Random square-triangle tilings

Abstract

An adequate update move for random square-triangle tilings rearranges a ‘zipper’ (a non-local, 1-D chain) of squares and triangles. Random canonical-cell tilings, upon which promising atomic models of icosahedral quasicrystals can be constructed, require an update move that is equally complex. A continuum theory is postulated for maximally random square-triangle tilings with small background phason strains. The entropy density of the theory includes terms up to 3rd order in the phason-strain field, and contains four unknown parameters: the entropy per unit area at zero phason strain, two 2nd-order elastic constants, and one 3rd-order elastic constant. By implementing the zipper update move, measuring phason-mode fluctuations, introducing a pseudo-Hamiltonian, and using the Ferrenberg-Swendsen histogram method, estimates of these four parameters are obtained.