# 21-8 step prism

21-8 step prism
File:21-8 step prism.png
Rank4
TypeIsogonal
SpaceSpherical
Elements
Cells42+42 phyllic disphenoids
Faces84 scalene triangles, 42+42 isosceles triangles
Edges21+42+42
Vertices21
Vertex figure10-vertex polyhedron with 16 triangular faces
Measures (edge length 1)
Edge lengths6-valence (21): ${\displaystyle 2\sqrt2\sin\frac\pi7 ≈ 1.22721}$
5-valence (42): ${\displaystyle 2\sqrt{\sin^2\frac{2\pi}{21}+\sin^2\frac{5\pi}{21}} ≈ 1.48259}$
4-valence (42): ${\displaystyle 2\sqrt{\sin^2\frac{\pi}{21}+\cos^2\frac{5\pi}{42}} ≈ 1.88546}$
Central density1
Related polytopes
Dual21-8 gyrochoron
Abstract & topological properties
Euler characteristic0
OrientableYes
Properties
SymmetryS2(I2(21)-8)×2R, order 84
ConvexYes
NatureTame

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:${\displaystyle \sqrt{\frac{11\cos\frac{2\pi}{7}-1}{12\cos\frac{2\pi}{7}-5}}}$ ≈ 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.