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arXiv · 1911.12752

Hexagon tilings of the plane that are not edge-to-edge

Dirk Frettlöh·Alexey Glazyrin·Zsolt Lángi
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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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Dirk Frettlöh, Alexey Glazyrin, Zsolt Lángi. 2019-11-28. Hexagon tilings of the plane that are not edge-to-edge. https://arxiv.org/abs/1911.12752

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