37-6 step prism

37-6 step prism
File:37-6 step prism.png (OFF file)
Rank	4
Type	Isogonal
Space	Spherical
Elements
Cells	37 tetragonal disphenoids, 74+74+74+74 phyllic disphenoids
Faces	148+148+148 scalene triangles, 74+148 isosceles triangles
Edges	74+74+74+74+74
Vertices	37
Vertex figure	20-vertex polyhedron with 36 triangular faces
Measures (edge length 1)
Central density	1
Related polytopes
Dual	37-6 gyrochoron
Abstract & topological properties
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	S2(I2(37)-6)×2I, order 148
Convex	Yes
Nature	Tame

The 37-6 step prism is a convex isogonal polychoron, member of the step prism family. It has 37 tetragonal disphenoids and 296 phyllic disphenoids of four kinds as cells. 4 tetragonal and 32 phyllic disphenoids join at each vertex. It is also the 37-31 step prism.

The ratio between the longest and shortest edges is 1:$\sqrt{\frac{4+2\cos\frac{\pi}{37}-2\cos\frac{6\pi}{37}}{4-2\cos\frac{2\pi}{37}-2\sin\frac{13\pi}{74}}}$ ≈ 1:2.08153.

Vertex coordinates

Coordinates for the vertices of a 37-6 step prism of circumradius √2 are given by:

• (sin(2πk/37), cos(2πk/37), sin(12πk/37), cos(12πk/37)),

where k is an integer from 0 to 36.