Date of Original Version
This is the author’s version of a work that was accepted for publication. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version is available at http://dx.doi.org/10.1016/j.tcs.2015.03.019
Abstract or Description
Random tilings are interesting as idealizations of atomistic models of quasicrystals and for their connection to problems in combinatorics and algorithms. Of particular interest is the tiling entropy density, which measures the relation of the number of distinct tilings to the number of constituent tiles. Tilings by squares and 45° rhombi receive special attention as presumably the simplest model that has not yet been solved exactly in the thermodynamic limit. However, an exact enumeration formula can be evaluated for tilings in finite regions with fixed boundaries. We implement this algorithm in an efficient manner, enabling the investigation of larger regions of parameter space than previously were possible. Our new results appear to yield monotone increasing and decreasing lower and upper bounds on the fixed boundary entropy density that converge towardS∞=0.36021(3).
Theoretical Computer Science, 598, 40-60.