Enumeration of octagonal tilings

TL;DR

This paper develops an efficient algorithm for enumerating octagonal tilings, providing new bounds on their entropy density and advancing understanding of quasicrystal models and combinatorial tiling problems.

Contribution

It introduces a novel, efficient enumeration method for octagonal tilings, enabling analysis of larger regions and more precise entropy bounds.

Findings

Bounds on entropy density converge to 0.36021(3)

Algorithm allows investigation of larger tiling regions

Provides insights into tiling entropy in quasicrystal models

Abstract

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 degree 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 toward S = 0.36021(3).