# Tetrahedral-octahedral honeycomb

Tetrahedral-octahedral honeycomb | |
---|---|

Rank | 4 |

Type | Uniform |

Space | Euclidean |

Notation | |

Bowers style acronym | Octet |

Coxeter diagram | |

Elements | |

Cells | 2N tetrahedra, N octahedra |

Faces | 8N triangles |

Edges | 6N |

Vertices | N |

Vertex figure | Cuboctahedron, edge length 1 |

Measures (edge length 1) | |

Vertex density | |

Dual cell volume | |

Related polytopes | |

Army | Octet |

Regiment | Octet |

Dual | Rhombic dodecahedral honeycomb |

Conjugate | None |

Topological properties | |

Orientable | Yes |

Properties | |

Symmetry | S_{4} |

Convex | Yes |

The **tetrahedral-octahedral honeycomb**, or **octet**, also known as the **alternated cubic honeycomb**, is a convex uniform honeycomb. 6 octahedra and 8 tetrahedra join at each vertex of this honeycomb, with a cuboctahedron as the vertex figure. As one of its names suggests, it can be formed by alternation of the cubic honeycomb. It is also the 3D simplectic honeycomb.

The vertex locations of this honeycomb are known as the **face-centered cubic** or FCC lattice, which has the important property that placing spheres at each of the points that touch each other results in a maximally dense packing of equal spheres. (There are infinitely many cubic close packings, but the FCC lattice is notable for its symmetry, along with the HCP lattice.) This geometric property makes the FCC lattice ubiquitous in chemistry, such as in the structure of sodium chloride crystals (as found in table salt). Taking the convex hull of a set of vertices enclosed by a sphere of any location or size results in a Waterman polyhedron.

## Vertex coordinates[edit | edit source]

The vertices of a tetrahedral-octahedral honeycomb of edge length 1 are given by

where i, j, and k are integers, and i+j+k is even.

## Representations[edit | edit source]

A tetrahedral-octahedral honeycomb has the following Coxeter diagrams:

- (full symmetry)
- (P4 symmetry, cyclotetrahedral honeycomb)
- (as alternated cubic honeycomb)
- (as alternated square prismatic honeycomb)
- (as alternated product of three diapeirogons)

## Gallery[edit | edit source]

## External links[edit | edit source]

- Klitzing, Richard. "octet".

- Wikipedia Contributors. "Tetrahedral-octahedral honeycomb".
- Binnendyk, Eric. "Category 1: Primaries" (#3).