14-3 step prism

14-3 step prism
(OFF file)
Rank	4
Type	Isogonal
Elements
Cells	14+14+14+14 phyllic disphenoids
Faces	28+28+28 scalene triangles, 14+14 isosceles triangles
Edges	14+14+14+14+14
Vertices	14
Vertex figure	10-vertex polyhedron with 16 triangular faces
Measures (circumradius ${\sqrt {2}}$, based on a uniform duoprism)
Edge lengths	8-valence (14): $2{\sqrt {\sin ^{2}{\frac {\pi }{14}}+\sin ^{2}{\frac {3\pi }{14}}}}\approx 1.32402$
	5-valence (14): $2\sin {\frac {\pi }{7}}{\sqrt {3+2\cos {\frac {2\pi }{7}}}}\approx 1.78831$
	4-valence (14): $2{\sqrt {\sin ^{2}{\frac {\pi }{14}}+\cos ^{2}{\frac {\pi }{7}}}}\approx 1.85608$
	3-valence (14): $2{\sqrt {\sin ^{2}{\frac {\pi }{7}}+\cos ^{2}{\frac {\pi }{14}}}}\approx 2.13423$
	4-valence (14): ${\sqrt {4+2\sin {\frac {3\pi }{14}}-2\sin {\frac {\pi }{14}}}}\approx 2.19133$
Central density	1
Related polytopes
Dual	14-3 gyrochoron
Abstract & topological properties
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	S2(I2(14)-3), order 28
Convex	Yes
Nature	Tame

The 14-3 step prism, also known as the 7-2 double step prism, is a convex isogonal polychoron and a member of the step prism family. It has 56 phyllic disphenoids of four kinds as cells, with 16 joining at each vertex. It can also be constructed as the 14-5 step prism, or as the convex hull of two opposite 7-2 step prisms.

Using the ratio method, the lowest possible ratio between the longest and shortest edges is 1:${\sqrt {\frac {1-\sin {\frac {\pi }{14}}+2\cos {\frac {\pi }{7}}}{\sin {\frac {\pi }{14}}+2\cos {\frac {\pi }{7}}-1}}}$ ≈ 1:1.58677.

Vertex coordinates

Coordinates for the vertices of a 14-3 step prism inscribed in a tetradecagonal duoprism with base lengths a and b are given by:

• $\left(a\sin {\frac {k\pi }{7}},\,a\cos {\frac {k\pi }{7}},\,b\sin {\frac {3k\pi }{7}},\,b\cos {\frac {3k\pi }{7}}\right)$,

where k is an integer from 0 to 13. If the edge length differences are to be minimized, the ratio of a:b must be equivalent to 1:${\sqrt {\frac {2\cos {\frac {\pi }{7}}+2\sin {\frac {\pi }{14}}-1}{1-2\sin {\frac {\pi }{14}}}}}$ ≈ 1:1.49899.

External links

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