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