Journal of Geometry

, Volume 102, Issue 1, pp 27–51

Optimal packings of up to six equal circles on a triangular flat torus


    • Department of MathematicsGrand Valley State University
  • Daniel Guillot
    • Department of MathematicsLouisiana State University
  • Anna Keaton
    • Department of Mathematical SciencesClemson University
  • Sandi Xhumari
    • Department of MathematicsUniversity of Connecticut

DOI: 10.1007/s00022-011-0099-6

Cite this article as:
Dickinson, W., Guillot, D., Keaton, A. et al. J. Geom. (2011) 102: 27. doi:10.1007/s00022-011-0099-6


For each n between 1 and 6, we prove that a certain arrangement of n equal circles is the unique optimally dense packing on a standard triangular flat torus (the quotient of the plane by the lattice generated by two unit vectors with a 60 angle). The packings of 1, 2, 3, 4 and 6 circles are based on either a toroidal triangular close packing or a toroidal triangular close packing with one circle removed. The packing of 5 circles is irregular. This proves two cases of a conjecture stronger than L. Fejes Toth’s conjecture about the strong solidity of the triangular close packing on the plane.

Mathematics Subject Classification (2010)

Primary 52C15


Equal circle packingflat toruspacking graphrigidity theory

Copyright information

© Springer Basel AG 2012