# Great stellated dodecahedron

Great stellated dodecahedron | |
---|---|

Rank | 3 |

Type | Regular |

Notation | |

Bowers style acronym | Gissid |

Coxeter diagram | x5/2o3o () |

Schläfli symbol | |

Elements | |

Faces | 12 pentagrams |

Edges | 30 |

Vertices | 20 |

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

Petrie polygons | 6 skew decagrams |

Measures (edge length 1) | |

Circumradius | |

Edge radius | |

Inradius | |

Volume | |

Dihedral angle | |

Central density | 7 |

Number of external pieces | 60 |

Level of complexity | 3 |

Related polytopes | |

Army | Doe, edge length |

Regiment | Gissid |

Dual | Great icosahedron |

Petrie dual | Petrial great stellated dodecahedron |

κ ^{?} | Petrial dodecahedron |

Conjugate | Dodecahedron |

Convex core | Dodecahedron |

Abstract & topological properties | |

Flag count | 120 |

Euler characteristic | 2 |

Schläfli type | {5,3} |

Orientable | Yes |

Genus | 0 |

Properties | |

Symmetry | H_{3}, order 120 |

Flag orbits | 1 |

Convex | No |

Nature | Tame |

History | |

Discovered by | Johannes Kepler^{[note 1]} |

First discovered | 1613 |

The **great stellated dodecahedron**, or **gissid**, is one of the four Kepler–Poinsot solids. It has 12 pentagrams as faces, joining 3 to a vertex.

It is the last stellation of the dodecahedron, from which its name is derived. It is also the only Kepler-Poinsot solid to share its vertices with the dodecahedron as opposed to the icosahedron. It has the smallest circumradius of any uniform polyhedron.

Great stellated dodecahedra appear as cells in two star regular polychora, namely the great stellated hecatonicosachoron and great grand stellated hecatonicosachoron.

## Vertex coordinates[edit | edit source]

The vertices of a great stellated dodecahedron of edge length 1, centered at the origin, are all sign changes of

- ,

along with all even permutations and all sign changes of

- .

The first set of vertices corresponds to a scaled cube which can be inscribed into the great stellated dodecahedron's vertices.

## Related polytopes[edit | edit source]

### Alternative realizations[edit | edit source]

The dodecahedron and the great stellated dodecahedron are conjugates. Thus they are realizations of the same underlying abstract regular polytope {5,3}. These are the only faithful symmetric realizations of this polytope in 3-dimensional Euclidean space, however there are many more skew faithful symmetric realizations. In total there are 28 faithful symmetric realizations, of which 3 are pure.

### In vertex figures[edit | edit source]

The great stellated dodecahedron appears as a vertex figure of one Schläfli–Hess polychoron.

Name | Picture | Schläfli symbol | Edge length |
---|---|---|---|

Great grand hecatonicosachoron | {5,5/2,3} |

## External links[edit | edit source]

- Bowers, Jonathan. "Polyhedron Category 1: Regulars" (#9).

- Klitzing, Richard. "gissid".
- Nan Ma. "Great stellated dodecahedron {5/2, 3}".

- Wikipedia contributors. "Great stellated dodecahedron".
- McCooey, David. "Great Stellated Dodecahedron"
- Hartley, Michael. "{5,3}*120".

## Notes[edit | edit source]

- ↑ Earlier authors drew shapes the Great stellated dodecahedron or similar shapes earlier, however Kepler was the first to recognize the Great stellated dodecahedron as regular, and explicitly describe it.