Between the Cracks: Filling Space with Polygonal Shapes
Christopher Ennis · American Mathematical Monthly · 2020
A proof is given, for a large class of polygons, of a conjecture of Shier concerning his algorithm for the disjoint but otherwise random placement of successively smaller copies of an arbitrary shape within a bounded region whose area is the infinite sum of all the shape areas. If the shapes decrease successively in size according to a power law with an exponent falling in a specified range of values, dependent on the particular shape, Shier conjectured there will always be sufficient room within the bounded region to place the next shape in the sequence, disjoint from all previous shapes. The conjecture implies the shapes would fill the bounded region, except for a possible set of measure zero. Our proof confirms the conjecture for all convex polygons, as well as those nonconvex polygons that satisfy a certain condition that we call double containment. Double containment seems to us a natural condition to place on the nonconvexity of polygons and may prove of interest beyond its use in this article. Empirically, Shier’s conjecture appears true for a much larger range of exponents and variety of shapes. Some possible reasons for this are discussed.