# Tetracontoctachoron

Tetracontoctachoron | |
---|---|

Rank | 4 |

Type | Uniform |

Space | Spherical |

Notation | |

Bowers style acronym | Cont |

Coxeter diagram | o3x4x3o () |

Elements | |

Cells | 48 truncated cubes |

Faces | 192 triangles, 144 octagons |

Edges | 576 |

Vertices | 288 |

Vertex figure | Tetragonal disphenoid, edge lengths 1 (base) and √2+√2 (sides) |

Edge figure | tic 8 tic 8 tic 3 |

Measures (edge length 1) | |

Circumradius | |

Inradius | |

Hypervolume | |

Dichoral angles | Tic–8–tic: 135° |

Tic–3–tic: 120° | |

Central density | 1 |

Number of pieces | 48 |

Level of complexity | 3 |

Related polytopes | |

Army | Cont |

Regiment | Cont |

Dual | Bitetracontoctachoron |

Conjugate | Great tetracontoctachoron |

Abstract properties | |

Euler characteristic | 0 |

Topological properties | |

Orientable | Yes |

Properties | |

Symmetry | F_{4}×2, order 2304 |

Convex | Yes |

Nature | Tame |

The **tetracontoctachoron**, or **cont**, also commonly called the **48-cell** or **bitruncated 24-cell**, is a convex noble uniform polychoron that consists of 48 truncated cubes as cells. Four cells join at each vertex. It is the medial stage of the truncation series between a regular icositetrachoron and itself. Alternatively, it is also the stellation core of the compound of two opposite icositetrachora, the stellated tetracontoctachoron.

It is the second in an infinite family of isochoric cubic swirlchora (the cubiswirlic tetracontoctachoron) and the first in an infinite family of isochoric chiral rhombic dodecahedral swirlchora (the rhombidodecaswirlic tetracontoctachoron). Its cells form 6 rings of 8 truncated cubes.

The solid angle at the vertex is 73/288.

It can form a non-Wythoffian uniform hyperbolic tiling with 64 tetracontoctachora at each vertex with an octagonal duotegum as the vertex figure, called a tetracontoctachoric tetracomb.

## Cross-sections[edit | edit source]

## Vertex coordinates[edit | edit source]

Coordinates for the vertices of a tetracontoctachoron of edge length 1 are all permutations of:

## Representations[edit | edit source]

A tetracontoctachoron has the following Coxeter diagrams:

- o3x4x3o (full symmetry)
- xo4xw3oo3wx&#zx (BC
_{4}symmetry) - xooxwUwxoox4xwwxoooxwwx3ooxwwxwwxoo&#xt (BC
_{3}axial, cell-first)

## Variations[edit | edit source]

The tetracontoctachoron has a semi-uniform variant with single symmetry called the icositetricositetrachoron, along with isochoric variants with swirlprismatic symmetry.

## Related polychora[edit | edit source]

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

Icositetrachoron | ico | ||

Truncated icositetrachoron | tico | ||

Rectified icositetrachoron | rico | ||

Tetracontoctachoron | cont | ||

Rectified icositetrachoron | rico | ||

Truncated icositetrachoron | tico | ||

Icositetrachoron | ico | ||

Small rhombated icositetrachoron | srico | ||

Great rhombated icositetrachoron | grico | ||

Small rhombated icositetrachoron | srico | ||

Great rhombated icositetrachoron | grico | ||

Small prismatotetracontoctachoron | spic | ||

Prismatorhombated icositetrachoron | prico | ||

Prismatorhombated icositetrachoron | prico | ||

Great prismatotetracontoctachoron | gippic | ||

Snub disicositetrachoron | sadi |

## Isogonal derivatives[edit | edit source]

Substitution by vertices of these following elements will produce these convex isogonal polychora:

- Truncated cube (48): Bitetracontoctachoron
- Triangle (192): Biambotetracontoctachoron
- Octagon (144): Small prismatotetracontoctachoron
- Edge (576): Rectified tetracontoctachoron

## External links[edit | edit source]

- Bowers, Jonathan. "Category 7: Bitruncates" (#300).

- Klitzing, Richard. "cont".

- Quickfur. "The Bitruncated 24-cell".

- Wikipedia Contributors. "Bitruncated 24-cell".
- Hi.gher.Space Wiki Contributors. "Xylomesochoron".