# Truncated cube

Truncated cube | |
---|---|

Rank | 3 |

Type | Uniform |

Notation | |

Bowers style acronym | Tic |

Coxeter diagram | x4x3o () |

Conway notation | tC |

Elements | |

Faces | 8 triangles, 6 octagons |

Edges | 12+24 |

Vertices | 24 |

Vertex figure | Isosceles triangle, edge lengths 1, √2+√2, √2+√2 |

Measures (edge length 1) | |

Circumradius | |

Volume | |

Dihedral angles | 8–3: |

8–8: 90° | |

Central density | 1 |

Number of external pieces | 14 |

Level of complexity | 3 |

Related polytopes | |

Army | Tic |

Regiment | Tic |

Dual | Triakis octahedron |

Conjugate | Quasitruncated hexahedron |

Abstract & topological properties | |

Flag count | 144 |

Euler characteristic | 2 |

Surface | Sphere |

Orientable | Yes |

Genus | 0 |

Properties | |

Symmetry | B_{3}, order 48 |

Convex | Yes |

Nature | Tame |

The **truncated cube**, the **truncated hexahedron**, or **tic**, is one of the 13 Archimedean solids. It consists of 8 triangles and 6 octagons. Each vertex joins one triangle and two octagons. As the name suggests, it can be obtained by truncation of the cube.

## Vertex coordinates[edit | edit source]

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

- .

## Representations[edit | edit source]

A truncated cube has the following Coxeter diagrams:

- x4x3o () (full symmetry)
- xwwx4xoox&#xt (BC2 axial, octagon-first)
- xwwxoo3ooxwwx&#xt (A2 axial, triangle-first)
- wx3oo3xw&#zx (A3 subsymmetry, as hull of 2 small rhombitetratetrahedra)
- wx xw4xo&#zx (BC2×A1 symmetry)
- wwx wxw xww&#zx (A1×A1×A1 symmetry)
- oxwUwxo xwwxwwx&#xt (A1×A1 axial)

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

The truncated cube has a semi-uniform variant of the form x4y3o that maintains its full symmetry. This variant has 8 triangles of size *y* and 6 ditetragons as faces.

With edges of length *a* (between two ditetragons) and *b* (between a ditetragon and a triangle), its circumradius is given by

and its volume is given by

- .

It has coordinates given by all permutations of:

- .

## Related polyhedra[edit | edit source]

A truncated cube can be augmented by attaching a square cupola to one of its octagonal faces, forming the augmented truncated cube. If a second square cupola is attached to the opposite octagonal face, the result is the biaugmented truncated cube.

The truncated rhombihedron is a uniform polyhedron compound composed of 5 truncated cubes.

## External links[edit | edit source]

- Bowers, Jonathan. "Polyhedron Category 2: Truncates" (#11).

- Klitzing, Richard. "tic".

- Quickfur. "The Truncated Cube".

- Wikipedia contributors. "Truncated cube".
- McCooey, David. "Truncated cube"