Two results in discrete geometry
This talk will concern polyhedra and polygons.
1. It's well known that there are exactly 5 platonic solids, but why are there only five? We'll observe that this follows from a remarkably short topological proof. Next we'll study "convex deltahedra", which is another collection of polyhedra, and observe that there are exactly 8 of them using a combinatorial version of the Gauss-Bonnet theorem.
2. Given an integer n, can we cut a square into n equal area triangles? This turns out to be impossible for all odd n, by application of graph theory and p-adics! We'll marvel at this awesome proof I learned about earlier this year.