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- Title
WORST-CASE OPTIMAL SQUARES PACKING INTO DISKS.
- Authors
Fekete, Sándor P.; Gurunathan, Vijaykrishna; Juneja, Kushagra; Keldenich, Phillip; Kleist, Linda; Scheffer, Christian
- Abstract
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 85+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 (π/(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.
- Subjects
PACKING problem (Mathematics); CONTAINERS; INTERVAL analysis; SUBDIVISION surfaces (Geometry); SQUARE
- Publication
Journal of Computational Geometry, 2022, Vol 13, Issue 2, p3
- ISSN
1920-180X
- Publication type
Article