Configuration space partitioning in tilings of a bounded region of the plane

Abstract

Given a finite collection of two-dimensional tile types, the field of study concerned with covering the plane with tiles of these types exclusively has a long history, having enjoyed great prominence in the last six to seven decades. Much of this interest has revolved around fundamental geometrical problems such as minimizing the variety of tile types to be used, and also around important applications in areas such as crystallography as well as others. All these applications are of course confined to finite spatial regions, but in many cases they refer back directly to progress in tiling the whole, unbounded plane. Tilings of bounded regions of the plane have also been actively studied, but in general the additional complications imposed by the boundary conditions tend to constrain progress to mostly indirect results, such as recurrence relations. Here we study the tiling of rectangular regions of the plane by rectangular tiles. The tile types we use are squares, dominoes, and straight tetraminoes. For this set of tile types, not even recurrence relations seem to be available. Our approach is to seek to characterize this complex system through some fundamental physical quantities. We do this on two parallel tracks, one analytical for what seems to be the most complex special case still amenable to such approach, the other based on the Wang-Landau method for state-density estimation. Given a simple energy function based solely on tile contacts, we have found either approach to lead to illuminating depictions of entropy, temperature, and above all partitions of the configuration space. The notion of a configuration, in this context, refers to how many tiles of each type are used. We have found that certain partitions help bind together different aspects of the system in question and conjecture that future applications will benefit from the possibilities they afford.