41-9 step prism

41-9 step prism
File:41-9 step prism.png (OFF file)
Rank	4
Type	Isogonal
Elements
Cells	82+82+82 phyllic disphenoids, 41 tetragonal disphenoids
Faces	164+164 scalene triangles, 82+164 isosceles triangles
Edges	82+82+82+82
Vertices	41
Vertex figure	16-vertex polyhedron with 28 triangular faces
Measures (edge length 1)
Central density	1
Related polytopes
Dual	41-9 gyrochoron
Abstract & topological properties
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	S2(I2(37)-6)×2I, order 164
Convex	Yes
Nature	Tame

The 41-9 step prism is a convex isogonal polychoron, member of the step prism family. It has 41 tetragonal disphenoids and 246 phyllic disphenoids of three kinds as cells. 4 tetragonal and 24 phyllic disphenoids join at each vertex. It is also the 41-32 step prism.

The ratio between the longest and shortest edges is 1:${\sqrt {\frac {4-2\cos {\frac {4\pi }{41}}+2\cos {\frac {5\pi }{41}}}{4-2\cos {\frac {8\pi }{41}}-2\cos {\frac {10\pi }{41}}}}}$ ≈ 1:2.06812.

Vertex coordinates

Coordinates for the vertices of a 41-9 step prism of circumradius √2 are given by:

• (sin(2πk/41), cos(2πk/41), sin(18πk/41), cos(18πk/41)),

where k is an integer from 0 to 40.