# Great grand stellated 120-cell

In geometry, the **great grand stellated 120-cell** or **great grand stellated polydodecahedron** is a regular star 4-polytope with Schläfli symbol {5/2,3,3}, one of 10 regular Schläfli-Hess 4-polytopes. It is unique among the 10 for having 600 vertices, and has the same vertex arrangement as the regular convex 120-cell.

Great grand stellated 120-cell | |
---|---|

Orthogonal projection | |

Type | Schläfli-Hess polychoron |

Cells | 120 {5/2,3} |

Faces | 720 {5/2} |

Edges | 1200 |

Vertices | 600 |

Vertex figure | {3,3} |

Schläfli symbol | {5/2,3,3} |

Coxeter-Dynkin diagram | |

Symmetry group | H_{4}, [3,3,5] |

Dual | Grand 600-cell |

Properties | Regular |

It is one of four *regular star polychora* discovered by Ludwig Schläfli. It is named by John Horton Conway, extending the naming system by Arthur Cayley for the Kepler-Poinsot solids, and the only one containing all three modifiers in the name.

With its dual, it forms the compound of great grand stellated 120-cell and grand 600-cell.

## Images

H_{4} |
A_{2} / B_{3} |
A_{3} / B_{2} |
---|---|---|

Great grand stellated 120-cell, {5/2,3,3} | ||

[10] | [6] | [4] |

120-cell, {5,3,3} | ||

## As a stellation

The **great grand stellated 120-cell** is the *final stellation* of the 120-cell, and is the only Schläfli-Hess polychoron to have the 120-cell for its convex hull. In this sense it is analogous to the three-dimensional great stellated dodecahedron, which is the final stellation of the dodecahedron and the only Kepler-Poinsot polyhedron to have the dodecahedron for its convex hull. Indeed, the great grand stellated 120-cell is dual to the grand 600-cell, which could be taken as a 4D analogue of the great icosahedron, dual of the great stellated dodecahedron.

The edges of the great grand stellated 120-cell are τ^{6} as long as those of the 120-cell core deep inside the polychoron, and they are τ^{3} as long as those of the small stellated 120-cell deep within the polychoron.

## See also

- List of regular polytopes
- Convex regular 4-polytope - Set of convex regular polychora
- Kepler-Poinsot solids - regular star polyhedron
- Star polygon - regular star polygons

## References

- Edmund Hess, (1883)
*Einleitung in die Lehre von der Kugelteilung mit besonderer Berücksichtigung ihrer Anwendung auf die Theorie der Gleichflächigen und der gleicheckigen Polyeder*. - H. S. M. Coxeter,
*Regular Polytopes*, 3rd. ed., Dover Publications, 1973. ISBN 0-486-61480-8. - John H. Conway, Heidi Burgiel, Chaim Goodman-Strass,
*The Symmetries of Things*2008, ISBN 978-1-56881-220-5 (Chapter 26, Regular Star-polytopes, pp. 404–408) - Klitzing, Richard. "4D uniform polytopes (polychora) o3o3o5/2x - gogishi".

## External links

- Regular polychora
- Discussion on names
- Reguläre Polytope
- The Regular Star Polychora
- Zome Model of the Final Stellation of the 120-cell