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Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.SoCG.2021.36
URN: urn:nbn:de:0030-drops-138356
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2021/13835/
Fekete, Sándor P. ;
Gurunathan, Vijaykrishna ;
Juneja, Kushagra ;
Keldenich, Phillip ;
Kleist, Linda ;
Scheffer, Christian
Packing Squares into a Disk with Optimal Worst-Case Density
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 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 (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.
BibTeX - Entry
@InProceedings{fekete_et_al:LIPIcs.SoCG.2021.36,
author = {Fekete, S\'{a}ndor P. and Gurunathan, Vijaykrishna and Juneja, Kushagra and Keldenich, Phillip and Kleist, Linda and Scheffer, Christian},
title = {{Packing Squares into a Disk with Optimal Worst-Case Density}},
booktitle = {37th International Symposium on Computational Geometry (SoCG 2021)},
pages = {36:1--36:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-184-9},
ISSN = {1868-8969},
year = {2021},
volume = {189},
editor = {Buchin, Kevin and Colin de Verdi\`{e}re, \'{E}ric},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2021/13835},
URN = {urn:nbn:de:0030-drops-138356},
doi = {10.4230/LIPIcs.SoCG.2021.36},
annote = {Keywords: Square packing, packing density, tight worst-case bound, interval arithmetic, approximation}
}
