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Low-Complexity Tilings of the Plane

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Descriptional Complexity of Formal Systems (DCFS 2019)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11612))

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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.

J. Kari—Research supported by the Academy of Finland grant 296018.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Turku, Turku, Finland

    Jarkko Kari

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  1. Jarkko Kari
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Correspondence to Jarkko Kari .

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Editors and Affiliations

  1. Slovak Academy of Sciences, Košice, Slovakia

    Michal Hospodár

  2. Slovak Academy of Sciences, Košice, Slovakia

    Galina Jirásková

  3. Saint Mary's University, Halifax, NS, Canada

    Stavros Konstantinidis

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Kari, J. (2019). Low-Complexity Tilings of the Plane. In: Hospodár, M., Jirásková, G., Konstantinidis, S. (eds) Descriptional Complexity of Formal Systems. DCFS 2019. Lecture Notes in Computer Science(), vol 11612. Springer, Cham. https://doi.org/10.1007/978-3-030-23247-4_2

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  • DOI: https://doi.org/10.1007/978-3-030-23247-4_2

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-23246-7

  • Online ISBN: 978-3-030-23247-4

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