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

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.