# Truncated octahedron

Truncated octahedron | |
---|---|

Rank | 3 |

Notation | |

Bowers style acronym | Toe |

Coxeter diagram | o4x3x () |

Conway notation | tO |

Stewart notation | K_{3} |

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 | |

Volume | |

Dihedral angles | 6–4: |

6–6: | |

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 | B_{3}, order 48 |

Flag orbits | 3 |

Convex | Yes |

Nature | Tame |

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 non-prismatic convex regular-faced polyhedron that can tile 3D space by itself.^{[1]}^{[2]} 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[edit | edit source]

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

- .

Dividing these by gives integral coordinates in 3-dimensions.

## Representations[edit | edit source]

A truncated octahedron has the following Coxeter diagrams:

- o4x3x () (full symmetry)
- x3x3x () (A
_{3}symmetry, as great rhombitetratetrahedron) - s4x3x () (as hexagon-alternated great rhombicuboctahedron)
- xuxux4ooqoo&#xt (B
_{2}axial, square-first) - xxux3xuxx&#xt (A
_{2}axial, hexagon-first) - Qqo xux4ooq&#zx (B
_{2}×A_{1}symmetry) - xu(wx)(wx)ux oq(oQ)(oQ)qo&#xt (K
_{2}axial, edge-first) - xu(xd)ux xu(dx)ux&#xt (square-first when seen as rectangle)

## Semi-uniform variant[edit | edit source]

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 and its volume is given by .

Generally, alternating these polyhedra gives a pyritohedral icosahedron.

It has coordinates given by all permutations of:

- .

## External links[edit | edit source]

- 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".

## References[edit | edit source]

- ↑ Steinhaus (1999:210)
- ↑ Wells (1991:232)

## Biblography[edit | edit source]

- Steinhaus, Hugo (1999).
*Mathematical Snapshots*(3 ed.). Dover. ISBN 0486409147. OL 41316M. - Wells, David (1991).
*The Penguin Dictionary of Curious and Interesting Geometry*. Illustrated by John Sharp. Penguin.