# Dodecateron

Dodecateron | |
---|---|

Rank | 5 |

Type | Uniform |

Space | Spherical |

Notation | |

Bowers style acronym | Dot |

Coxeter diagram | o3o3x3o3o () |

Elements | |

Tera | 12 rectified pentachora |

Cells | 30 tetrahedra, 30 octahedra |

Faces | 120 triangles |

Edges | 90 |

Vertices | 20 |

Vertex figure | Triangular duoprism, edge length 1 |

Measures (edge length 1) | |

Circumradius | |

Inradius | |

Hypervolume | |

Diteral angles | Rap–oct–rap: |

Rap–tet–rap: | |

Height | |

Central density | 1 |

Number of external pieces | 12 |

Level of complexity | 3 |

Related polytopes | |

Army | Dot |

Regiment | Dot |

Dual | Bidodecateron |

Conjugate | None |

Abstract & topological properties | |

Flag count | 4320 |

Euler characteristic | 2 |

Orientable | Yes |

Properties | |

Symmetry | A_{5}×2, order 1440 |

Convex | Yes |

Nature | Tame |

The **dodecateron**, also called the **birectified 5-simplex**, is a convex noble uniform polyteron. It consists of 12 rectified pentachora as facets. 6 rectified pentachora join at each triangular duoprismatic vertex. As the name suggests, it is the birectification of the hexateron. It is the medial stage in the series of truncations between the regular hexateron and its dual.

It can be seen as a segmentoteron as rectified pentachoron atop inverted rectified pentachoron, or a rectified pentachoric alterprism.

## Vertex coordinates[edit | edit source]

The vertices of a dodecateron of edge length 1 are given by the following points:

Much simpler coordinates can be given in six dimensions, as all permutations of:

## Representations[edit | edit source]

A dodecateron has the following Coxeter diagrams:

- o3o3x3o3o (full symmetry)
- oo3xo3ox3oo&#x (A
_{4}axial, rectified pentachoric alterprism) - oxoo3ooxo oxoo3ooxo&#xt (A
_{2}×A_{2}axial, vertex-first) - oxo xoo3oxooox&#xt (A
_{3}×A_{1}axial, tetrahedron-first) - oxoo3xoxo3ooox&#xt (A
_{3}symmetry)

## Gallery[edit | edit source]

## Related polytopes[edit | edit source]

The dodecateron is the colonel of a five-member regiment that also includes the biprismatododecateron, biprismatointercepted dodecateron, cellibiprismatohexateron, and cellidishexateron.

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

Hexateron | hix | ||

Rectified hexateron | rix | ||

Dodecateron | dot | ||

Rectified hexateron | rix | ||

Hexateron | hix | ||

Truncated hexateron | tix | ||

Bitruncated hexateron | bittix | ||

Bitruncated hexateron | bittix | ||

Truncated hexateron | tix | ||

Small rhombated hexateron | sarx | ||

Small birhombidodecateron | sibrid | ||

Small rhombated hexateron | sarx | ||

Great rhombated hexateron | garx | ||

Great birhombidodecateron | gibrid | ||

Great rhombated hexateron | garx | ||

Small prismated hexateron | spix | ||

Small prismated hexateron | spix | ||

Prismatotruncated hexateron | pattix | ||

Prismatorhombated hexateron | pirx | ||

Prismatorhombated hexateron | pirx | ||

Prismatotruncated hexateron | pattix | ||

Great prismated hexateron | gippix | ||

Great prismated hexateron | gippix | ||

Small cellidodecateron | scad | ||

Celliprismated hexateron | cappix | ||

Cellirhombidodecateron | card | ||

Celligreatorhombated hexateron | cograx | ||

Celliprismated hexateron | cappix | ||

Celliprismatotruncatododecateron | captid | ||

Celligreatorhombated hexateron | cograx | ||

Great cellidodecateron | gocad |

## External links[edit | edit source]

- Bowers, Jonathan. "Category 3: Rectates" (#32).

- Klitzing, Richard. "dot".

- Wikipedia Contributors. "Birectified 5-simplex".