Abstract
An irregular vertex in a tiling by polygons is a vertex of one tile and belongs to the interior of an edge of another tile. In this paper we show that for any integer \(k\geq 3\), there exists a normal tiling of the Euclidean plane by convex hexagons of unit area with exactly \(k\) irregular vertices. Using the same approach we show that there are normal edge-to-edge tilings of the plane by hexagons of unit area and exactly \(k\) many \(n\)-gons (\(n>6\)) of unit area. A result of Akopyan yields an upper bound for \(k\) depending on the maximal diameter and minimum area of the tiles. Our result complements this with a lower bound for the extremal case, thus showing that Akopyan’s bound is asymptotically tight.
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Acknowledgements
We thank BIRS-CMO and the organizers of the meeting "Soft Packings, Nested Clusters, and Condensed Matter" at Casa Matemática Oaxaca where we obtained the results in this paper.
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The first author is supported by the Research Centre for Mathematical Modelling (RCM2) of Bielefeld University.
The second author is partially supported by the Seed Grant Program of the College of Sciences of the University of Texas Rio Grande Valley.
The third author is partially supported by the NKFIH Hungarian Research Fund grant 134199, the János Bolyai Research Scholarship of the Hungarian Academy of Sciences, and grants BME FIKP-VĺZ and ÚNKP-20-5 New National Excellence Program by the Ministry of Innovation and Technology.
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Frettlöh, D., Glazyrin, A. & Lángi, Z. Hexagon tilings of the plane that are not edge-to-edge. Acta Math. Hungar. 164, 341–349 (2021). https://doi.org/10.1007/s10474-021-01155-5
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DOI: https://doi.org/10.1007/s10474-021-01155-5