Journal of Geometry

, Volume 102, Issue 1–2, pp 27–51

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

  • William Dickinson
  • Daniel Guillot
  • Anna Keaton
  • Sandi Xhumari


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 packing flat torus packing graph rigidity theory 


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Copyright information

© Springer Basel AG 2012

Authors and Affiliations

  • William Dickinson
    • 1
  • Daniel Guillot
    • 2
  • Anna Keaton
    • 3
  • Sandi Xhumari
    • 4
  1. 1.Department of MathematicsGrand Valley State UniversityAllendaleUSA
  2. 2.Department of MathematicsLouisiana State UniversityBaton RougeUSA
  3. 3.Department of Mathematical SciencesClemson UniversityClemsonUSA
  4. 4.Department of MathematicsUniversity of ConnecticutStorrsUSA

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