Bard College Mathematics, Computer Science, and Physics Talks

The angle defect, which goes back to Descartes, is a very simple way of measuring the curvature at the vertices of a polyhedral surface in Euclidean space. The angle defect is the polyhedral (and much simpler) analog of Gaussian curvature, as studied in differential geometry. Although the angle defect is the only plausible definition of curvature at the vertices of a polyhedral surface, it turns out that there is more than one possible way to generalize this definition to arbitrary finite 2-dimensional polyhedra, and to higher dimensional polyhedra. This talk will present a few different such generalizations, and will discuss a way to compare these different generalizations in dimension 2. The talk will be elementary, though a willingness to consider higher dimensional polyhedra is required.