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

TL;DR

This paper constructs convex hexagon tilings of the plane with a specified number of irregular vertices, demonstrating the range of such tilings and analyzing bounds related to tile geometry.

Contribution

It shows the existence of plane tilings by convex hexagons with any number of irregular vertices and extends to edge-to-edge tilings with polygons larger than hexagons, complementing existing bounds.

Findings

Existence of tilings with any number of irregular vertices for convex hexagons.

Construction of edge-to-edge tilings with hexagons and larger polygons with specified irregular vertices.

Bounds on irregular vertices related to tile diameter and area, showing asymptotic tightness.

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.