# 17-2 step prism

17-2 step prism
Rank4
TypeIsogonal
SpaceSpherical
Elements
Cells17+17+17+17+17+17+17 phyllic disphenoids
Faces34+34+34+34+34+34 scalene triangles, 17+17 isosceles triangles
Edges17+17+17+17+17+17+17+17
Vertices17
Vertex figureRidge-quintitriakis bi-apiculated tetrahedron
Measures (circumradius ${\displaystyle \sqrt2}$, based on a uniform duoprism)
Edge lengths15-valence (17): ${\displaystyle 2\sin\frac{\pi}{17}\sqrt{3+2\cos\frac{2\pi}{17}} ≈ 0.81058}$
3-valence (17): ${\displaystyle 2\sqrt{\sin^2\frac{2\pi}{17}+\sin^2\frac{4\pi}{17}} ≈ 1.52887}$
4-valence (17): ${\displaystyle 2\sqrt{\sin^2\frac{\pi}{17}+\cos^2\frac{\pi}{34}} ≈ 2.02509}$
4-valence (17): ${\displaystyle \sqrt{4-2\sin\frac{5\pi}{34}+2\sin\frac{7\pi}{34}} ≈ 2.07697}$
4-valence (17): ${\displaystyle \sqrt{4-2\sin\frac{5\pi}{34}+2\cos\frac{3\pi}{17}} ≈ 2.19293}$
4-valence (17): ${\displaystyle \sqrt{4+2\sin\frac{3\pi}{34}+2\sin\frac{7\pi}{34}} ≈ 2.39846}$
4-valence (17): ${\displaystyle \sqrt{4-2\sin\frac{\pi}{34}+2\cos\frac{\pi}{17}} ≈ 2.40446}$
4-valence (17): ${\displaystyle \sqrt{4+2\sin\frac{3\pi}{34}+2\cos\frac{3\pi}{17}} ≈ 2.49955}$
Central density1
Related polytopes
Dual17-2 gyrochoron
Abstract & topological properties
Euler characteristic0
OrientableYes
Properties
SymmetryS2(I2(17)-2), order 34
ConvexYes
NatureTame

The 17-2 step prism is a convex isogonal polychoron and a member of the step prism family. It has 119 phyllic disphenoids of seven kinds as cells, with 28 joining at each vertex. It can also be constructed as the 17-8 step prism.

Using the ratio method, the lowest possible ratio between the longest and shortest edges is 1:${\displaystyle \frac{\sin\frac{4\pi}{17}\sqrt{2+2\sin\frac{\pi}{34}+2\cos\frac{\pi}{17}-2\cos\frac{2\pi}{17}}}{\sin\frac{\pi}{34}+\sin\frac{3\pi}{34}}}$ ≈ 1:2.78329.

## Vertex coordinates

Coordinates for the vertices of a 17-2 step prism inscribed in a heptadecagonal duoprism with base lengths a and b are given by:

• (a*sin(2πk/17), a*cos(2πk/17), b*sin(4πk/17), b*cos(4πk/17)),

where k is an integer from 0 to 16. If the edge length differences are to be minimized, the ratio of a:b must be equivalent to 1:${\displaystyle \frac{1}{\sqrt{2\cos\frac{\pi}{17}-2\cos\frac{2\pi}{17}}}}$ ≈ 1:3.14656.