# 20-6 step prism

20-6 step prism
File:20-6 step prism.png
Rank4
TypeIsogonal
SpaceSpherical
Elements
Cells20+20+20+20 phyllic disphenoids, 10 rhombic disphenoids
Faces40+40+40+40 scalene triangles, 20 isosceles triangles
Edges10+20+20+20+20+20
Vertices20
Vertex figure11-vertex polyhedron with 18 triangular faces
Measures (circumradius $\sqrt2$ , based on a uniform duoprism)
Edge lengths8-valence (20): $\sqrt{\frac{7-\sqrt5-\sqrt{10-2\sqrt5}}{2}} ≈ 1.09836$ 6-valence (20): $\sqrt{\frac{7+\sqrt5-\sqrt{10+2\sqrt5}}{2}} ≈ 1.64801$ 4-valence (20): $\sqrt{5-\sqrt5} ≈ 1.66251$ 4-valence (20): $\sqrt{\frac{7-\sqrt5+\sqrt{10-2\sqrt5}}{2}} ≈ 1.88614$ 3-valence (20): 2
4-valence (10): 2
Central density1
Related polytopes
Dual20-6 gyrochoron
Abstract & topological properties
Euler characteristic0
OrientableYes
Properties
SymmetryS2(I2(20)-6), order 40
ConvexYes
NatureTame

The 20-6 step prism is a convex isogonal polychoron and a member of the step prism family. It has 10 rhombic disphenoids and 80 phyllic disphenoids of four kinds as cells, with 18 (2 rhombic and 16 phyllic disphenoids) joining at each vertex.

Using the ratio method, the lowest possible ratio between the longest and shortest edges is 1:$\frac{\sqrt{589+95\sqrt5+19\sqrt{250+82\sqrt5}}}{19}$ ≈ 1:1.82090.

## Vertex coordinates

Coordinates for the vertices of a 20-6 step prism inscribed in an icosagonal duoprism with base lengths a and b are given by:

• (a*sin(2πk/20), a*cos(2πk/20), b*sin(12πk/20), b*cos(12πk/20)),

where k is an integer from 0 to 19. If the edge length differences are to be minimized, the ratio of a:b must be equivalent to 1:1.