Type Abstract regular 4-polytope
Cells 57 hemi-dodecahedra
Faces 171 {5}
Edges 171
Vertices 57
Vertex figure {3,5}/2
Schläfli symbol {(5,3)/2,5}
Symmetry group [(5,3)/2,5], order 3420
Abstract L2(19)
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.

The symmetry order is 3420, from the product of the number of cells (57) and the symmetry of each cell (60). The symmetry abstract structure is the projective special linear group, L2(19).

It has Schläfli symbol {(5,3)/2,5} with 5 hemi-dodecahedral cells, {5,3}/2, around each edge. It was discovered by H. S. M. Coxeter (1982).

Perkel graphEdit

Perkel graphs with 19-fold symmetry

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 alsoEdit

  • 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.)


  • Coxeter, H. S. M. (1982), "Ten toroids and fifty-seven hemidodecahedra", Geometriae Dedicata, 13 (1): 87–99, doi:10.1007/BF00149428, MR 0679218, S2CID 120672023.
  • 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 (PDF), SIGGRAPH '07, New York, NY, USA: ACM, doi:10.1145/1278780.1278784, S2CID 37594016
  • The Classification of Rank 4 Locally Projective Polytopes and Their Quotients, 2003, Michael I Hartley

External linksEdit