Configuration space partitioning in tilings of a bounded region of the plane
arxiv(2023)
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.
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