Transitional 10-3 double gyrostep prism

Transitional 10-3 double gyrostep prism
File:Transitional 10-3 double gyrostep prism.png (OFF file)
Rank	4
Type	Isogonal
Space	Spherical
Elements
Cells	10 tetragonal disphenoids, 10 chiral gyrobifastigia
Faces	40 isosceles triangles, 20 kites
Edges	20+40
Vertices	20
Vertex figure	Tetragonal antiwedge
Measures (based on unit decachoron)
Edge lengths	4-valence (20): 1
	3-valence (40): $\sqrt3 ≈ 1.73205$
Circumradius	$\sqrt2 ≈ 1.41421$
Central density	1
Related polytopes
Dual	Transitional 10-3 antibigyrochoron
Abstract & topological properties
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	S2(I2(10)-3)×2I, order 40
Convex	Yes
Nature	Tame

The transitional 10-3 double gyrostep prism is a convex isogonal polychoron that consists of 10 chiral gyrobifastigia and 10 tetragonal disphenoids. 2 chiral gyrobifastigia and 4 tetragonal disphenoids join at each vertex. It can be obtained as the convex hull of two orthogonal stretched 10-3 step prisms.

It can also be constructed as a subsymmetrical faceting of the decachoron, formed by removing a gyrochoron-symmetric subset of 10 vertices. The disphenoids are the decachoron's vertex figures, while the gyrobifastigium cells are subsymmetric facetings of the decachoron's truncated tetrahedron cells.

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

Vertex coordinates

Coordinates for the vertices of a transitional 10-3 double gyrostep prism are given by:

• (a*sin(2πk/10), a*cos(2πk/10), b*sin(6πk/10), b*cos(6πk/10)),
• (b*sin(2πk/10), b*cos(2πk/10), -a*sin(6πk/10), -a*cos(6πk/10)),

where a = $\sqrt{\frac{5-2\sqrt5}{5}}$, b = $\sqrt{\frac{5+2\sqrt5}{5}}$, and k is an integer from 0 to 9.

External links

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