Math 583GA
Polygons, Tilings and Polyhedra

Branko Grünbaum

Spring 2003, Monday/Wednesday/Friday 2:30-3:20

This course will investigate polyhedra in 3-dimensional Euclidean space, from various points of view. These include combinatorial, topological, geometric and metric aspects -- each approach necessitates specific tools, that give a certain amount of information and lead to additional problems and questions. Although polyhedra have been the topic of mathematical study since the very beginnings of geometric science in antiquity, very simple questions are still unsolved. We shall encounter many of these; one, which can be easily understood without any prerequisites is the following: How to decide, given a quadrangle, whether there exists a polyhedron all faces of which are congruent to that quadrangle. I do not know the answer, and even if the question is modified by restricting the polyhedra and/or the quadrangles in question to be convex or satisfy some other reasonable condition -- I still do not know the answer.

A major emphasis will be on polyhedra which are not necessarily convex, but exhibit reasonably strong symmetry properties. This includes various classes that have been studied for a long time (such as Platonic, Archimedean, Kepler-Poinsot or uniform polyhedra), and others that are still largely unexplored (for example, isogonal, isohedral, noble, or regular polyhedra).

As a preliminary to the study of polyhedra in 3-space we shall first spend some time discussing polygons in the plane, and spherical polygons (on the 2-sphere in 3-space). One of the reasons for doing this is that we shall need the information about polygons in order to be able to investigate polyhedra: after all, the best way to think about polyhedra is as special collections of polygons. Another reason is that polygons exhibit -- in a much simpler setting -- many of the problems and difficulties which make polyhedra such a challenging topic. Tilings can be understood as a limiting case of polyhedra. As such, they share many properties and problems - but also lead to questions which are specific to them.

There is no book or monograph that presents in organized form the material I expect to cover in this course. Therefore I'll try to keep producing lecture notes that will parallel my presentation; at times, I shall probably resort to distributing copies of various articles or sections from books. The only formal prerequisites for the course are the basics of linear algebra, group theory, and calculus; highly desirable is curiosity about the space we inhabit.

There will be weekly homework assignments, and a take-home final.