A framework for the construction of self-replicating tilings (Q1042460)

A lattice tiling of \({\mathbb R}^2\) is a self-replicating tiling if all tiles are translates of each other by elements of the lattice and there is an expanding linear map of the plane that maps each tile to a union of other tiles. \textit{F. M. Dekking} [J. Comb. Theory, Ser. A 32, 315--320 (1982; Zbl 0492.05019)] gave a method to compute the boundaries of self-replicating tilings as a `recurrent set' on a free group of a finite alphabet. The author generalizes Dekking's construction to deal with multi-tiles and generations by both translations and rotations. He presents examples including self-replicating tiles for crystallographic tilings and aperiodic tilings.