57-cell

57-cell
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	L2(19) (order 3420)
Dual	self-dual
Properties	Regular

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 faces. Its symmetry group is the projective special linear group L₂(19), so it has 3420 symmetries.

It has Schläfli symbol {5,3,5} with 5 hemi-dodecahedral cells around each edge. It was discovered by H. S. M. Coxeter in 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 in 1979 by Manley Perkel.

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

• Peter McMullen, Egon Schulte, Abstract Regular Polytopes, Cambridge University Press, 2002. ISBN 0-521-81496-0
• PDF The Regular 4-Dimensional 57-Cell, Carlo H. Séquin and James F. Hamlin, CS Division, U.C. Berkeley
• M. Perkel, Bounding the valency of polygonal graphs with odd girth, Canad. J. Math. 31 (1979) 1307-1321

External links

• Siggraph 2007: 11-cell and 57-cell by Carlo Sequin
• Weisstein, Eric W., "Perkel graph", MathWorld.
• Perkel graph
• Richard Klitzing, Explanations, Grünbaum-Coxeter Polytopes