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Deformation Theory Seminar

Monday, December 6, 2021 - 2:00pm

Satyan Devadoss and `Sam" Shimian Zhang

U San Diego and U Colorado Boulder


University of Pennsylvania

The talk will be of interest at almost all levels, bright undergrads to emeritus faculty

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.  

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.