# Small stellated hecatonicosachoron

Small stellated hecatonicosachoron | |
---|---|

Rank | 4 |

Type | Regular |

Space | Spherical |

Bowers style acronym | Sishi |

Info | |

Coxeter diagram | x5/2o5o3o |

Schläfli symbol | {5/2,5,3} |

Symmetry | H4, order 14400 |

Army | Ex |

Regiment | Sishi |

Elements | |

Vertex figure | Dodecahedron, edge length (√5–1)/2 |

Cells | 120 small stellated dodecahedra |

Faces | 720 pentagrams |

Edges | 1200 |

Vertices | 120 |

Measures (edge length 1) | |

Circumradius | 1 |

Edge radius | |

Face radius | |

Inradius | |

Hypervolume | |

Dichoral angle | 144° |

Central density | 4 |

Euler characteristic | -480 |

Number of pieces | 1440 |

Level of complexity | 4 |

Related polytopes | |

Dual | Faceted hexacosichoron |

Conjugate | Great grand hecatonicosachoron |

Properties | |

Convex | No |

Orientable | Yes |

Nature | Tame |

The **small stellated hecatonicosachoron**, or **sishi**, also commonly called the **small stellated 120-cell**, is one of the 10 regular Schläfli–Hess polychora. It has 120 small stellated dodecahedra as cells, joining 3 to an edge and 12 to a vertex in a dodecahedral arrangement.

## Cross-sections[edit | edit source]

## Vertex coordinates[edit | edit source]

The vertices of a small stellated hecatonicosachoron of edge length 1, centered at the origin, are given by all permutations of:

and all even permutations of:

The first two sets of vertices form a unit icositetrachoron that can be inscribed into the small stellated hecatonicosachoron. This corresponds to the fact there is a compound of 25 icositetrachora with the same vertices and edges as the small stellated hecatonicosachoron.

## Related polychora[edit | edit source]

The small stellated hecatonicosachoron is the colonel of a regiment containing 26 uniform polychora. Of these, one other is regular (the great grand hecatonicosachoron), three are semiregular (the small ditrigonary hexacosihecatonicosachoron, ditrigonary dishecatonicosachoron, and great ditrigonary hexacosihecatonicosachoron), 3 are noble (the invertidodecahedronary hecatonicosachoron, pentagrammic prismatic heptacosicosachoron, and pentagonal prismatic heptacosicosachoron), 8 others have full symmetry, and the other 10 (including 4 discovered in 2020) have lower symmetries. The regiment also includes at least 6 uniform compounds, 8 fissary uniforms, and an unknown number (at least several hundred) of scaliforms (all but 2 of the latter have been discovered in 2020 or later).

A significant number of unit-edge uniform polychora can be vertex-inscribed into the small stellated hecatonicosachoron. These include the icositetrachoron, the tesseract, the small prismatodecachoron, the small ditrigonary icosidodecahedral prism, the great duoantiprism, and the pentagonal-pentagrammic duoprism.

## External links[edit | edit source]

- Bowers, Jonathan. "Category 1: Regular Polychora" (#10).

- Klitzing, Richard. "Sishi".

- Wikipedia Contributors. "Small stellated 120-cell".