Journal of Geometry

, Volume 102, Issue 1, pp 27-51

First online:

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

  • William DickinsonAffiliated withDepartment of Mathematics, Grand Valley State University Email author 
  • , Daniel GuillotAffiliated withDepartment of Mathematics, Louisiana State University
  • , Anna KeatonAffiliated withDepartment of Mathematical Sciences, Clemson University
  • , Sandi XhumariAffiliated withDepartment of Mathematics, University of Connecticut

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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