| 57-cell | |
|---|---|
| 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 | |
In mathematics, the 57-cell 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.
Contents |
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. [1]
See also
- 11-cell - abstract regular polytope with hemi-icosahedral 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
- [2] 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
This entry is from Wikipedia, the leading user-contributed encyclopedia. It may not have been reviewed by professional editors (see full disclaimer)
