# 10-2 step prism

10-2 step prism
Rank4
TypeIsogonal
SpaceSpherical
Elements
Cells10+10+10 phyllic disphenoids, 5 rhombic disphenoids
Faces20+20+20 scalene triangles, 10 isosceles triangles
Edges5+10+10+10+10
Vertices10
Vertex figureRidge-ditriakis notch
Measures (circumradius ${\displaystyle \sqrt2}$, based on a uniform duoprism)
Edge lengths8-valence (10): ${\displaystyle \sqrt{4-\sqrt5} ≈ 1.32813}$
4-valence (5): 2
3-valence (10): ${\displaystyle \sqrt5 ≈ 2.23607}$
4-valence (10): ${\displaystyle \sqrt5 ≈ 2.23607}$
4-valence (10): ${\displaystyle \sqrt{4+\sqrt5} ≈ 2.49721}$
Central density1
Related polytopes
Dual10-2 gyrochoron
Abstract & topological properties
Euler characteristic0
OrientableYes
Properties
SymmetryS2(I2(10)-2), order 20
ConvexYes
NatureTame

The 10-2 step prism is a convex isogonal polychoron and a member of the step prism family. It has 5 rhombic disphenoids and 30 phyllic disphenoids of three kinds as cells, with 14 (2 rhombic and 12 phyllic disphenoids) joining at each vertex.

It is one of 4 isogonal polychora with 10 vertices, and the only one to have only step prism symmetry in its highest symmetry form.

Using the ratio method, the lowest possible ratio between the longest and shortest edges is 1:${\displaystyle \frac{\sqrt{26+10\sqrt5}}{4}}$ ≈ 1:1.73855.

## Vertex coordinates

Coordinates for the vertices of a 10-2 step prism inscribed in a decagonal duoprism with base lengths a and b are given by:

• (a*sin(πk/5), a*cos(πk/5), b*sin(2πk/5), b*cos(2πk/5)),

where k is an integer from 0 to 9. If the edge length differences are to be minimized, the ratio of a:b must be equivalent to 1:${\displaystyle \frac{1+\sqrt5}{2}}$ ≈ 1:1.61803.