# 57-cell

In mathematics, the **57-cell** (**pentacontakaiheptachoron**) is a self-dual abstract regular 4-polytope (four-dimensional polytope). Its 57 cells are hemi-dodecahedra. It also has 57 vertices, 171 edges and 171 two-dimensional faces. Its symmetry group is the projective special linear group L_{2}(19), so it has 3420 symmetries.

57-cell | |
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Some drawings of the Perkel graph. | |

Type | Abstract regular 4-polytope |

Cells | 57 hemi-dodecahedra |

Faces | 171 {5} |

Edges | 171 |

Vertices | 57 |

Vertex figure | (hemi-icosahedron) |

Schläfli symbol | {5,3,5} |

Symmetry group | L_{2}(19) (order 3420) |

Dual | self-dual |

Properties | Regular |

It has Schläfli symbol {5,3,5} with 5 hemi-dodecahedral cells around each edge. It was discovered by H. S. M. Coxeter (1982).

## Perkel graph

The vertices and edges form the Perkel graph, the unique distance-regular graph with intersection array {6,5,2;1,1,3}, discovered by Manley Perkel (1979).

## See also

- 11-cell – abstract regular polytope with hemi-icosahedral cells.
- 120-cell – regular 4-polytope with dodecahedral cells
- Order-5 dodecahedral honeycomb - regular hyperbolic honeycomb with same Schläfli symbol {5,3,5}. (The 57-cell can be considered as being derived from it by identification of appropriate elements.)

## References

- Coxeter, H. S. M. (1982), "Ten toroids and fifty-seven hemidodecahedra",
*Geometriae Dedicata*,**13**(1): 87–99, doi:10.1007/BF00149428, MR 0679218. - McMullen, Peter; Schulte, Egon (2002),
*Abstract Regular Polytopes*, Encyclopedia of Mathematics and its Applications,**92**, Cambridge: Cambridge University Press, pp. 185–186, 502, doi:10.1017/CBO9780511546686, ISBN 0-521-81496-0, MR 1965665 - Perkel, Manley (1979), "Bounding the valency of polygonal graphs with odd girth",
*Canadian Journal of Mathematics*,**31**(6): 1307–1321, doi:10.4153/CJM-1979-108-0, MR 0553163. - Séquin, Carlo H.; Hamlin, James F. (2007), "The Regular 4-dimensional 57-cell" (PDF),
*ACM SIGGRAPH 2007 Sketches*(PDF), SIGGRAPH '07, New York, NY, USA: ACM, doi:10.1145/1278780.1278784

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