Truncated octahedron

The truncated octahedron or toe is one of the 13 Archimedean solids. It consists of 6 squares and 8 ditrigons. Each vertex joins one square and two hexagons. As the name suggests, it can be obtained by the truncation of the octahedron. It is also the omnitruncate of the tetrahedral family.

Truncated octahedron
(3D model) (OFF file)
Rank	3
Space	Spherical
Notation
Bowers style acronym	Toe
Coxeter diagram	o4x3x ()
Elements
Faces	6 squares, 8 hexagons
Edges	12+24
Vertices	24
Vertex figure	Isosceles triangle, edge lengths √2, √3, √3
Measures (edge length 1)
Circumradius	$\frac{\sqrt{10}}{2} ≈ 1.58113$
Volume	$8\sqrt2 ≈ 11.31371$
Dihedral angles	6–4: $\arccos\left(-\frac{\sqrt3}{3}\right) ≈ 125.26439^\circ$
	6–6: $\arccos\left(-\frac13\right) ≈ 109.47122^\circ$
Central density	1
Number of external pieces	14
Level of complexity	3
Related polytopes
Army	Toe
Regiment	Toe
Dual	Tetrakis hexahedron
Conjugate	None
Abstract & topological properties
Flag count	144
Euler characteristic	2
Surface	Sphere
Orientable	Yes
Genus	0
Properties
Symmetry	B3, order 48
Convex	Yes
Nature	Tame

It is the only Archimedean solid that can tile 3D space by itself. This results in the bitruncated cubic honeycomb.

It can be alternated into the icosahedron after all edge lengths are made equal.

It is the 4th-order permutohedron.

Vertex coordinates

A truncated octahedron of edge length 1 has vertex coordinates given by all permutations of

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

Representations

A truncated octahedron has the following Coxeter diagrams:

• o4x3x (full symmetry)
• x3x3x (A3 symmetry, as great rhombitetratetrahedron)
• s4x3x (as hexagon-alternated great rhombicuboctahedron)
• xuxux4ooqoo&#xt (BC2 axial, square-first)
• xxux3xuxx&#xt (A2 axial, hexagon-first)
• Qqo xux4ooq&#zx (BC2×A1 symmetry)
• xu(wx)(wx)ux oq(oQ)(oQ)qo&#xt (A1×A1 axial, edge-first)
• xu(xd)ux xu(dx)ux&#xt (square-first when seen as rectangle)

Semi-uniform variant

Truncated tetratetrahedron, a variant of the truncated octahedron with tetrahedral symmetry

The truncated octahedron has a semi-uniform variant of the form o4y3x that maintains its full symmetry. This variant has 6 squares of size y and 8 ditrigons as faces.

With edges of length a (between two ditrigons) and b (between a ditrigon and a square), its circumradius is given by $\frac{\sqrt{2a^2+4b^2+4ab}}{2}$ and its volume is given by $(a^3+6a^2b+12ab^2+5b^3)\frac{\sqrt2}{3}$.

Generally, alternating these polyhedra gives a pyritohedral icosahedron.

It has coordinates given by all permutations of:

• $\left(±(a+b)\frac{\sqrt2}{2},\,±b\frac{\sqrt2}{2},\,0\right).$

Related polyhedra

o4o3o truncations

Name	OBSA	Schläfli symbol	CD diagram	Picture
Cube	cube	{4,3}	x4o3o
Truncated cube	tic	t{4,3}	x4x3o
Cuboctahedron	co	r{4,3}	o4x3o
Truncated octahedron	toe	t{3,4}	o4x3x
Octahedron	oct	{3,4}	o4o3x
Small rhombicuboctahedron	sirco	rr{4,3}	x4o3x
Great rhombicuboctahedron	girco	tr{4,3}	x4x3x
Snub cube	snic	sr{4,3}	s4s3s

o3o3o truncations

Name	OBSA	Schläfli symbol	CD diagram	Picture
Tetrahedron	tet	{3,3}	x3o3o
Truncated tetrahedron	tut	t{3,3}	x3x3o
Tetratetrahedron = Octahedron	oct	r{3,3}	o3x3o
Truncated tetrahedron	tut	t{3,3}	o3x3x
Tetrahedron	tet	{3,3}	o3o3x
Small rhombitetratetrahedron = Cuboctahedron	co	rr{3,3}	x3o3x
Great rhombitetratetrahedron = Truncated octahedron	toe	tr{3,3}	x3x3x
Snub tetrahedron = Icosahedron	ike	sr{3,3}	s3s3s

External links

• Bowers, Jonathan. "Polyhedron Category 2: Truncates" (#12).

• Klitzing, Richard. "toe".

• Quickfur. "The Truncated Octahedron".

• Wikipedia Contributors. "Truncated octahedron".
• McCooey, David. "Truncated Octahedron"

• Hi.gher.Space Wiki Contributors. "Octahedral truncate".