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 two-dimensional 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 (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 679218 .
• 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 553163 .
• Séquin, Carlo H.; Hamlin, James F. (2007), "The Regular 4-dimensional 57-cell" (PDF), ACM SIGGRAPH 2007 Sketches, SIGGRAPH '07, New York, NY, USA: ACM, doi:10.1145/1278780.1278784 

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