Small 10-3 double step prism

Small 10-3 double step prism
File:Small 10-3 double step prism.png (OFF file)
Rank	4
Type	Isogonal
Space	Spherical
Elements
Cells	10 tetragonal disphenoids, 20+20 phyllic disphenoids, 40 irregular tetrahedra
Faces	20 isosceles triangles, 40+40+40+40 scalene triangles
Edges	10+20+20+20+40
Vertices	20
Vertex figure	11-vertex polyhedron with 18 triangular faces
Measures (edge length 1)
Central density	1
Related polytopes
Dual	Small 10-3 bigyrochoron
Abstract & topological properties
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	I2(10)+×4×I, order 40
Convex	Yes
Nature	Tame

The small 10-3 double step prism is a convex isogonal polychoron that consists of 10 tetragonal disphenoids, 40 phyllic disphenoids of two kinds, and 40 irregular tetrahedra. 2 tetragonal disphenoids, 8 phyllic disphenoids, and 8 irregular tetrahedra join at each vertex. It can be obtained as one of several polychora formed as the convex hull of two orthogonal 10-3 step prisms.

This polychoron cannot be optimized using the ratio method, because the solution $\frac{a}{b} = \frac{3-\sqrt5}{2}$ would yield a biambodecachoron instead.

Vertex coordinates

Coordinates for the vertices of a small 10-3 double step prism are given by:

• $\left(a\sin\left(\frac{2\pi k}{10}\right),\,a\cos\left(\frac{2\pi k}{10}\right),\,b\sin\left(\frac{6\pi k}{10}\right),\,b\cos\left(\frac{6\pi k}{10}\right)\right),$
• $\left(b\sin\left(\frac{2\pi k}{10}\right),\,b\cos\left(\frac{2\pi k}{10}\right),\,a\sin\left(\frac{6\pi k}{10}\right),\,a\cos\left(\frac{6\pi k}{10}\right)\right),$

where a/b is greater than $\frac{3-\sqrt5}{2}$ but less than 1, and k is an integer from 0 to 9.

External links

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