Two Dimensional Tilings in Two and Three Dimensions

Dana Randall
Georgia Institute of Technology

A domino tiling is a covering of some finite region of Z^2 with dominoes, with each domino covering exactly two adjacent squares. These tilings have attracted attention across disciplines because of their rich underlying structure. For example, physicists study domino tilings on square regions to study physical properties of systems of diatomic molecules -- square regions can be shown to have maximal entropy. Tilings of the so-called Aztec diamond have been shown to exhibit beautiful combinatorial properties (which is related to the fact that Aztec diamonds have lower entropy).

In this talk I will present an unusual extension of "domino tilings" to three-dimensions which consists of tiling special three-dimensional regions with triangular prisms. While it is counter-intuitive that triangular prisms are related to dominoes, the connection arises from a compelling bijection, due to Thurston, which maps domino tilings to surfaces. The set of three-dimensional tilings with prisms can be decomposed into a set of nested surfaces S1, S2, ..., Sk, where each surfaces corresponds to a domino tiling (via Thurston's bijection) and the surfaces are forced to satisfy a partial order S1 <= S2 <= ... <= Sk (the partial order just refers to one surface being "higher" than another).

We will focus on tiling two types of regions: the first is an "octagonal pyramid" and the set of prism tilings corresponds bijectively to nested domino tilings of square regions. The second is an "Aztec pyramid" where the set of tilings corresponds to nested tilings of the Aztec diamond. While the square and Aztec diamond have different entropy in two-dimensions, we will show that these three-dimensional analogues have the same entropy. Finally, we will discuss Markov chains which can be used to sample tilings in both two and three dimensions.