57-cell

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57-cell
Perkel graph embeddings.svg
Some drawings of the Perkel graph.
Type Abstract regular 4-polytope
Cells 57 hemi-dodecahedra
Hemi-dodecahedron.png
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 in 1982.

Perkel graph[edit]

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[edit]

  • 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[edit]

  • 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[edit]