}} ≈ 1.05859$ 3-valence (18): $\sqrt3 ≈ 1.73205$ 4-valence (18): $\sqrt{3+2\sin\frac{\pi}{18}} ≈ 1.82956$ 3-valence (18): $\sqrt{3+4\sin^2\frac\pi9} ≈ 1.86223$ 4-valence (18): $\sqrt{5-2\cos\frac{\pi}{18}} ≈ 2.15701$ 4-valence (18): $\sqrt5 ≈ 2.23607$ Central density1
Euler characteristic0
Related polytopes
Dual18-3 gyrochoron
Properties
ConvexYes
OrientableYes
NatureTame

The 18-3 step prism, also known as the 9-3 double step prism, is an isogonal polychoron and a member of the step prism family. It has 6 triangular antiprisms and 72 phyllic disphenoids of four kinds as cells, with 16 disphenoids and 2 antiprisms joining at each vertex.

## Vertex coordinates

Coordinates for the vertices of an 18-3 step prism inscribed in an octadecagonal duoprism with base lengths a and b are given by:

• (a*sin(πk/9), a*cos(πk/9), b*sin(πk/3), b*cos(πk/3)),

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:$\sqrt{1+2\cos\frac\pi9}$ ≈ 1:1.69688.