13-3 step prism

13-3 step prism
(OFF file)
Rank	4
Type	Isogonal
Elements
Cells	13+13+13+13 phyllic disphenoids
Faces	26+26+26 scalene triangles, 13+13 isosceles triangles
Edges	13+13+13+13+13
Vertices	13
Vertex figure	10-vertex polyhedron with 16 triangular faces
Measures (circumradius ${\sqrt {2}}$, based on a uniform duoprism)
Edge lengths	8-valence (13): $2{\sqrt {\sin ^{2}{\frac {\pi }{13}}+\sin ^{2}{\frac {3\pi }{13}}}}\approx 1.40997$
	6-valence (13): $2{\sqrt {\sin ^{2}{\frac {\pi }{13}}+\sin ^{2}{\frac {4\pi }{13}}}}\approx 1.71415$
	3-valence (13): ${\sqrt {4+2\cos {\frac {3\pi }{13}}-2\cos {\frac {4\pi }{13}}}}\approx 2.08827$
	4-valence (13): ${\sqrt {4+2\sin {\frac {3\pi }{26}}-2\sin {\frac {\pi }{26}}}}\approx 2.11380$
	3-valence (13): ${\sqrt {4+2\cos {\frac {\pi }{13}}-2\cos {\frac {4\pi }{13}}}}\approx 2.19220$
Central density	1
Related polytopes
Dual	13-3 gyrochoron
Abstract & topological properties
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	S2(I2(13)-3), order 26
Convex	Yes
Nature	Tame

The 13-3 step prism is a convex isogonal polychoron and member of the step prism family. It has 52 phyllic disphenoids of four kinds as cells, with 16 joining at each vertex. It can also be constructed as the 13-4 step prism.

Using the ratio method, the lowest possible ratio between the longest and shortest edges is 1:${\sqrt {\frac {11+22\sin {\frac {\pi }{26}}-12\sin {\frac {3\pi }{26}}-18\cos {\frac {\pi }{13}}+2\cos {\frac {2\pi }{13}}}{9+16\sin {\frac {\pi }{26}}-6\sin {\frac {3\pi }{26}}-10\cos {\frac {\pi }{13}}-2\cos {\frac {2\pi }{13}}}}}$ ≈ 1:1.53455.

Vertex coordinates

Coordinates for the vertices of a 13-3 step prism inscribed in a tridecagonal duoprism with base lengths a and b are given by:

• (a*sin(2πk/13), a*cos(2πk/13), b*sin(6πk/13), b*cos(6πk/13)),

where k is an integer from 0 to 12. If the edge length differences are to be minimized, the ratio of a:b must be equivalent to :${\sqrt {\frac {\cos {\frac {2\pi }{13}}+\sin {\frac {3\pi }{26}}}{\cos {\frac {2\pi }{13}}-\sin {\frac {\pi }{26}}}}}$ ≈ 1:1.27325.

External links

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