This project is related to symplectic geometry, which is a type of geometry that shows up in physics all of the way from classical mechanics to string theory. One of the great things about symplectic geometry is that certain four-dimensional symplectic manifolds can actually be represented by two dimensional objects, called Delzant polygons, so problems that are in principle difficult can actually beapproached using familiar two-dimensional techniques.
In this project we will investigate certain "packing capacities" of Delzant polygons, which essentially means we will try to measure how many triangles of a certain type can fit into each Delzant polygon. To do this we will have to learn what a Delzant polygon is, and what kinds of triangle packings are admissible. Any sort of polygon packing result in this context will actually imply some interesting results about ball-packing the associated four-dimensional manifold, but we won't have to actually do any four-dimensional geometry.
Some knowledge of linear algebra would be useful. We will be using 2 x 2 integer matrices.