# 8-2 step prism

8-2 step prism
Rank4
TypeIsogonal
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 ${\displaystyle {\sqrt {2}}}$, based on a uniform duoprism)
Edge lengths6-valence (8): ${\displaystyle {\sqrt {4-{\sqrt {2}}}}\approx 1.60804}$
4-valence (4): 2
4-valence (8): ${\displaystyle {\sqrt {4+{\sqrt {2}}}}\approx 2.32685}$
3-valence (8): ${\displaystyle {\sqrt {6}}\approx 2.44949}$
Central density1
Related polytopes
Dual8-2 gyrochoron
Abstract & topological properties
Flag count480
Euler characteristic0
OrientableYes
Properties
SymmetryS2(I2(8)-2), order 16
Flag orbits30
ConvexYes
NatureTame

The 8-2 step prism is a convex isogonal polychoron and a step prism. 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:${\displaystyle {\frac {\sqrt {6+2{\sqrt {2}}}}{2}}}$ ≈ 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:${\displaystyle {\frac {\sqrt {4+2{\sqrt {2}}}}{2}}}$ ≈ 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