Digonal-scalenohedral 8-3 double step prism

Digonal-scalenohedral 8-3 double step prism
File:Digonal-scalenohedral 8-3 double step prism.png (OFF file)
Rank	4
Type	Isogonal
Space	Spherical
Elements
Cells	16 phyllic disphenoids, 16 tetragonal disphenoids, 8 chiral digonal scalenohedra
Faces	32 scalene triangles, 32+32 isosceles triangles
Edges	8+16+16+32
Vertices	16
Vertex figure	9-vertex polyhedron with 3 tetragons and 8 triangles
Measures (based on unit rectified tesseract)
Edge lengths	6-valence (16): 1
	4-valence (16): $\sqrt2 ≈ 1.41421$
	4-valence (8): $\sqrt3 ≈ 1.73205$
	3-valence (32): $\sqrt3 ≈ 1.73205$
Circumradius	$\frac{\sqrt6}{2} ≈ 1.22475$
Central density	1
Related polytopes
Dual	Trapezoprismatic intersected 8-3 bigyrochoron
Abstract & topological properties
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	S2(I2(8)-3)×2R, order 32
Convex	Yes
Nature	Tame

The digonal-scalenohedral 8-3 double step prism is a convex isogonal polychoron, formed as a convex hull of two 8-3 step prisms in a way that leaves six corealmic vertices corresponding to a digonal scalenohedron. It consists of 8 chiral digonal scalenohedra, 16 tetragonal disphenoids, and 16 phyllic disphenoids. 3 digonal scalenohedra, 4 tetragonal disphenoids, and 4 phyllic disphenoids join at each vertex.

It can be constructed by removing an inscribed digonal-scalenohedral 8-3 double step prism from a rectified tesseract.

The ratio between the longest and shortest edges is 1:$\sqrt3$ ≈ 1:1.73205.

Vertex coordinates

Coordinates for the vertices of a digonal-scalenohedral 8-3 double step prism are given by:

• (a*sin(2πk/8), a*cos(2πk/8), b*sin(6πk/8), b*cos(6πk/8)),
• (b*sin(2πk/8), b*cos(2πk/8), a*sin(6πk/8), a*cos(6πk/8)),

where a = $\frac{\sqrt2-1}{2}$, b = $\frac{1+\sqrt2}{2}$, and k is an integer from 0 to 7.

External links

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