10-2 double step prism

10-2 double step prism
File:10-2 double step prism.png (OFF file)
Rank	4
Type	Isogonal
Elements
Cells	40 phyllic disphenoids, 20 rhombic disphenoids, 20 tetragonal disphenoids
Faces	80 scalene triangles, 80 isosceles triangles
Edges	20+40+40
Vertices	20
Vertex figure	Tetrakis parallelogramic alterprism
Measures (longest edge length 1)
Edge lengths	6-valence (40): ${\frac {\sqrt {4-{\sqrt {5}}}}{2}}\approx 0.66407$
	4-valence (40): ${\frac {\sqrt {3}}{2}}\approx 0.86603$
	4-valence (20): 1
Central density	1
Related polytopes
Dual	10-2 bigyrochoron
Abstract & topological properties
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	S2(I2(10)×2-2)×2I, order 80
Convex	Yes
Nature	Tame

The 10-2 double step prism is a convex isogonal polychoron that consists of 20 tetragonal disphenoids, 20 rhombic disphenoids, and 40 phyllic disphenoids. 4 tetragonal disphenoids, 4 rhombic disphenoids, and 8 phyllic disphenoids join at each vertex. It can be obtained as the convex hull of two opposite 10-2 step prisms.

The ratio between the longest and shortest edges is 1:${\frac {\sqrt {176+44{\sqrt {5}}}}{11}}$ \approx 1:1.50588.

Vertex coordinates

Coordinates for the vertices of a 10-2 double step prism are given by:

• $(a\sin(2\pi k/10),a\cos(2\pi k/10),a\sin(6\pi k/10),a\cos(6\pi k/10))$,
• $(a\sin(2\pi k/10),a\cos(2\pi k/10),-a\sin(6\pi k/10),-a\cos(6\pi k/10))$,

where a = 1/2 and k is an integer from 0 to 9.

External links

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