# 13-5 step prism

13-5 step prism
File:13-5 step prism.png
Rank4
TypeIsogonal
Elements
Cells26 phyllic disphenoids, 13 tetragonal disphenoids
Faces26+52 isosceles triangles
Edges26+26
Vertices13
Vertex figureProdigonal antiprismatic symmetric snub disphenoid
Measures (circumradius ${\displaystyle {\sqrt {2}}}$, based on a uniform duoprism)
Edge lengths5-valence (26): ${\displaystyle 2{\sqrt {\sin ^{2}{\frac {2\pi }{13}}+\sin ^{2}{\frac {3\pi }{13}}}}\approx 1.61951}$
4-valence (26): ${\displaystyle {\sqrt {4+2\cos {\frac {3\pi }{13}}-2\cos {\frac {2\pi }{13}}}}\approx 1.93031}$
Central density1
Related polytopes
Dual13-5 gyrochoron
Abstract & topological properties
Euler characteristic0
OrientableYes
Properties
SymmetryS2(I2(13)-5)×2I, order 52
ConvexYes
NatureTame

The 13-5 step prism is a convex isogonal polychoron and a member of the step prism family. It has 13 tetragonal disphenoids and 26 phyllic disphenoids as cells. 4 tetragonal and 8 phyllic disphenoids join at each vertex. It is also the pentagonal funk tegum.

Compared to other 13-vertex step prisms, this polychoron has doubled symmetry, because 13 is a factor of 52+1 = 26.

Its vertex figure is topologically equivalent to the Johnson solid snub disphenoid.

The ratio between the longest and shortest edges is 1:${\displaystyle {\sqrt {\frac {2-\cos {\frac {2\pi }{13}}+\cos {\frac {3\pi }{13}}}{2-\sin {\frac {\pi }{26}}-\sin {\frac {5\pi }{26}}}}}}$ ≈ 1:1.19192.

## Vertex coordinates

Coordinates for the vertices of a 13-5 step prism of circumradius 2 are given by:

• (sin(2πk/13), cos(2πk/13), sin(10πk/13), cos(10πk/13)),

where k is an integer from 0 to 12.

## Isogonal derivatives

Substitution by vertices of these following elements will produce these convex isogonal polychora: