Abstract or Additional Information
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.
This talk will describe what is known about the solution to Reinhardt's
1934 conjecture. (Hint: we are closing in on the answer.) I will avoid
technicalities to make the talk accessible to students, but we will
cover the basics and also make some excursions into non-Euclidean
geometry, ordinary differential equations, optimal control theory, and
Hamiltonian mechanics.
Here is a blog post by John Baez about the problem:
http://blogs.ams.org/visualinsight/2014/11/01/packing-smoothed-octagons/