13-5 step prism

13-5 step prism
File:13-5 step prism.png (OFF file)
Rank	4
Type	Isogonal
Space	Spherical
Elements
Cells	26 phyllic disphenoids, 13 tetragonal disphenoids
Faces	26+52 isosceles triangles
Edges	26+26
Vertices	13
Vertex figure	Prodigonal antiprismatic symmetric snub disphenoid
Measures (circumradius $\sqrt2$, based on a uniform duoprism)
Edge lengths	5-valence (26): $2\sqrt{\sin^2\frac{2\pi}{13}+\sin^2\frac{3\pi}{13}} ≈ 1.61951$
	4-valence (26): $\sqrt{4+2\cos\frac{3\pi}{13}-2\cos\frac{2\pi}{13}} ≈ 1.93031$
Central density	1
Related polytopes
Dual	13-5 gyrochoron
Abstract & topological properties
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	S2(I2(13)-5)×2I, order 52
Convex	Yes
Nature	Tame

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:$\sqrt{\frac{2-\cos\frac{2\pi}{13}+\cos\frac{3\pi}{13}}{2-\sin\frac{\pi}{26}-\sin\frac{5\pi}{26}}}$ ≈ 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:

• Tetragonal disphenoid (13): 13-5 step prism
• Edge (26): Small 13-5 double step prism
• Edge (26): Small 13-5 double gyrostep prism

External links

• Bowers, Jonathan. "Four Dimensional Dice Up To Twenty Sides".