20-5 step prism

20-5 step prism
(OFF file)
Rank	4
Type	Isogonal
Space	Spherical
Elements
Cells	20+20 phyllic disphenoids, 4 pentagonal gyroprisms
Faces	40+40 scalene triangles, 20 isosceles triangles, 4 pentagons
Edges	20+20+20+20
Vertices	20
Vertex figure	Base-triakis tetragonal antiwedge
Measures (circumradius $\sqrt2$, based on a uniform duoprism)
Edge lengths	3-valence (20): $\sqrt{\frac{5-\sqrt5}{2}} ≈ 1.17557$
	6-valence (20): $\sqrt{\frac{8-\sqrt{10+2\sqrt5}}{2}} ≈ 1.44841$
	4-valence (20): $\sqrt{\frac{8-\sqrt{10-2\sqrt5}}{2}} ≈ 1.68060$
	3-valence (20): $\sqrt{\frac{11-\sqrt5}{2}} ≈ 2.09331$
Central density	1
Related polytopes
Dual	20-5 gyrochoron
Abstract & topological properties
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	S2(I2(20)-5), order 40
Convex	Yes
Nature	Tame

The 20-5 step prism is a convex isogonal polychoron and a member of the step prism family. It has 4 pentagonal gyroprisms and 40 phyllic disphenoids of two kinds as cells, with 8 phyllic disphenoids and 2 pentagonal gyroprisms joining at each vertex.

Using the ratio method, the lowest possible ratio between the longest and shortest edges is 1:$\frac{\sqrt{50-50\sqrt5+40\sqrt{25+10\sqrt5}}}{10}$ ≈ 1:1.46107.

Vertex coordinates

Coordinates for the vertices of a 20-5 step prism inscribed in an icosagonal duoprism with base lengths a and b are given by:

• (a*sin(πk/10), a*cos(πk/10), b*sin(πk/2), b*cos(πk/2)),

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:$\frac{\sqrt{1-\sqrt5+\sqrt{10+2\sqrt5}}}{2}$ ≈ 1:0.80127.

External links

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