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The 57-Cell
The 57-Cell
A cube is made from squares. A tesseract is made from cubes. The four-dimensional 57-cell is made from 57 hemi-dodecahedra, Coxeter's name for a Petersen graph with six faces. The skeleton of this polytope is the Perkel graph. The top set of sliders changes the symmetrical embedding for the underlying graph. The cells can overlap (in green) with zero, one, or five edges (a face). Each cell shares a face with six other cells.
As an option, part of a 3D representation of the Perkel graph can be seen. Seven more abstract points are needed. A ring of ten yellow points is parallel to each set of pentagonal faces on the outer dodecahedron. Five of these yellow points connect to blue lines aligned with the ring; connect these to a new point. Connect these six new points to a final point at infinity.