Math 553A: Polyhedra
Branko Grünbaum
Spring 2001, Monday/Wednesday/Friday 2:30-3:20
The aim of the course is to present the beginnings of a general theory
of polyhedra and other complexes consisting of flat parts. Polygons,
polyhedra, configurations and other examples of this type of objects have
been investigated for a long time. However, most of the research
has been characterized by timidity in the definition of concepts, and the
reluctance to adequately consider the interplay between the geometric objects
and the underlying combinatorial structures. Algebra would be unthinkable
without homomorphisms, or without group representations that are not faithful.
However, in the geometry of the polyhedral objects the basic ideas have
since ancient times been formulated in such a way as to automatically impose
an isomorphism between the geometric and the combinatorial aspects.
The main purpose of the course is the elimination of this shortcoming in
the theory of polyhedra. As introduction and motivation for the course,
a survey of the various kinds of polyhedra studied since antiquity will
be given before the development of the new approaches.
The more general point of view allows both the unrestricted preservation
of continuity in representations of combinatorial structures, and the consistent
application of duality. This leads to a complete reconsideration
of many classical topics, including the regular polygons, regular (Platonic)
polyhedra, isohedral or isogonal tilings and polyhedra, and to a variety
of new results and open problems.