{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/packing-squares-into-a-disk-with-optimal","title":"Packing Squares into a Disk with Optimal Worst-Case Density","arxiv_id":"2103.07258","date":"2021-03-12","proceeding":null,"authors":["Sándor P. Fekete","Vijaykrishna Gurunathan","Kushagra Juneja","Phillip Keldenich","Linda Kleist","Christian Scheffer"],"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 $\\delta=\\frac{8}{5\\pi}\\approx 0.509$. This implies that any set of (not necessarily equal) squares of total area $A \\leq \\frac{8}{5}$ can always be packed into a disk with radius 1; in contrast, for any $\\varepsilon>0$ there are sets of squares of total area $\\frac{8}{5}+\\varepsilon$ 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 $\\left(\\frac{1}{2}\\right)$, circles in a square $\\left(\\frac{\\pi}{(3+2\\sqrt{2})}\\approx 0.539\\right)$ and circles in a circle $\\left(\\frac{1}{2}\\right)$ 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.","url_abs":"https://arxiv.org/abs/2103.07258v3","url_pdf":"https://arxiv.org/pdf/2103.07258v3.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"packing-squares-into-a-disk-with-optimal","repo_url":"https://github.com/phillip-keldenich/squares-in-disk","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}