# Small rhombated hexateron

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Small rhombated hexateron | |
---|---|

Rank | 5 |

Type | Uniform |

Space | Spherical |

Notation | |

Bowers style acronym | Sarx |

Coxeter diagram | x3o3x3o3o () |

Elements | |

Tera | 15 tetrahedral prisms, 6 rectified pentachora, 6 small rhombated pentachora |

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

Faces | 20+60+120 triangles, 90 squares |

Edges | 60+180 |

Vertices | 60 |

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

Measures (edge length 1) | |

Circumradius | |

Hypervolume | |

Diteral angles | Rap–tet–tepe: |

Srip–trip–tepe: | |

Srip–oct–rap: | |

Srip–co–srip: | |

Central density | 1 |

Number of external pieces | 27 |

Level of complexity | 16 |

Related polytopes | |

Army | Sarx |

Regiment | Sarx |

Dual | Triangular-pyramidal tegmatic hexecontateron |

Conjugate | None |

Abstract & topological properties | |

Flag count | 11520 |

Euler characteristic | 2 |

Orientable | Yes |

Properties | |

Symmetry | A_{5}, order 720 |

Convex | Yes |

Nature | Tame |

The **small rhombated hexateron**, or **sarx**, also called the **cantellated 5-simplex**, is a convex uniform polyteron. It consists of 15 tetrahedral prisms, 6 rectified pentachora, and 6 small rhombated pentachora. One rectified pentachoron, 2 tetrahedral prisms, and 3 small rhombated pentachora join at each vertex. As the name suggests, it is the cantellation of the hexateron.

## Vertex coordinates[edit | edit source]

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

## Representations[edit | edit source]

A small rhombated hexateron has the following Coxeter diagrams:

- x3o3x3o3o (full symmetry)
- oxx3xxo3oox3ooo&#xt (A
_{4}axial, rectified pentachoron-first) - x(uo)xo x(ou)xx3o(xo)xo3o(oo)ox&#xt (A
_{3}×A_{1}axial, tetrahedral prism-first) - ox(ou)xx3oo(xo)xo xx(uo)xo3ox(ox)oo&#xt (A
_{2}×A_{2}axial, triangle-first)

## Gallery[edit | edit source]

## Related polytopes[edit | edit source]

The small rhombated hexateron is the colonel of a regiment that includes 15 uniform members.

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. "sarx".

- Wikipedia Contributors. "Cantellated 5-simplex".