Tilings from graph directed iterated function systems

A new method for constructing self-referential tilings of Euclidean space from a graph directed iterated function system (GIFS), based on a combinatorial structure we call a pre-tree, is introduced. For each GIFS, a family of tilings is constructed indexed by a parameter. For what we call a commensurate GIFS, our method is used to define what we refer to as balanced tilings. Under mild conditions on the commensurate GIFS and the parameter, the resulting balanced tilings have a finite set of prototiles, are self-similar, and are quasiperiodic. A notion of rigidity is defined for a commensurate GIFS, and a necessary and sufficient condition for two rigid balanced tilings to be congruent is provided. For a given rigid GIFS, there are uncountably many balanced tilings, corresponding to uncountably many parameters. All rigid balanced tilings are non-periodic.