---
OA_place: repository
OA_type: green
_id: '22198'
abstract:
- lang: eng
  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."
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Robert
  full_name: Connelly, Robert
  last_name: Connelly
- first_name: Matthew
  full_name: Funkhouser, Matthew
  last_name: Funkhouser
- first_name: Vivian Zieve
  full_name: Kuperberg, Vivian Zieve
  id: c3bac823-112d-11f0-a3f5-c264f852e697
  last_name: Kuperberg
- first_name: Evan
  full_name: Solomonides, Evan
  last_name: Solomonides
citation:
  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>
  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>
  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>.
  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.
  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.
  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>.
  short: R. Connelly, M. Funkhouser, V.Z. Kuperberg, E. Solomonides, Discrete &#38;
    Computational Geometry 58 (2017) 614–642.
date_created: 2026-06-29T12:58:50Z
date_published: 2017-01-09T00:00:00Z
date_updated: 2026-07-14T11:14:51Z
day: '09'
doi: 10.1007/s00454-016-9843-x
extern: '1'
external_id:
  arxiv:
  - '1512.08762'
intvolume: '        58'
issue: '3'
language:
- iso: eng
main_file_link:
- url: https://doi.org/10.48550/arXiv.1512.08762
month: '01'
oa_version: Preprint
page: 614-642
publication: Discrete & Computational Geometry
publication_identifier:
  eissn:
  - 1432-0444
  issn:
  - 0179-5376
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
scopus_import: '1'
status: public
title: Packings of equal disks in a square torus
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 58
year: '2017'
...