# Small prismated hexateron

Small prismated hexateron | |
---|---|

Rank | 5 |

Type | Uniform |

Space | Spherical |

Notation | |

Bowers style acronym | Spix |

Coxeter diagram | x3o3o3x3o () |

Elements | |

Tera | 20 triangular duoprisms, 15 octahedral prisms, 6 rectified pentachora, 6 small prismatodecachora |

Cells | 15+30 tetrahedra, 60+60+60 triangular prisms, 30 octahedra |

Faces | 60+60+120 triangles, 180 squares |

Edges | 90+180 |

Vertices | 60 |

Vertex figure | Triangular antifastegium, edge lengths 1 (base triangular prism and top triangle) and √2 (sides) |

Measures (edge length 1) | |

Circumradius | |

Hypervolume | |

Diteral angles | Ope–trip–triddip: 135° |

Rap–oct–ope: | |

Spid–trip–triddip: | |

Spid–trip–ope: | |

Spid–tet–rap: | |

Spid–tet–spid: | |

Central density | 1 |

Number of external pieces | 47 |

Level of complexity | 25 |

Related polytopes | |

Army | Spix |

Regiment | Spix |

Dual | Antitegmatonotched hexecontateron |

Conjugate | None |

Abstract & topological properties | |

Flag count | 18000 |

Euler characteristic | 2 |

Orientable | Yes |

Properties | |

Symmetry | A_{5}, order 720 |

Convex | Yes |

Nature | Tame |

The **small prismated hexateron**, or **spix**, also called the **runcinated 5-simplex**, is a convex uniform polyteron. It consists of 20 triangular duoprisms, 15 octahedral prisms, 6 rectified pentachora, and 6 small prismatodecachora. One rectified pentachoron, 2 small prismatodecachora, 3 octahedral prisms, and 3 triangular duoprisms join at each vertex. As the name suggests, it is the runcination of the hexateron.

## Vertex coordinates[edit | edit source]

The vertices of a small prismated hexateron of edge length 1 can be given in 6 dimensions as all permutations of:

## Representations[edit | edit source]

A small prismated hexateron has the following Coxeter diagrams:

- x3o3o3x3o (full symmetry)
- oox3xxo3ooo3oxx&#xt (A
_{4}axial, rectified pentachoron-first)

## Gallery[edit | edit source]

## Related polytopes[edit | edit source]

The regiment of the small prismated hexateron includes a total of 37 members plus one fissary member. This makes it the largest regiment of uniform polytera with A5 symmetry.

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]

- Klitzing, Richard. "spix".

- Wikipedia Contributors. "Runcinated 5-simplex".