Math 553B: Polyhedra

Robert Connelly

Winter 2000, MWF 1:30-2:20

This is a course that will cover some of the geometric properties of mostly low-dimensional convex polyhedra and a few non-convex polyhedra.  Some of the topics that I hope to cover will be:
  1. Cauchy's Theorem that says that a convex three-dimensional polytope has a unique convex realization with its faces kept congruent.  In particular, convex polyhedra are rigid.
  2. Examples of embedded non-convex two-dimensional polyhedral surfaces that are not rigid.
  3. Dehn's Theorem that shows the infinitesimal rigidity of the boundary of a triangulated three-dimensional polytope.
  4. The theorem of J. C. Maxwell and L. Cremona that expresses a correspondence between stresses on a graph in the plane and lifts of that graph to polyhedral surfaces in three-space.
  5. Steinitz' Theorem that says that any vertex three-connected planar graph is the graph of the edges of a convex three-dimensional polytope.  This is also related to Tutte's Theorem on how to draw a planar graph.
  6. A discussion of A. D. Alexandrov's Theorem that says that any intrinsically convex polyhedral metric on a two-dimensional sphere can be realized as the metric of the boundary of a convex three-dimensional polytope (possibly degenerate).
Other topics can be included according to the interests of the students.