# Low-Complexity Tilings of the Plane

## TL;DR

This paper reviews an algebraic approach to analyze periodicity in low-complexity two-dimensional grid colorings, where the number of patterns is bounded by the shape size, contributing to understanding their structure.

## Contribution

It introduces an algebraic method to study the periodicity of low-complexity configurations in the infinite grid.

## Key findings

- Algebraic techniques effectively identify periodic patterns.
- Low complexity configurations exhibit structured, predictable patterns.
- The method simplifies analysis of pattern repetition in grid colorings.

## Abstract

A two-dimensional configuration is a coloring of the infinite grid Z^2 with finitely many colors. For a finite subset D of 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.