The 57-cell, also called the pentacontaheptachoron, is a regular self-dual locally projective polytope with 57 hemidodecahedral facets described by Coxeter and also constructed by Vanden Cruyce. It has 57 vertices, 171 edges, 171 faces, and 57 cells. 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. 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.
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