# Truncated octahedron

Truncated octahedron
Rank3
SpaceSpherical
Notation
Bowers style acronymToe
Coxeter diagramo4x3x ()
Elements
Faces6 squares, 8 hexagons
Edges12+24
Vertices24
Vertex figureIsosceles triangle, edge lengths 2, 3, 3
Measures (edge length 1)
Circumradius${\displaystyle \frac{\sqrt{10}}{2} ≈ 1.58113}$
Volume${\displaystyle 8\sqrt2 ≈ 11.31371}$
Dihedral angles6–4: ${\displaystyle \arccos\left(-\frac{\sqrt3}{3}\right) ≈ 125.26439^\circ}$
6–6: ${\displaystyle \arccos\left(-\frac13\right) ≈ 109.47122^\circ}$
Central density1
Number of pieces14
Level of complexity3
Related polytopes
ArmyToe
RegimentToe
DualTetrakis hexahedron
ConjugateNone
Abstract properties
Flag count144
Euler characteristic2
Topological properties
SurfaceSphere
OrientableYes
Genus0
Properties
SymmetryB3, order 48
ConvexYes
NatureTame

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.

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

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

## Representations

A truncated octahedron has the following Coxeter diagrams:

## 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 ${\displaystyle \frac{\sqrt{2a^2+4b^2+4ab}}{2}}$ and its volume is given by ${\displaystyle (a^3+6a^2b+12ab^2+5b^3)\frac{\sqrt2}{3}}$.

Generally, alternating these polyhedra gives a pyritohedral icosahedron.

It has coordinates given by all permutations of:

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

## 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