@article{22198,
  abstract     = {Packings of equal disks in the plane are known to have density at most
π/
√
12, although this density is never achieved in the square torus, which is what we
call the plane modulo the square lattice. We find packings of disks in a square torus
that we conjecture to be the most dense for certain numbers of packing disks, using
continued fractions to approximate 1/
√
3 and 2 −
√
3. We also define a constant to
measure the efficiency of a packing motived by a related constant due to Markov for
continued fractions. One idea is to use the unique factorization property of Gaussian
integers to prove that there is an upper bound for the Markov constant for grid-like
packings. By way of contrast, we show that an upper bound by Gruber [In many cases
optimal configurations are almost regular hexagonal, vol. 65, pp. 121–145, 1999;Geom
Dedicata 84(1–3):271–320, 2001] for the error for the limiting density of a packing
of equal disks in a planar square, which is on the order of 1/
√
N, is the best possible,
whereas for our examples for the square torus, the error for the limiting density is on
the order of 1/N, where N is the number of packing disks.},
  author       = {Connelly, Robert and Funkhouser, Matthew and Kuperberg, Vivian Zieve and Solomonides, Evan},
  issn         = {1432-0444},
  journal      = {Discrete & Computational Geometry},
  number       = {3},
  pages        = {614--642},
  publisher    = {Springer Nature},
  title        = {{Packings of equal disks in a square torus}},
  doi          = {10.1007/s00454-016-9843-x},
  volume       = {58},
  year         = {2017},
}