Truncated octahedral prism

Truncated octahedral prism
(OFF file)
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	${\frac {\sqrt {11}}{2}}\approx 1.65831$
Hypervolume	$8{\sqrt {2}}\approx 11.31371$
Dichoral angles	Cube–4–hip: $\arccos \left(-{\frac {\sqrt {3}}{3}}\right)\approx 125.26439^{\circ }$
	Hip–4–hip: $\arccos \left(-{\frac {1}{3}}\right)\approx 109.47122^{\circ }$
	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	B3×A1, 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 K-4.89 on Richard Klitzing's list).

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

Gallery

• 

  Card with cell counts, verf, and cross-sections

• 

  Segmentochoron display, toe atop toe

• 

  Net

Vertex coordinates

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

• $\left(\pm {\sqrt {2}},\,\pm {\frac {\sqrt {2}}{2}},\,0,\,\pm {\frac {1}{2}}\right).$

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)

External links

• Bowers, Jonathan. "Category 19: Prisms" (#900).

• Klitzing, Richard. "Tope".
• Wikipedia contributors. "Truncated octahedral prism".