# 8-2 step prism

8-2 step prism Rank4
TypeIsogonal
SpaceSpherical
Elements
Cells8+8 phyllic disphenoids, 4 rhombic disphenoids
Faces16+16 scalene triangles, 8 isosceles triangles
Edges4+8+8+8
Vertices8
Vertex figureRidge-triakis notch
Measures (circumradius $\sqrt2$ , based on a uniform duoprism)
Edge lengths6-valence (8): $\sqrt{4-\sqrt2} ≈ 1.60804$ 4-valence (4): 2
4-valence (8): $\sqrt{4+\sqrt2} ≈ 2.32685$ 3-valence (8): $\sqrt6 ≈ 2.44949$ Central density1
Related polytopes
Dual8-2 gyrochoron
Abstract & topological properties
Euler characteristic0
OrientableYes
Properties
SymmetryS2(I2(8)-2), order 16
ConvexYes
NatureTame

The 8-2 step prism is a convex isogonal polychoron and a member of the step prism family. It has 4 rhombic disphenoids and 16 phyllic disphenoids of two kinds as cells, with a total of 10 (2 rhombic and 8 phyllic disphenoids) joining at each vertex. It is one of 3 isogonal polychora with 8 vertices, and the only one not to be uniform (the other 2 are the hexadecachoron and tetrahedral prism).

Using the ratio method, the lowest possible ratio between the longest and shortest edges is 1:$\frac{\sqrt{6+2\sqrt2}}{2}$ ≈ 1:1.48563.

## Vertex coordinates

Coordinates for the vertices of an 8-2 step prism inscribed in an octagonal duoprism with base lengths a and b are given by:

• (a*sin(πk/4), a*cos(πk/4), b*sin(πk/2), b*cos(πk/2)),

where k is an integer from 0 to 7. If the edge length differences are to be minimized, the ratio of a:b must be equivalent to 1:$\frac{\sqrt{4+2\sqrt2}}{2}$ ≈ 1:1.30656.

## Isogonal derivatives

Substitution by vertices of these following elements will produce these convex isogonal polychora:

• Phyllic disphenoid (8): 8-2 step prism
• Scalene triangle (8): 8-2 step prism
• Scalene triangle (16): 16-2 step prism
• Edge (8): 8-2 step prism