Mathematics Colloquia and Seminars

Return to Colloquia & Seminar listing

It is easier to pack some shapes in the plane than others. For example, identical squares tile the plane, with no wasted space, but even the best packing of identical circular disks leaves about 10% of the plane unfilled. This talk will discuss the problem of finding the worst possible convex shape for packing. Do some shapes leave even more unfilled space than circles do? In 1934, Reinhardt made a surprising conjecture about what the worst shape should be. The conjecture is still unresolved. Ulam conjectured what the worst shape should be in three dimensions, and that conjecture is also still unresolved. We will describe what is known about the solution to Reinhardt's 1934 conjecture. Hint: we are closing in on the answer. We will cover the basics and also make some excursions into hyperbolic geometry, ordinary differential equations, optimal control theory, and Hamiltonian mechanics.

Part of the Thurston Lecture Series.