The 57-cell, also called the pentacontaheptachoron, is a regular self-dual locally projective polytope with 57 hemidodecahedral facets described by Coxeter (1982) and also constructed by Vanden Cruyce (1985; Hartley and Leemans 2004). It has 57 vertices, 171 edges, 171 faces, and 57 cells (Coxeter 1982). It cannot be represented in 3-dimensional space in any reasonable way and is highly self-intersecting even in 4-dimensional space because its boundary cells are single-sided manifolds such as a Möbius strip or Klein bottle (Séquin and Hamlin 2007). Its symmetry group is the projective special linear group L_2(19), of order 3420.

The skeleton of the 57-cell is the Perkel graph.

See also

11-Cell, 16-Cell, 24-Cell, 120-Cell, 600-Cell, Pentatope, Perkel Graph, Tesseract

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Coxeter, H. S. M. "Ten Toroids and Fifty-Seven Hemi-Dodecahedra." Geom. Dedicata 13, 87-99, 1982.Hartley, M. I. and Leemans, D. "Quotients of a Universal Locally Projective Polytope of Type {5,3,5}." Math. Z. 247, 663-674, 2004.McMullen, P. and Schulte, E. Abstract Regular Polytopes. New York: Cambridge University Press, pp. 35-36, 2002.Séquin, C. H. and Hamlin, J. F. "The Regular 4-Dimensional 57-Cell." International Conference on Computer Graphics and Interactive Techniques: ACM SIGGRAPH 2007 Sketches. New York: ACM, 2007.Vanden Cruyce, P. "Geometries Related to PSL(2, 19)." Eur. J. Combin. 6, 163-173, 1985.

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Weisstein, Eric W. "57-Cell." From MathWorld--A Wolfram Web Resource.

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