57-cell

In mathematics, the 57-cell (or pentacontaheptachoron) 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. [http://www.math.wright.edu/People/manley.perkel/Vita]

See also

* 11-cell - abstract regular polytope with hemi-icosahedral cells.
* Order-5 dodecahedral honeycomb - regular honeycomb with same Schläfli symbol {5,3,5}.

References

* Peter McMullen, Egon Schulte, Abstract Regular Polytopes, Cambridge University Press, 2002. ISBN 0-521-81496-0
* [http://www.cs.berkeley.edu/~sequin/PAPERS/2007_SIGGRAPH_57Cell.pdf] 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

* [http://www.cs.berkeley.edu/~sequin/TALKS/2007_SIGGRAPH_57Cell.ppt Siggraph 2007: 11-cell and 57-cell by Carlo Sequin]
*
* [http://www.win.tue.nl/~aeb/graphs/Perkel.html Perkel graph]