# 14-4 step prism

14-4 step prism Rank4
TypeIsogonal
SpaceSpherical
Elements
Cells14+14+14 phyllic disphenoids, 7 rhombic disphenoids
Faces28+28+28 scalene triangles, 14 isosceles triangles
Edges7+14+14+14+14
Vertices14
Vertex figure9-vertex polyhedron with 14 triangular faces
Measures (circumradius $\sqrt2$ , based on a uniform duoprism)
Edge lengths6-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 density1
Related polytopes
Dual14-4 gyrochoron
Abstract & topological properties
Euler characteristic0
OrientableYes
Properties
SymmetryS2(I2(14)-4), order 28
ConvexYes
NatureTame

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.