  These are the lecture notes from my 2014 PCMI graduate summer school
lectures. In these lectures, we'll study simple models of materials from
several different perspectives: geometry (packing problems), information theory
(error-correcting codes), and physics (ground states of interacting particle
systems). These perspectives each shed light on some of the same problems and
phenomena, while highlighting different techniques and connections. One
noteworthy phenomenon is the exceptional symmetry that is found in certain
special cases, and we'll examine when and why it occurs. The overall theme of
the lectures is thus order vs. disorder. How much symmetry can we expect to see
in optimal geometric structures?
