Complex quasiperiodic self-similar tilings: their parameterization, boundaries, complexity, growth and symmetry

A class of quasiperiodic tilings of the complex plane is discussed. These tilings are based on beta-expansions corresponding to cubic irrationalities. There are three classes of tilings: Q(3), Q(4) and Q(5). These classes consist of three, four and five pairwise similar prototiles, respectively. A simple algorithm for construction of these tilings is considered. This algorithm uses greedy expansions of natural numbers on some sequence. Weak and strong parameterizations for tilings are obtained. Layerwise growth, the complexity function and the structure of fractal boundaries of tilings are studied. The parameterization of vertices and boundaries of tilings, and also similarity transformations of tilings, are considered.