# Tetrahedral symmetry

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## Convex polytopes with A

## Wythoffians with A

Tetrahedral symmetry | |
---|---|

Rank | 3 |

Space | Spherical |

Order | 24 |

Info | |

Coxeter diagram | |

Elements | |

Axes | 3 × (BC_{2}×A_{1})/2, 4 × A_{2}×I |

Related polytopes | |

Omnitruncate | Great rhombitetratetrahedron |

**Tetrahedral symmetry**, also known as **tettic symmetry** and notated **A _{3}**, is a 3D spherical Coxeter group. It is the symmetry group of the regular tetrahedron.

## Subgroups[edit | edit source]

- Chiral tetrahedral symmetry (maximal)
- Triangular pyramidal symmetry (maximal)
- Chiral triangular pyramidal symmetry
- Digonal antiprismatic symmetry (maximal)
- Prodigonal antiprismatic symmetry
- Chiral digonal prismatic symmetry
- Rectangular pyramidal symmetry
- Chiral digonal pyramidal symmetry
- Reflection symmetry
- Identity symmetry

## Convex polytopes with A_{3} symmetry[edit | edit source]

- Tetrahedron (regular)
- Tetratetrahedron (isogonal)/Rhombic hexahedron (isotopic)
- Truncated tetrahedron (isogonal)/Triakis tetrahedron (isotopic)
- Rhombitetratetrahedron (isogonal)/Deltoidal dodecahedron (isotopic)
- Great rhombitetratetrahedron (isogonal)/Disdyakis hexahedron (isotopic)

## Wythoffians with A_{3} symmetry[edit | edit source]

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 () |