# Truncated octahedral prism

The truncated octahedral prism or tope is a prismatic uniform polychoron that consists of 2 truncated octahedra, 6 cubes, and 8 hexagonal prisms. Each vertex joins 1 truncated octahedron, 1 cube, and 2 hexagonal prisms. As the name suggests, it is a prism based on the truncated octahedron. As such it is also a convex segmentochoron (designated K-4.89 on Richard Klitzing's list).

Truncated octahedral prism
Rank4
TypeUniform
SpaceSpherical
Notation
Bowers style acronymTope
Coxeter diagramx o4x3x ()
Elements
Cells6 cubes, 8 hexagonal prisms, 2 truncated octahedra
Faces12+12+24 squares, 16 hexagons
Edges24+24+48
Vertices48
Vertex figureSphenoid, edge lengths 2, 3, 3 (base), 2 (legs)
Measures (edge length 1)
Circumradius${\displaystyle \frac{\sqrt{11}}{2} ≈ 1.65831}$
Hypervolume${\displaystyle 8\sqrt2 ≈ 11.31371}$
Dichoral anglesCube–4–hip: ${\displaystyle \arccos\left(-\frac{\sqrt3}{3}\right) ≈ 125.26439°}$
Hip–4–hip: ${\displaystyle \arccos\left(-\frac13\right) ≈ 109.47122°}$
Toe–4–cube: 90°
Toe–6–hip: 90°
Height1
Central density1
Number of external pieces16
Level of complexity12
Related polytopes
ArmyTope
RegimentTope
DualTetrakis hexahedral tegum
ConjugateNone
Abstract & topological properties
Flag count1152
Euler characteristic0
OrientableYes
Properties
SymmetryB3×A1, order 96
ConvexYes
NatureTame

This polychoron can be alternated into a pyritohedral icosahedral antiprism, although it cannot be made uniform.

## Vertex coordinates

The vertices of a truncated octahedral prism of edge length 1 are given by all permutations of the first three coordinates of:

• ${\displaystyle \left(±\sqrt2,\,±\frac{\sqrt2}{2},\,0,\,±\frac12\right).}$

## Representations

A truncated octahedral prism has the following Coxeter diagrams:

• x o4x3x (full symmetry)
• x x3x3x (       ) (bases have A3 symmetry)
• s2s4x3x (       ) (bases have A3 symmetry, as snub)
• oo4xx3xx&#x (bases considered separately)
• xx3xx3xx&#x (bases separately under A3)
• xxxxx xuxux4ooqoo&#xt (BC2×A1 axial, cube-first)
• xxxx xuxx3xxux&#xt (A2×A1 axial, hexagonal prism-first)