# Icositetrachoric tetracomb

The **icositetrachoric tetracomb** or **icot**, also known as the **24-cell tetracomb** or **24-cell honeycomb**, is one of three regular tetracombs or tessellations of 4D Euclidean space. 3 icositetrachora join at each face, and 8 join at each vertex of this honeycomb.

Icositetrachoric tetracomb | |
---|---|

Rank | 5 |

Type | Regular |

Space | Euclidean |

Notation | |

Bowers style acronym | Icot |

Coxeter diagram | x3o4o3o3o () |

Schläfli symbol | {3,4,3,3} |

Elements | |

Tera | N icositetrachora |

Cells | 12N octahedra |

Faces | 32N triangles |

Edges | 24N |

Vertices | 3N |

Vertex figure | Tesseract, edge length 1 |

Measures (edge length 1) | |

Vertex density | |

Related polytopes | |

Army | Icot |

Regiment | Icot |

Dual | Hexadecachoric tetracomb |

Petrie dual | Petrial 24-cell honeycomb |

Conjugate | None |

Abstract & topological properties | |

Orientable | Yes |

Properties | |

Symmetry | U_{5} |

Convex | Yes |

Nature | Tame |

It can be obtained from the tesseractic tetracomb by decomposing alternate tesseracts into 8 cubic pyramids and attaching those to the neighbouring tesseract, making it an icositetrachoron. It can also be obtained as a birectified tesseractic tetracomb, or a rectified hexadecachoric tetracomb.

## Vertex coordinates edit

The vertices of an icositetrachoric tetracomb of edge length 1 are given by all permutations of:

- ,

where i , j , k , and l range over the integers.

An alternate set of coordinates can be given by:

- ,
- for i +j +k +l even

where i , j , k , l are integers.

## Representations edit

An icositetrachoric tetracomb has the following Coxeter diagrams:

- x3o4o3o3o ( ) (full symmetry)
- o3x3o4o3o ( ) (U
_{5}symmetry, as rectified hexadecachoric tetracomb) - o4o3x3o4o ( ) (R
_{5}symmetry, birectified tesseractic tetracomb) - o4o3x3o *c3o ( ) (S
_{5}symmetry, rectified demitesseractic tetracomb) - o3x3o *b3o *b3o ( ) (Q
_{5}symmetry, cells of four different types)

## External links edit

- Klitzing, Richard. "icot".
- Wikipedia contributors. "24-cell honeycomb".