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 , of order 3420.
Coxeter, H. S. M. "Ten Toroids and Fifty-Seven Hemi-Dodecahedra." Geom. Dedicata13, 87-99, 1982.Hartley,
M. I. and Leemans, D. "Quotients of a Universal Locally Projective Polytope
of Type ." 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.
, of order 3420.
." Math. Z. 247,
663-674, 2004.McMullen, P. and Schulte, E.