Enumeration of Octagonal Tilings with Application to Quasicrystal Entropy

Maxwell Hutchinson, University of Chicago

Quasicrystals can be modeled as rhombic tilings of octagonal spaces in two dimensions. The size of the state-space of tilings is related to the entropy of the quasicrystal, making the enumeration of valid tilings as relevant as it is challenging. Various computational techniques yield improvements in preexisting algorithms, exposing previously intractable tiling sizes. The entropy density converges in the limit of large tilings of constant shape, allowing the practical application of these techniques to large systems. The relationship of the entropy density to the shape is found to be approximately Gaussian with respect to the logarithm of the ratio of edge lengths.

Abstract Author(s): Max Hutchinson