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.
Monday, December 6, 2021 - 2:00pm
Satyan Devadoss and `Sam" Shimian Zhang
U San Diego and U Colorado Boulder
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 all unfoldings of the n-simplex, the n-cube and the 4-orthoplex yield nets, but demonstrate its surprising failure for any orthoplex of higher dimension. This is joint work with numerous authors, and the talk is highly visual and interactive, at the intersection of topology, geometry, and combinatorics.