2021/03/12 by Fekete, Sándor P., Gurunathan, Vijaykrishna, Juneja, Kushagra +3
#Computational Geometry (cs.CG) #F.2.2 #FOS: Computer and information sciences
paper · doi:10.48550/arxiv.2103.07258
We provide a tight result for a fundamental problem arising from packing squares into a circular container: The critical density of packing squares into a disk is δ=(8)/(5π)≈ 0.509. This implies that any set of (not necessarily equal) squares of total area A ≤ (8)/(5) can always be packed into a disk with radius 1; in contrast, for any ε>0 there are sets of squares of total area (8)/(5)+ε that cannot be packed, even if squares may be rotated. This settles the last (and arguably, most elusive) case of packing circular or square objects into a circular or square container: The critical densities for squares in a square ((1)/(2)), circles in a square (\fracπ(3+2√(2))≈ 0.539) and circles in a circle ((1)/(2)) have already been established, making use of recursive subdivisions of a square container into pieces bounded by straight lines, or the ability to use recursive arguments based on similarity of objects and container; neither of these approaches can be applied when packing squares into a circular container. Our proof uses a careful manual analysis, complemented by a computer-assisted part that is based on interval arithmetic. Beyond the basic mathematical importance, our result is also useful as a blackbox lemma for the analysis of recursive packing algorithms. At the same time, our approach showcases the power of a general framework for computer-assisted proofs, based on interval arithmetic.