29-12 step prism

29-12 step prism
File:29-12 step prism.png (OFF file)
Rank	4
Type	Isogonal
Space	Spherical
Elements
Cells	58+58 phyllic disphenoids, 29 tetragonal disphenoids
Faces	116 scalene triangles, 58+116 isosceles triangles
Edges	58+58+58
Vertices	29
Vertex figure	Biorthowedged rhombic gyroprism
Measures (circumradius $\sqrt2$, based on a uniform duoprism)
Edge lengths	7-valence (58): $2\sqrt{\sin^2\frac{2\pi}{29}+\sin^2\frac{5\pi}{29}} ≈ 1.11715$
	4-valence (58): $2\sqrt{\sin^2\frac{3\pi}{29}+\sin^2\frac{7\pi}{29}} ≈ 1.51642$
	4-valence (58): $2\sqrt{\sin^2\frac{\pi}{29}+\cos^2\frac{5\pi}{58}} ≈ 1.93919$
Central density	1
Related polytopes
Dual	29-12 gyrochoron
Abstract & topological properties
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	S2(I2(29)-12)×2I, order 116
Convex	Yes
Nature	Tame

The 29-12 step prism is a convex isogonal polychoron, member of the step prism family. It has 29 tetragonal disphenoids and 116 phyllic disphenoids of two kinds as cells, with 20 (4 tetragonal and 16 phyllic disphenoids) joining at each vertex.

The ratio between the longest and shortest edges is 1:$\sqrt{\frac{2-\cos\frac{2\pi}{29}+\cos\frac{5\pi}{29}}{2-\sin\frac{9\pi}{58}-\cos\frac{4\pi}{29}}}$ ≈ 1:1.73583.

Compared to other 29-vertex step prisms, this polychoron has doubled symmetry, because 29 is a factor of 12²+1 = 145.

Vertex coordinates

Coordinates for the vertices of a 29-12 step prism of circumradius √2 are given by:

• (sin(2πk/29), cos(2πk/29), sin(24πk/29), cos(24πk/29)),

where k is an integer from 0 to 28.