Truncated octahedral prism
Truncated octahedral prism  

Rank  4 
Type  Uniform 
Notation  
Bowers style acronym  Tope 
Coxeter diagram  x o4x3x () 
Elements  
Cells  6 cubes, 8 hexagonal prisms, 2 truncated octahedra 
Faces  12+12+24 squares, 16 hexagons 
Edges  24+24+48 
Vertices  48 
Vertex figure  Sphenoid, edge lengths √2, √3, √3 (base), √2 (legs) 
Measures (edge length 1)  
Circumradius  
Hypervolume  
Dichoral angles  Cube–4–hip: 
Hip–4–hip:  
Toe–4–cube: 90°  
Toe–6–hip: 90°  
Height  1 
Central density  1 
Number of external pieces  16 
Level of complexity  12 
Related polytopes  
Army  Tope 
Regiment  Tope 
Dual  Tetrakis hexahedral tegum 
Conjugate  None 
Abstract & topological properties  
Flag count  1152 
Euler characteristic  0 
Orientable  Yes 
Properties  
Symmetry  B_{3}×A_{1}, order 96 
Convex  Yes 
Nature  Tame 
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 K4.89 on Richard Klitzing's list).
This polychoron can be alternated into a pyritohedral icosahedral antiprism, although it cannot be made uniform.
Gallery[edit  edit source]

Card with cell counts, verf, and crosssections

Segmentochoron display, toe atop toe

Net
Vertex coordinates[edit  edit source]
The vertices of a truncated octahedral prism of edge length 1 are given by all permutations of the first three coordinates of:
Representations[edit  edit source]
A truncated octahedral prism has the following Coxeter diagrams:
 x o4x3x (full symmetry)
 x x3x3x () (bases have A_{3} symmetry)
 s2s4x3x () (bases have A_{3} symmetry, as snub)
 oo4xx3xx&#x (bases considered separately)
 xx3xx3xx&#x (bases separately under A_{3})
 xxxxx xuxux4ooqoo&#xt (BC_{2}×A_{1} axial, cubefirst)
 xxxx xuxx3xxux&#xt (A_{2}×A_{1} axial, hexagonal prismfirst)
External links[edit  edit source]
 Bowers, Jonathan. "Category 19: Prisms" (#900).
 Klitzing, Richard. "Tope".
 Wikipedia contributors. "Truncated octahedral prism".