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A note on discrete lattice-periodic sets with an application to Archimedean tilings

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Abstract

Cao and Yuan obtained a Blichfeldt-type result for the vertex set of the edge-to-edge tiling of the plane by regular hexagons. Observing that the vertex set of every Archimedean tiling is the union of translates of a fixed lattice, we take a more general viewpoint and investigate basic questions for such point sets about the homogeneous and inhomogeneous problem in the geometry of numbers. The Archimedean tilings nicely exemplify our results.

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Authors and Affiliations

  1. Institute for Mathematics, École Polytechnique Fédérale de Lausanne, 1015, Lausanne, Switzerland

    Matthias Schymura

  2. College of Mathematics and Information Science, Hebei Normal University, Shijiazhuang, 050016, China

    Liping Yuan

Authors
  1. Matthias Schymura
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  2. Liping Yuan
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Correspondence to Matthias Schymura.

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The first author was supported by the Freie Universität Berlin within the Excellence Initiative of the German Research Foundation. The second author gratefully acknowledges financial support by National Natural Science Foundation of China (no. 11871192 and no. 11471095), Outstanding Youth Science Foundation of Hebei Province (no. 2013205189), Program for Excellent Talents in University, Hebei Province (no. GCC2014043) and Key Project of the Education Department of Hebei Province (no. zd2017043).

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Schymura, M., Yuan, L. A note on discrete lattice-periodic sets with an application to Archimedean tilings. Beitr Algebra Geom 60, 749–759 (2019). https://doi.org/10.1007/s13366-019-00446-x

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  • DOI: https://doi.org/10.1007/s13366-019-00446-x

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