Convex Platonic polyhedra in arbitrary dimension were classified in the nineteenth century. The interesting feature of this classification is that there are exactly 5 ``exotic'' polyhedra: two in dimension 3 (the icosahedron and the dodecahedron) and three in dimension 4 (the 24-cell, the 120-cell and the 600-cell). A Platonic cell complex is a cell complex whose group of combinatorial symmetries is transitive on complete flags of cells of the form: (vertex, edge, 2-face, ...). Platonic solids correspond to the case where the underlying cell complex is a sphere. The case where the cell complex is a manifold is also well understood. The answer is also known when the cell complex is 2-dimensional. However, for general cell complexes there is no such classification. We plan to discuss the theory and classification in the case of manifolds and then have the working group look at some open problems in the general case.

To sign up for this working group, enroll for 3 credits in 693 (Davis), call number 12591-4