# 42-13 step prism

42-13 step prism
File:42-13 step prism.png
Rank4
TypeIsogonal
SpaceSpherical
Elements
Cells84+84+84 phyllic disphenoids, 42 rhombic disphenoids
Faces168+168+168 scalene triangles, 84 isosceles triangles
Edges42+42+84+84+84
Vertices42
Vertex figure16-vertex polyhedron with 28 triangular faces
Measures (edge length 1)
Central density1
Related polytopes
Dual42-13 gyrochoron
Abstract & topological properties
Euler characteristic0
OrientableYes
Properties
SymmetryS2(I2(42)-13)×2R, order 168
ConvexYes
NatureTame

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:${\displaystyle \sqrt{\frac{7+46\cos\frac{2\pi}{7}}{16\cos\frac{2\pi}{7}-6}}}$ ≈ 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.