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Low-Complexity Tilings of the Plane

  • Jarkko KariEmail author
Conference paper
  • 151 Downloads
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11612)

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.

Keywords

Pattern complexity Periodicity Nivat’s conjecture Low complexity configurations Low complexity subshifts Commutative algebra Algebraic subshifts Domino problem 

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Copyright information

© IFIP International Federation for Information Processing 2019

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of TurkuTurkuFinland

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