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≥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.
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
Acta Mathematica Hungarica