The Department of Mathematics and Statistics' Nagle Lecture Series (NLS), held in honor of the late R. Kent Nagle, featured Dr. Cristopher Bishop, of Stony Brook University. He discussed how to decompose complex geometric objects into smaller pieces by cutting them into triangles.
The purpose of mathematics, claimed SUNY Distinguished Professor Christopher Bishop, is to take in the universe around us and reduce it so it will fit in our heads. At the April 9 R. Kent Nagle Lecture, his example was to break polygons into triangles and polyhedral into tetrahedra.
Two centuries ago, William Wallace, Janos Bolyai, and Paul Gerwien independently proved that given any two polygons of the same area, one could be cut into finitely many triangles and those triangles could be assembled into the other. Professor Bishop then asked if the same was true of pairs of polyhedra of the same volume: could one be chopped into finitely many tetrahedra, which then could be rearranged to be the other? The answer turned out to be “no”: a cube cannot be dismantled and reassembled into a regular tetrahedron of the same volume. However, every polyhedron has a vector called its Dehn Invariant, and given two polyhedra, one can be cut into tetrahedra and reassembled into the other if and only if they have the same Dehn Invariant.
From there, Professor Bishop turned to computer graphics. To create an image of a rabbit, a torus, or a fantasy warrior, one can create a mesh of acute triangles and make a polyhedron with triangular faces in the form of the rabbit, torus, or fantasy warrior. Most computer graphics are generated by generating polyhedra with triangular or quadrilateral faces.
The R. Kent Nagle Lecture Series brings distinguished mathematicians to USF to speak to the USF community about significant mathematical topics of the day.