20-6 step prism

20-6 step prism
File:20-6 step prism.png (OFF file)
Rank	4
Type	Isogonal
Space	Spherical
Elements
Cells	20+20+20+20 phyllic disphenoids, 10 rhombic disphenoids
Faces	40+40+40+40 scalene triangles, 20 isosceles triangles
Edges	10+20+20+20+20+20
Vertices	20
Vertex figure	11-vertex polyhedron with 18 triangular faces
Measures (circumradius $\sqrt2$, based on a uniform duoprism)
Edge lengths	8-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 density	1
Related polytopes
Dual	20-6 gyrochoron
Abstract & topological properties
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	S2(I2(20)-6), order 40
Convex	Yes
Nature	Tame

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.

External links

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