I made a demo on the 57-cell. I seem to be the first to notice that the skeleton is the Perkel graph. http://demonstrations.wolfram.com/The57Cell/ I tried to build the 57-cell in 4D. I thought I would start by finding a really nice hemi-dodecahedron, or Petersen graph. These six faces work really well: {{{0,0,2,0},{-1,-1,-1,Sqrt[5]},{2,0,0,0}, {-1,1,1,Sqrt[5]}, {0,0,-2,0}}, {{0,0,2,0}, {1,1,-1,Sqrt[5]}, {-2,0,0,0},{1,-1,1,Sqrt[5]}, {0,0,-2,0}}, {{0,2,0,0}, {-1,-1,-1,Sqrt[5]},{0,0,2,0}, {1,1,-1,Sqrt[5]}, {0,-2,0,0}}, {{0,2,0,0}, {1,-1,1,Sqrt[5]}, {0,0,-2,0},{-1,1,1,Sqrt[5]}, {0,-2,0,0}}, {{2,0,0,0}, {-1,-1,-1,Sqrt[5]},{0,2,0,0}, {1,-1,1,Sqrt[5]}, {-2,0,0,0}}, {{2,0,0,0}, {-1,1,1,Sqrt[5]}, {0,-2,0,0},{1,1,-1,Sqrt[5]}, {-2,0,0,0}}} Connected vertices are at distance 4. Non-connected vertices are at distance 2Sqrt[2]. Any two diagonals are at right angles to each other, through the origin. All faces have a bizarre regularity. Unfortunately, these warped faces don't reflect to create new points, so I couldn't figure out how to glue two of these weird faces together. --Ed Pegg Jr