Restricted square-triangle tilings

Abstract

Abstract. Restricted square-triangle tilings are tilings of regular squares and triangles where the number of vertex types has been limited to a subset of all possible vertex types. Such structures have recently been observed in several places: for example as strip phases in colloidal patterns on laser fields with pentagonal symmetry, and as square-pair-free tilings in patterns formed by dipolar magnetic binary colloidal mixtures. In general there are four vertex configurations: A (4 squares, 4 4 ), Z (6 triangles, 3 6 ), H (4 2 3 3 ), and S (43433). Strip phases belong to type AHZ, square-pair-free phases to SZ, standard inflation generated quasicrystals to HSZ, and unrestricted dodecagonal random tilings to AHSZ. We study several vertex type combinations and describe the properties of the patterns generated. AHZ tilings can be mapped to binary necklaces and classified completely. A similar mapping is possible for many HS tilings. Defect-free SZ tilings can be transformed into a triangle-square-hexagon (TSH) supertiling with matching rules. Monte-Carlo (MC) simulations are used to examine the properties of several classes of restricted tilings.