57-cell

In mathematics, the 57-cell (pentacontaheptachoron) is a self-dual abstract regular 4-polytope (four-dimensional polytope).

It also has 57 vertices, 171 edges and 171 two-dimensional faces.

The symmetry abstract structure is the projective special linear group of the 2-dimensional vector space over the finite field of 19 elements, L2(19).

It has Schläfli type {5,3,5} with 5 hemi-dodecahedral cells around each edge.

The vertices and edges form the Perkel graph, the unique distance-regular graph with intersection array {6,5,2;1,1,3}, discovered by Manley Perkel (1979).