Small polyhedral models of the torus, the projective plane and the Klein bottle

author: Branko Grünbaum, University of Washington
published: Sept. 7, 2007,   recorded: September 2007,   views: 4236

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Models of these manifolds have been studied at least since the work of Moebius, with increasing depth and many results in more recent times. The models range from purely combinatorial to various types of geometric representations, such as by topological complexes, by planar-faced polyhedra (convex or nor necessarily convex), or by smooth manifolds. The talk will give a survey of available results, and then concentrate on what seems to be a new direction – models that admit as faces selfintersecting polygons. One of the unexpected results is that in some cases such models are simpler and more readily visualized than the more traditional ones, and that in other cases they are the only possible ones. The understanding of the role of selfintersecting polygons as faces sheds light, among other things, on the relations between the Platonic solids and the Kepler-Poinsot regular polyhedra. Many open problems remain, both in the traditional framework and in the new one.

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