14-4 step prism

14-4 step prism
(OFF file)
Rank	4
Type	Isogonal
Space	Spherical
Elements
Cells	14+14+14 phyllic disphenoids, 7 rhombic disphenoids
Faces	28+28+28 scalene triangles, 14 isosceles triangles
Edges	7+14+14+14+14
Vertices	14
Vertex figure	9-vertex polyhedron with 14 triangular faces
Measures (circumradius $\sqrt2$, based on a uniform duoprism)
Edge lengths	6-valence (14): $2\sqrt{\sin^2\frac{\pi}{14}+\sin^2\frac{2\pi}{7}} ≈ 1.62576$
	6-valence (14): $2\sqrt{\sin^2\frac\pi7+\sin^2\frac{3\pi}{14}} ≈ 1.51920$
	4-valence (14): $2\sin\frac\pi7\sqrt{3+2\cos\frac{2\pi}{7}} ≈ 1.78831$
	4-valence (7): 2
	3-valence (14): $2\sqrt{\sin^2\frac\pi7+\cos^2\frac{\pi}{14}} ≈ 2.13423$
Central density	1
Related polytopes
Dual	14-4 gyrochoron
Abstract & topological properties
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	S2(I2(14)-4), order 28
Convex	Yes
Nature	Tame

The 14-4 step prism is a convex isogonal polychoron and a member of the step prism family. It has 7 rhombic disphenoids and 42 phyllic disphenoids of three kinds as cells, with 14 (2 rhombic and 12 phyllic disphenoids) joining at each vertex.

Using the ratio method, the lowest possible ratio between the longest and shortest edges is 1:$\sqrt{\frac{4}{{5-2\sin\frac{\pi}{14}+2\cos\frac{2\pi}{7}-4\cos\frac{\pi}{7}}}}$ ≈ 1:1.34899.

Vertex coordinates

Coordinates for the vertices of a 14-4 step prism inscribed in a tetradecagonal duoprism with base lengths a and b are given by:

• (a*sin(πk/7), a*cos(πk/7), b*sin(3πk/7), b*cos(3πk/7)),

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:$\sqrt{\frac{1+\cos\frac{2\pi}{7}}{1+\cos\frac\pi7}}$ ≈ 1:0.92414.

External links

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