[{"quality_controlled":"1","doi":"10.1007/s00454-016-9843-x","title":"Packings of equal disks in a square torus","publication_status":"published","year":"2017","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","author":[{"first_name":"Robert","full_name":"Connelly, Robert","last_name":"Connelly"},{"full_name":"Funkhouser, Matthew","first_name":"Matthew","last_name":"Funkhouser"},{"last_name":"Kuperberg","id":"c3bac823-112d-11f0-a3f5-c264f852e697","first_name":"Vivian Zieve","full_name":"Kuperberg, Vivian Zieve"},{"last_name":"Solomonides","first_name":"Evan","full_name":"Solomonides, Evan"}],"day":"09","date_published":"2017-01-09T00:00:00Z","arxiv":1,"article_processing_charge":"No","abstract":[{"text":"Packings of equal disks in the plane are known to have density at most\r\nπ/\r\n√\r\n12, although this density is never achieved in the square torus, which is what we\r\ncall the plane modulo the square lattice. We find packings of disks in a square torus\r\nthat we conjecture to be the most dense for certain numbers of packing disks, using\r\ncontinued fractions to approximate 1/\r\n√\r\n3 and 2 −\r\n√\r\n3. We also define a constant to\r\nmeasure the efficiency of a packing motived by a related constant due to Markov for\r\ncontinued fractions. One idea is to use the unique factorization property of Gaussian\r\nintegers to prove that there is an upper bound for the Markov constant for grid-like\r\npackings. By way of contrast, we show that an upper bound by Gruber [In many cases\r\noptimal configurations are almost regular hexagonal, vol. 65, pp. 121–145, 1999;Geom\r\nDedicata 84(1–3):271–320, 2001] for the error for the limiting density of a packing\r\nof equal disks in a planar square, which is on the order of 1/\r\n√\r\nN, is the best possible,\r\nwhereas for our examples for the square torus, the error for the limiting density is on\r\nthe order of 1/N, where N is the number of packing disks.","lang":"eng"}],"OA_place":"repository","_id":"22198","type":"journal_article","language":[{"iso":"eng"}],"external_id":{"arxiv":["1512.08762"]},"month":"01","status":"public","scopus_import":"1","issue":"3","date_updated":"2026-07-14T11:14:51Z","oa_version":"Preprint","extern":"1","page":"614-642","date_created":"2026-06-29T12:58:50Z","citation":{"short":"R. Connelly, M. Funkhouser, V.Z. Kuperberg, E. Solomonides, Discrete &#38; Computational Geometry 58 (2017) 614–642.","mla":"Connelly, Robert, et al. “Packings of Equal Disks in a Square Torus.” <i>Discrete &#38; Computational Geometry</i>, vol. 58, no. 3, Springer Nature, 2017, pp. 614–42, doi:<a href=\"https://doi.org/10.1007/s00454-016-9843-x\">10.1007/s00454-016-9843-x</a>.","ama":"Connelly R, Funkhouser M, Kuperberg VZ, Solomonides E. Packings of equal disks in a square torus. <i>Discrete &#38; Computational Geometry</i>. 2017;58(3):614-642. doi:<a href=\"https://doi.org/10.1007/s00454-016-9843-x\">10.1007/s00454-016-9843-x</a>","ieee":"R. Connelly, M. Funkhouser, V. Z. Kuperberg, and E. Solomonides, “Packings of equal disks in a square torus,” <i>Discrete &#38; Computational Geometry</i>, vol. 58, no. 3. Springer Nature, pp. 614–642, 2017.","chicago":"Connelly, Robert, Matthew Funkhouser, Vivian Zieve Kuperberg, and Evan Solomonides. “Packings of Equal Disks in a Square Torus.” <i>Discrete &#38; Computational Geometry</i>. Springer Nature, 2017. <a href=\"https://doi.org/10.1007/s00454-016-9843-x\">https://doi.org/10.1007/s00454-016-9843-x</a>.","apa":"Connelly, R., Funkhouser, M., Kuperberg, V. Z., &#38; Solomonides, E. (2017). Packings of equal disks in a square torus. <i>Discrete &#38; Computational Geometry</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s00454-016-9843-x\">https://doi.org/10.1007/s00454-016-9843-x</a>","ista":"Connelly R, Funkhouser M, Kuperberg VZ, Solomonides E. 2017. Packings of equal disks in a square torus. Discrete &#38; Computational Geometry. 58(3), 614–642."},"intvolume":"        58","publisher":"Springer Nature","publication":"Discrete & Computational Geometry","OA_type":"green","article_type":"original","main_file_link":[{"url":"https://doi.org/10.48550/arXiv.1512.08762"}],"publication_identifier":{"issn":["0179-5376"],"eissn":["1432-0444"]},"volume":58}]