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 {5,3,5} with 5 hemi-dodecahedral cells around each edge. It was discovered by . 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 by . See also 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 . . The Classification of Rank 4 Locally Projective Polytopes and Their Quotients, 2003, Michael I Hartley External links Siggraph 2007: 11-cell and 57-cell by Carlo Sequin Perkel graph 4-polytopes