57-cell

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In mathematics, the 57-cell is a four-dimensional self-dual abstract regular polytope. Its facets are hemi-dodecahedra, and as its name implies, are 57 in number. It also has 57 vertices, 171 edges and 171 ridges. Its symmetry group is the projective special linear group L₂(19), so it has 3420 symmetries. It has Schläfli symbol {5,3,5}. It was discovered by H. S. M. Coxeter in 1982.