---
res:
  bibo_abstract:
  - "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.@eng"
  bibo_authorlist:
  - foaf_Person:
      foaf_givenName: Robert
      foaf_name: Connelly, Robert
      foaf_surname: Connelly
  - foaf_Person:
      foaf_givenName: Matthew
      foaf_name: Funkhouser, Matthew
      foaf_surname: Funkhouser
  - foaf_Person:
      foaf_givenName: Vivian Zieve
      foaf_name: Kuperberg, Vivian Zieve
      foaf_surname: Kuperberg
      foaf_workInfoHomepage: http://www.librecat.org/personId=c3bac823-112d-11f0-a3f5-c264f852e697
  - foaf_Person:
      foaf_givenName: Evan
      foaf_name: Solomonides, Evan
      foaf_surname: Solomonides
  bibo_doi: 10.1007/s00454-016-9843-x
  bibo_issue: '3'
  bibo_volume: 58
  dct_date: 2017^xs_gYear
  dct_isPartOf:
  - http://id.crossref.org/issn/0179-5376
  - http://id.crossref.org/issn/1432-0444
  dct_language: eng
  dct_publisher: Springer Nature@
  dct_title: Packings of equal disks in a square torus@
...