# 17-4 step prism

17-4 step prism
File:17-4 step prism.png
Rank4
TypeIsogonal
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 ${\displaystyle {\sqrt {2}}}$, based on a uniform duoprism)
Edge lengths8-valence (34): ${\displaystyle 2{\sqrt {\sin ^{2}{\frac {\pi }{17}}+\sin ^{2}{\frac {4\pi }{17}}}}\approx 1.39661}$
4-valence (34): ${\displaystyle {\sqrt {4+2\sin {\frac {3\pi }{34}}-2\sin {\frac {5\pi }{34}}}}\approx 1.91203}$
3-valence (34): ${\displaystyle {\sqrt {4+2\cos {\frac {\pi }{17}}-2\cos {\frac {4\pi }{17}}}}\approx 2.11847}$
Central density1
Related polytopes
Dual17-4 gyrochoron
Abstract & topological properties
Flag count2040
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:

• ${\displaystyle \left(\sin {\frac {2k\pi }{17}},\,\cos {\frac {2k\pi }{17}},\,\sin {\frac {8k\pi }{17}},\,\cos {\frac {8k\pi }{17}}\right)}$,

where k is an integer from 0 to 16.