20-4 step prism

20-4 step prism
(OFF file)
Rank	4
Type	Isogonal
Elements
Cells	20+20+20 phyllic disphenoids, 5 square gyroprisms
Faces	40+40+40 scalene triangles, 20 isosceles triangles, 5 squares
Edges	20+20+20+20+20
Vertices	20
Vertex figure	Base-ditriakis tetragonal antiwedge
Measures (circumradi ${\sqrt {2}}$, based on a uniform duoprism)
Edge lengths	8-valence (20): ${\sqrt {\frac {9-{\sqrt {5}}-{\sqrt {10+2{\sqrt {5}}}}}{2}}}\approx 1.21649$
	3-valence (20): ${\sqrt {2}}\approx 1.41421$
	4-valence (15): ${\sqrt {5-{\sqrt {5}}}}\approx 1.66251$
	3-valence (20): 2
	4-valence (20): ${\sqrt {\frac {9+{\sqrt {5}}-{\sqrt {10-2{\sqrt {5}}}}}{2}}}\approx 2.10772$
Central density	1
Related polytopes
Dual	20-4 gyrochoron
Abstract & topological properties
Flag count	1760
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	S2(I2(20)-4), order 40
Convex	Yes
Nature	Tame

The 20-4 step prism is an isogonal polychoron and a member of the step prism family. It has 5 square gyroprisms and 60 phyllic disphenoids of three kinds as cells, with 12 phyllic disphenoids and 2 square gyroprisms joining at each vertex.

Using the ratio method, the lowest possible ratio between the longest and shortest edges is 1:${\frac {\sqrt {4+2{\sqrt {10+2{\sqrt {5}}}}}}{2}}$ ≈ 1:1.70356.

Vertex coordinates

Coordinates for the vertices of a 20-4 step prism inscribed in an icosagonal duoprism with base lengths a and b are given by:

• $\left(a\sin {\frac {k\pi }{10}},\,a\cos {\frac {k\pi }{10}},\,b\sin {\frac {2k\pi }{5}},\,b\cos {\frac {2k\pi }{5}}\right)$,

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:${\frac {\sqrt[{4}]{625+250{\sqrt {5}}}}{5}}$ ≈ 1:1.17319.

External links

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