18-2 step prism

18-2 step prism
(OFF file)
Rank	4
Type	Isogonal
Elements
Cells	18+18+18+18+18+18+18 phyllic disphenoids, 9 rhombic disphenoids
Faces	36+36+36+36+36+36+36 scalene triangles, 18 isosceles triangles
Edges	9+18+18+18+18+18+18+18+18
Vertices	18
Vertex figure	Ridge-sextitriakis notch
Measures (circumradius ${\sqrt {2}}$, based on a uniform duoprism)
Edge lengths	16-valence (18): $2\sin {\frac {\pi }{18}}{\sqrt {3+2\cos {\frac {\pi }{9}}}}\approx 0.76715$
	3-valence (18): $2\sin {\frac {\pi }{9}}{\sqrt {3+2\cos {\frac {2\pi }{9}}}}\approx 1.45623$
	4-valence (9+18): 2
	4-valence (18): $2{\sqrt {\sin ^{2}{\frac {\pi }{9}}+\cos ^{2}{\frac {\pi }{18}}}}\approx 2.08502$
	4-valence (18): ${\sqrt {4+2\cos {\frac {2\pi }{9}}-2\sin {\frac {\pi }{18}}}}\approx 2.27701$
	4-valence (18): ${\sqrt {4+2\cos {\frac {\pi }{9}}-2\sin {\frac {\pi }{18}}}}\approx 2.35204$
	4-valence (18): ${\sqrt {6}}\approx 2.44949$
	4-valence (18): ${\sqrt {4+2\cos {\frac {\pi }{9}}+2\sin {\frac {\pi }{18}}}}\approx 2.49533$
Central density	1
Related polytopes
Dual	18-2 gyrochoron
Abstract & topological properties
Flag count	3240
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	S2(I2(18)-2), order 36
Convex	Yes
Nature	Tame

The 18-2 step prism is a convex isogonal polychoron and a member of the step prism family. It has 9 rhombic disphenoids and 126 phyllic disphenoids of six kinds as cells, with 30 (2 rhombic and 28 phyllic disphenoids) joining at each vertex.

Using the ratio method, the lowest possible ratio between the longest and shortest edges is 1:${\frac {\sqrt {13+18\cos {\frac {\pi }{9}}+6\cos {\frac {2\pi }{9}}}}{2}}$ ≈ 1:2.93729.

Vertex coordinates

Coordinates for the vertices of an 18-2 step prism inscribed in an octadecagonal duoprism with base lengths a and b are given by:

• $\left(a\sin {\frac {k\pi }{9}},\,a\cos {\frac {k\pi }{9}},\,b\sin {\frac {2k\pi }{9}},\,b\cos {\frac {2k\pi }{9}}\right)$,

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:$1+2\cos {\frac {\pi }{9}}$ ≈ 1:2.87939.

External links

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