# 29-12 step prism

29-12 step prism
File:29-12 step prism.png
Rank4
TypeIsogonal
SpaceSpherical
Elements
Cells58+58 phyllic disphenoids, 29 tetragonal disphenoids
Faces116 scalene triangles, 58+116 isosceles triangles
Edges58+58+58
Vertices29
Vertex figureBiorthowedged rhombic gyroprism
Measures (circumradius $\sqrt2$ , based on a uniform duoprism)
Edge lengths7-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 density1
Related polytopes
Dual29-12 gyrochoron
Abstract & topological properties
Euler characteristic0
OrientableYes
Properties
SymmetryS2(I2(29)-12)×2I, order 116
ConvexYes
NatureTame

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.