42-13 step prism

42-13 step prism
File:42-13 step prism.png (OFF file)
Rank	4
Type	Isogonal
Space	Spherical
Elements
Cells	84+84+84 phyllic disphenoids, 42 rhombic disphenoids
Faces	168+168+168 scalene triangles, 84 isosceles triangles
Edges	42+42+84+84+84
Vertices	42
Vertex figure	16-vertex polyhedron with 28 triangular faces
Measures (edge length 1)
Central density	1
Related polytopes
Dual	42-13 gyrochoron
Abstract & topological properties
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	S2(I2(42)-13)×2R, order 168
Convex	Yes
Nature	Tame

The 42-13 step prism is a convex isogonal polychoron, member of the step prism family. It has 42 rhombic disphenoids and 252 phyllic disphenoids of three kinds as cells. 4 rhombic and 24 phyllic disphenoids join at each vertex. It is also the 42-29 step prism.

The ratio between the longest and shortest edges is 1:$\sqrt{\frac{7+46\cos\frac{2\pi}{7}}{16\cos\frac{2\pi}{7}-6}}$ ≈ 1:2.99572.

Vertex coordinates

Coordinates for the vertices of a 42-13 step prism of circumradius √2 are given by:

• (sin(πk/21), cos(πk/21), sin(13πk/21), cos(13πk/21)),

where k is an integer from 0 to 40.