14-2 step prism

14-2 step prism
(OFF file)
Rank	4
Type	Isogonal
Space	Spherical
Elements
Cells	14+14+14+14+14 phyllic disphenoids, 7 rhombic disphenoids
Faces	28+28+28+28+28 scalene triangles, 14 isosceles triangles
Edges	7+14+14+14+14+14+14
Vertices	14
Vertex figure	Ridge-quadritriakis notch
Measures (circumradius $\sqrt2$, based on a uniform duoprism)
Edge lengths	12-valence (14): $2\sin\frac{\pi}{14}\sqrt{3+2\cos\frac\pi7} ≈ 0.97523$
	3-valence (14): $2\sin\frac\pi7\sqrt{3+2\cos\frac{2\pi}{7}} ≈ 1.78831$
	4-valence (7): 2
	4-valence (14): $2\sqrt{\sin^2\frac\pi7+\cos^2\frac{\pi}{14}} ≈ 2.13423$
	4-valence (14): $\sqrt{4+2\cos\frac\pi7-2\sin\frac{\pi}{14}} ≈ 2.31450$
	4-valence (14): $\sqrt{4+2\sin\frac{\pi}{14}+2\cos\frac{3\pi}{14}} ≈ 2.38580$
	4-valence (14): $\sqrt{4+2\cos\frac\pi7+2\sin\frac{\pi}{14}} ≈ 2.49940$
Central density	1
Related polytopes
Dual	14-2 gyrochoron
Abstract properties
Euler characteristic	0
Topological properties
Orientable	Yes
Properties
Symmetry	S2(I2(14)-2), order 28
Convex	Yes
Nature	Tame

The 14-2 step prism is a convex isogonal polychoron and a member of the step prism family. It has 7 rhombic disphenoids and 70 phyllic disphenoids of five kinds as cells, with 22 (2 rhombic and 20 phyllic disphenoids) joining at each vertex.

Using the ratio method, the lowest possible ratio between the longest and shortest edges is 1:$\sqrt{\frac{83}{60-72\cos\frac{2\pi}{7}}-\frac1{12}}$ ≈ 1:2.35298.

Vertex coordinates

Coordinates for the vertices of a 14-2 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(2πk/7), b*cos(2πk/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:$1+2\cos\frac{2\pi}{7}$ ≈ 1:2.24698.

External links

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