Partitioning Regular Polygons into Circular Pieces II:Nonconvex Partitions

In [DO03] we explored partitioning regular k-gons into “circular” convex pieces. Circularity of a polygon is measured by the aspect ratio: the ratio of the diameters of the smallest circumscribing circle to the largest inscribed disk. We seek partitions with aspect ratio close to 1, ideally the optimal ratio. Although we start with regular polygons, most of the machinery developed extends to arbitrary polygons. For convex pieces, we showed in [DO03] that optimality can be achieved for an equilateral triangle only by an infinite partition, and that for all k ≥ 5, the 1-piece partition is optimal. We left the difficult case of a square unsettled, narrowing the optimal ratio to a small range. Here we turn our attention to partitions that permit the pieces to be nonconvex. The results are cleanest if we do not demand that the pieces be polygonal, but rather permit curved sides to the pieces. The results change dramatically. The equilateral triangle has an optimal 4-piece partition, the square an optimal 13-piece partition, the pentagan an optimal partition with more than 20 thousand pieces. For hexagons and beyond, we provide a general algorithm that approaches optimality, but does not achieve it.