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 (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 by Manley Perkel (1979).

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]

  • Coxeter, H. S. M. (1982), "Ten toroids and fifty-seven hemidodecahedra", Geometriae Dedicata, 13 (1): 87–99, doi:10.1007/BF00149428, MR 0679218.
  • 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 0553163.
  • 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[edit]