# Icositetrachoric tetracomb

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

Rank | 5 |

Type | Regular |

Space | Euclidean |

Notation | |

Bowers style acronym | Icot |

Coxeter diagram | |

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 |

Conjugate | None |

Abstract & topological properties | |

Orientable | Yes |

Properties | |

Symmetry | U_{5} |

Convex | Yes |

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.

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 | edit source]

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 | edit source]

An icositetrachoric tetracomb has the following Coxeter diagrams:

- (full symmetry)
- (U5 symmetry, as rectified hexadecachoric tetracomb)
- (R5 symmetry, birectified tesseractic tetracomb)
- (S5 symmetry, rectified demitesseractic tetracomb)
- (Q5 symmetry, cells of four different types)

## Related polytopes[edit | edit source]

Name | OBSA | CD diagram | Picture |
---|---|---|---|

Tesseractic tetracomb | test | ||

Rectified tesseractic tetracomb | rittit | ||

Icositetrachoric tetracomb | icot | ||

Truncated tesseractic tetracomb | tattit | ||

Bitruncated tesseractic tetracomb | batitit | ||

Small rhombated tesseractic tetracomb | srittit | ||

Rectified icositetrachoric tetracomb | ricot | ||

Great rhombated tesseractic tetracomb | grittit | ||

Truncated icositetrachoric tetracomb | ticot | ||

Small prismated tesseractic tetracomb | sidpitit | ||

Prismatotruncated tesseractic tetracomb | potatit | ||

Prismatorhombated tesseractic tetracomb | prittit | ||

Great prismated tesseractic tetracomb | gippittit | ||

Tesseractic tetracomb | test | ||

Celliprismated tesseractic tetracomb | capotat | ||

Cellirhombated tesseractic tetracomb | scartit | ||

Celligreatorhombated tesseractic tetracomb | gicartit | ||

Celliprismatotruncated tesseractic tetracomb | captatit | ||

Great cellated tesseractic tetracomb | otatit |

## External links[edit | edit source]

- Klitzing, Richard. "icot".

- Wikipedia Contributors. "24-cell honeycomb".