17-2 step prism

17-2 step prism
(OFF file)
Rank	4
Type	Isogonal
Space	Spherical
Elements
Cells	17+17+17+17+17+17+17 phyllic disphenoids
Faces	34+34+34+34+34+34 scalene triangles, 17+17 isosceles triangles
Edges	17+17+17+17+17+17+17+17
Vertices	17
Vertex figure	Ridge-quintitriakis bi-apiculated tetrahedron
Measures (circumradius $\sqrt2$, based on a uniform duoprism)
Edge lengths	15-valence (17): $2\sin\frac{\pi}{17}\sqrt{3+2\cos\frac{2\pi}{17}} ≈ 0.81058$
	3-valence (17): $2\sqrt{\sin^2\frac{2\pi}{17}+\sin^2\frac{4\pi}{17}} ≈ 1.52887$
	4-valence (17): $2\sqrt{\sin^2\frac{\pi}{17}+\cos^2\frac{\pi}{34}} ≈ 2.02509$
	4-valence (17): $\sqrt{4-2\sin\frac{5\pi}{34}+2\sin\frac{7\pi}{34}} ≈ 2.07697$
	4-valence (17): $\sqrt{4-2\sin\frac{5\pi}{34}+2\cos\frac{3\pi}{17}} ≈ 2.19293$
	4-valence (17): $\sqrt{4+2\sin\frac{3\pi}{34}+2\sin\frac{7\pi}{34}} ≈ 2.39846$
	4-valence (17): $\sqrt{4-2\sin\frac{\pi}{34}+2\cos\frac{\pi}{17}} ≈ 2.40446$
	4-valence (17): $\sqrt{4+2\sin\frac{3\pi}{34}+2\cos\frac{3\pi}{17}} ≈ 2.49955$
Central density	1
Related polytopes
Dual	17-2 gyrochoron
Abstract & topological properties
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	S2(I2(17)-2), order 34
Convex	Yes
Nature	Tame

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:$\frac{\sin\frac{4\pi}{17}\sqrt{2+2\sin\frac{\pi}{34}+2\cos\frac{\pi}{17}-2\cos\frac{2\pi}{17}}}{\sin\frac{\pi}{34}+\sin\frac{3\pi}{34}}$ ≈ 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:$\frac{1}{\sqrt{2\cos\frac{\pi}{17}-2\cos\frac{2\pi}{17}}}$ ≈ 1:3.14656.

External links

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