21-8 step prism

21-8 step prism
File:21-8 step prism.png (OFF file)
Rank	4
Type	Isogonal
Space	Spherical
Elements
Cells	42+42 phyllic disphenoids
Faces	84 scalene triangles, 42+42 isosceles triangles
Edges	21+42+42
Vertices	21
Vertex figure	10-vertex polyhedron with 16 triangular faces
Measures (edge length 1)
Edge lengths	6-valence (21): $2\sqrt2\sin\frac\pi7 ≈ 1.22721$
	5-valence (42): $2\sqrt{\sin^2\frac{2\pi}{21}+\sin^2\frac{5\pi}{21}} ≈ 1.48259$
	4-valence (42): $2\sqrt{\sin^2\frac{\pi}{21}+\cos^2\frac{5\pi}{42}} ≈ 1.88546$
Central density	1
Related polytopes
Dual	21-8 gyrochoron
Abstract & topological properties
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	S2(I2(21)-8)×2R, order 84
Convex	Yes
Nature	Tame

The 21-8 step prism is a convex isogonal polychoron, member of the step prism family. It has 84 phyllic disphenoids of two kinds as cells, with 16 joining at each vertex..

The ratio between the longest and shortest edges is 1:$\sqrt{\frac{11\cos\frac{2\pi}{7}-1}{12\cos\frac{2\pi}{7}-5}}$ ≈ 1:1.53638.

Compared to other 21-vertex step prisms, this polychoron has doubled symmetry, because 21 is a factor of 8²-1 = 63.

Vertex coordinates

Coordinates for the vertices of a 21-8 step prism of circumradius √2 are given by:

• (sin(2πk/21), cos(2πk/21), sin(16πk/21), cos(15πk/21)),

where k is an integer from 0 to 20.