Math 583GB
Polyhedra
Branko Grünbaum
Spring 2005, Monday/Wednesday/Friday 2:30
The aim of the course is to present a general theory of
polyhedra and other complexes consisting of flat parts. Polygons,
polyhedra, configurations and other examples of this type of object 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 sway 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 general approach will enable a
far-reaching development of the traditional topics, such as regular
polygons, regular (Platonic) polyhedra, isohedra or isogonal tilings and
polyhedra. It also leads to a variety of new results and open problems.
The more general point of view allows both the unrestricted preservation of
continuity in representations of combinatorial structures and the
consistent application of duality.