Fletcher Jones Professor of Applied Mathematics
University of San Diego
Unfoldings and Nets of Polyhedra and Polytopes
Before arriving at the University of San Diego, he was a professor at Williams College since 2002, along with holding visiting positions at Ohio State, UC Berkeley, MSRI, Harvey Mudd, and Stanford. He is an inaugural Fellow of the American Mathematical Society, and recipient of two national teaching awards from the Mathematical Association of America. His works and thoughts have appeared in venues such as NPR, the Times of London, the Washington Post, the Los Angeles Times, and Forbes, with support over the years by the National Science Foundation, the Mellon Foundation, the John Templeton Foundation, and the Department of Defense.
The study of unfolding polyhedra was popularized by Albrecht Dürer in the early 16th century who first recorded examples of polyhedra nets (connected edge unfoldings of polyhedra that lay flat on the plane without overlap). It was conjectured that every convex polyhedron can be cut along certain edges to admit at least one net. This claim remains tantalizingly open and has resulted in numerous areas of exploration, from origami foldings to airbag designs.
Over a decade ago, it was shown that *every* edge unfolding of the Platonic solids yielding a valid net. We consider this property for regular polytopes in higher dimensions, notably the simplex, cube, and orthoplex. We prove that things works beautifully for the cube and simplex but surprising fail for any orthoplex of dimension greater than four. This is joint work with numerous authors, and the talk is highly visual and interactive, at the intersection of computational geometry, algorithms, and combinatorics.