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

Frettlöh, Dirk and Glazyrin, Alexey and Lángi, Zsolt (2021) Hexagon tilings of the plane that are not edge-to-edge. ACTA MATHEMATICA HUNGARICA, 164. pp. 341-349. ISSN 0236-5294

Text 10474_2021_1155_OnlinePDF.pdf Restricted to Repository staff only Download (431kB)

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≥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.

Actions (login required)

Edit Item