10-2 step prism

10-2 step prism
(OFF file)
Rank	4
Type	Isogonal
Space	Spherical
Elements
Cells	10+10+10 phyllic disphenoids, 5 rhombic disphenoids
Faces	20+20+20 scalene triangles, 10 isosceles triangles
Edges	5+10+10+10+10
Vertices	10
Vertex figure	Ridge-ditriakis notch
Measures (circumradius $\sqrt2$, based on a uniform duoprism)
Edge lengths	8-valence (10): $\sqrt{4-\sqrt5} ≈ 1.32813$
	4-valence (5): 2
	3-valence (10): $\sqrt5 ≈ 2.23607$
	4-valence (10): $\sqrt5 ≈ 2.23607$
	4-valence (10): $\sqrt{4+\sqrt5} ≈ 2.49721$
Central density	1
Related polytopes
Dual	10-2 gyrochoron
Abstract & topological properties
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	S2(I2(10)-2), order 20
Convex	Yes
Nature	Tame

The 10-2 step prism is a convex isogonal polychoron and a member of the step prism family. It has 5 rhombic disphenoids and 30 phyllic disphenoids of three kinds as cells, with 14 (2 rhombic and 12 phyllic disphenoids) joining at each vertex.

It is one of 4 isogonal polychora with 10 vertices, and the only one to have only step prism symmetry in its highest symmetry form.

Using the ratio method, the lowest possible ratio between the longest and shortest edges is 1:$\frac{\sqrt{26+10\sqrt5}}{4}$ ≈ 1:1.73855.

Vertex coordinates

Coordinates for the vertices of a 10-2 step prism inscribed in a decagonal duoprism with base lengths a and b are given by:

• (a*sin(πk/5), a*cos(πk/5), b*sin(2πk/5), b*cos(2πk/5)),

where k is an integer from 0 to 9. If the edge length differences are to be minimized, the ratio of a:b must be equivalent to 1:$\frac{1+\sqrt5}{2}$ ≈ 1:1.61803.

External links

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