Low-Complexity Tilings of the Plane

Abstract : A two-dimensional configuration is a coloring of the infinite grid $$\mathbb {Z}^2$$ with finitely many colors. For a finite subset D of $$\mathbb {Z}^2$$, the D-patterns of a configuration are the colored patterns of shape D that appear in the configuration. The number of distinct D-patterns of a configuration is a natural measure of its complexity. A configuration is considered having low complexity with respect to shape D if the number of distinct D-patterns is at most |D|, the size of the shape. This extended abstract is a short review of an algebraic method to study periodicity of such low complexity configurations.