  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.
