# Truncated tetrahedron

Truncated tetrahedron | |
---|---|

Rank | 3 |

Type | Uniform |

Space | Spherical |

Notation | |

Bowers style acronym | Tut |

Coxeter diagram | x3x3o () |

Stewart notation | T_{3} |

Elements | |

Faces | 4 triangles, 4 hexagons |

Edges | 6+12 |

Vertices | 12 |

Vertex figure | Isosceles triangle, edge lengths 1, √3, √3 |

Measures (edge length 1) | |

Circumradius | |

Volume | |

Dihedral angles | 6–3: |

6–6: | |

Central density | 1 |

Number of pieces | 8 |

Level of complexity | 3 |

Related polytopes | |

Army | Tut |

Regiment | Tut |

Dual | Triakis tetrahedron |

Conjugate | None |

Abstract properties | |

Flag count | 72 |

Euler characteristic | 2 |

Topological properties | |

Surface | Sphere |

Orientable | Yes |

Genus | 0 |

Properties | |

Symmetry | A_{3}, order 24 |

Convex | Yes |

Nature | Tame |

The **truncated tetrahedron**, or **tut**, is one of the 13 Archimedean solids, and the only one with tetrahedral symmetry. It consists of 4 triangles and 4 hexagons. Each vertex joins one triangle and two hexagons. As the name suggests, it can be obtained by truncation of the tetrahedron.

## Vertex coordinates[edit | edit source]

A truncated tetrahedron of edge length 1 has vertex coordinates given by all permutations and even sign changes of:

## Representations[edit | edit source]

A truncated tetrahedron has the following Coxeter diagrams:

- x3x3o (full symmetry)
- s4o3x () (as triangle-alternated small rhombicuboctahedron)
- xux3oox&#xt (A
_{2}axial, triangle-first) - xuxo oxux&#xt (A
_{1}×A_{1}axial, edge-first)

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

The truncated tetrahedron has a semi-uniform variant of the form x3y3o that maintains its full symmetry. This variant has 4 triangles of size y and 4 ditrigons as faces.

With edges of length a (between two ditrigons) and b (between a ditrigon and a triangle), its circumradius is given by and its volume is given by .

It has coordinates given by all permutations and even sign changes of:

These semi-uniform truncated tetrahedra occur as vertex figures of two uniform polychora, the small ditetrahedronary hexacosihecatonicosachoron and ditetrahedronary dishecatonicosachoron.

## Related polyhedra[edit | edit source]

It is possible to augment one of the hexagonal faces of the truncated tetrahedron with a triangular cupola to form the augmented truncated tetrahedron.

A number of uniform polyhedron compounds are composed of truncated tetrahedra:

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[edit | edit source]

- Bowers, Jonathan. "Polyhedron Category 2: Truncates" (#10).

- Klitzing, Richard. "tut".

- Quickfur. "The Truncated Tetrahedron".

- Wikipedia Contributors. "Truncated tetrahedron".
- McCooey, David. "Truncated Tetrahedron"

- Hi.gher.Space Wiki Contributors. "Tetrahedral truncate".