The **icositetrachoron** is a four-dimensional regular polytope composed of 24 octahedral cells, three of which join at each edge. It is also called a **24-cell** because of this. Its Bowers acronym is "**ico**", and it can also be called a **xylochoron**. It is one of the three regular polychora that tile 4-dimensional space, leading to the icositetrachoric tetracomb.

## Construction

One way to construct the icositetrachoron is to augment cubic pyramids onto the cells of the tesseract. If the sides of the pyramids are exactly the same edge length as the tesseract, the pyramidal cells merge into octahedra.

Another way is via the rectification of the hexadecachoron. The tetrahedral cells will rectify into octahedra, while the vertices become further octahedral cells. This can be represented as .

## Structure and Sections

### Structure

The icositetrachoron is composed of octahedra joining six to a vertex. Seen from a cell, we first have one, then the eight joining to it. The next layer is six octahedra sharing only a vertex with the top, then another eight, and finally the opposite cell.

### Subfacets

- 24 points (0D)
- 96 line segments (1D)
- 96 triangles (2D)
- 24 octahedra (3D)
- 1 icositetrachoron

### Hypervolumes

## See also

Octahedral dichoron | Icositetrachoron
| Order-4 octahedral honeycomb |

Cubic hosochoron | Icositetrachoron
| Order-3 square tiling honeycomb |

Regular | Rectified | Birectified | Trirectified | Truncated | Bitruncated | Tritruncated | |
---|---|---|---|---|---|---|---|

Icositetrachoron
| Rectified icositetrachoron | Rectified icositetrachoron | Icositetrachoron
| Truncated icositetrachoron | Bitruncated icositetrachoron | Truncated icositetrachoron | |

Cantellated | Bicantellated | Cantitruncated | Bicantitruncated | Runcinated | Runcicantellated | Runcitruncated | Runcicantitruncated |

Cantellated pentachoron | Cantellated icositetrachoron | Cantitruncated icositetrachoron | Cantitruncated icositetrachoron | Runcinated icositetrachoron | Runcitruncated icositetrachoron | Runcitruncated icositetrachoron | Omnitruncated icositetrachoron |

Regular | Rectified | Birectified | Trirectified | Truncated | Bitruncated | Tritruncated | |
---|---|---|---|---|---|---|---|

Tesseract | Rectified tesseract | Icositetrachoron
| Hexadecachoron | Truncated tesseract | Bitruncated tesseract | Truncated hexadecachoron | |

Cantellated | Bicantellated | Cantitruncated | Bicantitruncated | Runcinated | Runcicantellated | Runcitruncated | Runcicantitruncated |

Cantellated tesseract | Rectified icositetrachoron | Cantitruncated tesseract | Truncated icositetrachoron | Runcinated tesseract | Runcitruncated hexadecachoron | Runcitruncated tesseract | Omnitruncated tesseract |

Regular | Rectified | Birectified | Trirectified | Truncated | Bitruncated | Tritruncated | |
---|---|---|---|---|---|---|---|

Hexadecachoron | Icositetrachoron
| Hexadecachoron | Hexadecachoron | Truncated Hexadecachoron | Truncated Hexadecachoron | Rectified tesseract | |

Cantellated | Bicantellated | Cantitruncated | Bicantitruncated | Runcinated | Runcicantellated | Runcitruncated | Runcicantitruncated |

Rectified tesseract | Truncated hexadecachoron | Bitruncated tesseract | Bitruncated tesseract | Rectified tesseract | Rectified icositetrachoron | Bitruncated tesseract | Truncated icositetrachoron |

Rectified Octahedral hosochoron | Icositetrachoron
| Rectified cubic honeycomb | Rectified order-4 dodecahedral honeycomb | Rectified order-4 Hexagonal tiling honeycomb |

Rectified tetrahedral dichoron | Rectified pentachoron | Icositetrachoron
| Rectified hexacosichoron | Rectified order-6 tetrahedral honeycomb |