# 17-4 step prism

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17-4 step prism
File:17-4 step prism.png
Rank4
TypeIsogonal
SpaceSpherical
Elements
Cells34+34 phyllic disphenoids, 17 tetragonal disphenoids
Faces68 scalene triangles, 34+68 isosceles triangles
Edges34+34+34
Vertices17
Vertex figureParatetraaugmented triakis tetragonal disphenoid
Measures (circumradius ${\displaystyle \sqrt2}$, based on a uniform duoprism)
Edge lengths8-valence (34): ${\displaystyle 2\sqrt{\sin^2\frac{\pi}{17}+\sin^2\frac{4\pi}{17}} ≈ 1.39661}$
4-valence (34): ${\displaystyle \sqrt{4+2\sin\frac{3\pi}{34}-2\sin\frac{5\pi}{34}} ≈ 1.91203}$
3-valence (34): ${\displaystyle \sqrt{4+2\cos\frac{\pi}{17}-2\cos\frac{4\pi}{17}} ≈ 2.11847}$
Central density1
Related polytopes
Dual17-4 gyrochoron
Abstract & topological properties
Euler characteristic0
OrientableYes
Properties
SymmetryS2(I2(17)-4)×2I, order 68
ConvexYes
NatureTame

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:${\displaystyle \frac{\sqrt{871+13\sqrt{17}+13\sqrt{1802+134\sqrt{17}}}}{26}}$ ≈ 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.