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.

多边形拼贴中的不规则顶点是一个拼贴的顶点，属于另一拼贴边缘的内部。在本文中，我们证明对于任何整数\(k\geq 3\)，都存在欧几里得平面的法向平铺，由单位面积的凸六边形与恰好\(k\) 个不规则顶点构成。使用相同的方法，我们证明了平面的正常边到边平铺由单位面积的六边形和恰好\(k\)多个\(n\) -gons ( \(n>6\) ) 单位区域。Akopyan 的结果产生了\(k\)的上限取决于瓷砖的最大直径和最小面积。我们的结果用极值情况的下限对此进行了补充，从而表明 Akopyan 的边界是渐近紧的。