18-5 step prism

18-5 step prism
(OFF file)
Rank	4
Type	Isogonal
Space	Spherical
Elements
Cells	18+18+18+18 phyllic disphenoids
Faces	36+36+36 scalene triangles, 18+18 isosceles triangles
Edges	18+18+18+18+18
Vertices	18
Vertex figure	Metabitriakis snub disphenoid
Measures (circumradius $\sqrt2$, based on a uniform duoprism)
Edge lengths	6-valence (18): $\sqrt2 ≈ 1.41421$
	5-valence (18): $2\sin\frac\pi9\sqrt{3+2\cos\frac{2\pi}{9}} ≈ 1.45623$
	6-valence (18): $2\sqrt{\sin^2\frac{\pi}{18}+\cos^2\frac{2\pi}{9}} ≈ 1.57096$
	4-valence (18): $2\sqrt{\sin^2\frac{\pi}{18}+\cos^2\frac\pi9} ≈ 1.91120$
	3-valence (18): $2\sqrt{\sin^2\frac\pi9+\cos^2\frac{\pi}{18}} ≈ 2.08502$
Central density	1
Related polytopes
Dual	18-5 gyrochoron
Abstract & topological properties
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	S2(I2(18)-5), order 36
Convex	Yes
Nature	Tame

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.

Using the ratio method, the lowest possible ratio between the longest and shortest edges is 1:$\sqrt2$ ≈ 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:

• (a*sin(πk/9), a*cos(πk/9), b*sin(5πk/9), b*cos(5πk/9)),

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:$\sqrt{\frac{\cos\frac{2\pi}{9}}{\cos\frac\pi9}}$ ≈ 1:0.90289.

External links

• Bowers, Jonathan. "Four Dimensional Dice Up To Twenty Sides".