17-4 step prism

17-4 step prism
File:17-4 step prism.png (OFF file)
Rank	4
Type	Isogonal
Space	Spherical
Elements
Cells	34+34 phyllic disphenoids, 17 tetragonal disphenoids
Faces	68 scalene triangles, 34+68 isosceles triangles
Edges	34+34+34
Vertices	17
Vertex figure	Paratetraaugmented triakis tetragonal disphenoid
Measures (circumradius $\sqrt2$, based on a uniform duoprism)
Edge lengths	8-valence (34): $2\sqrt{\sin^2\frac{\pi}{17}+\sin^2\frac{4\pi}{17}} ≈ 1.39661$
	4-valence (34): $\sqrt{4+2\sin\frac{3\pi}{34}-2\sin\frac{5\pi}{34}} ≈ 1.91203$
	3-valence (34): $\sqrt{4+2\cos\frac{\pi}{17}-2\cos\frac{4\pi}{17}} ≈ 2.11847$
Central density	1
Related polytopes
Dual	17-4 gyrochoron
Abstract & topological properties
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	S2(I2(17)-4)×2I, order 68
Convex	Yes
Nature	Tame

The 17-4 step prism is a convex isogonal polychoron, member of the step prism family. It has 17 tetragonal disphenoids and 68 phyllic disphenoids of two kinds as cells. 2 tetragonal and 16 phyllic disphenoids join at each vertex.

Compared to other 17-vertex step prisms, this polychoron has doubled symmetry, because 17 is a factor of 4²+1 = 17.

The ratio between the longest and shortest edges is 1:$\frac{\sqrt{871+13\sqrt{17}+13\sqrt{1802+134\sqrt{17}}}}{26}$ ≈ 1:1.51687.

Vertex coordinates

Coordinates for the vertices of a 17-4 step prism of circumradius √2 are given by:

• (sin(2πk/17), cos(2πk/17), sin(8πk/17), cos(8πk/17)),

where k is an integer from 0 to 16.

External links

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