Compact Packings of the Plane with Three Sizes of Discs

  • Thomas FerniqueEmail author
  • Amir Hashemi
  • Olga Sizova


A compact packing is a set of non-overlapping discs where all the holes between discs are curvilinear triangles. There is only one compact packing by discs of size 1. There are exactly nine values of r which allow a compact packing by discs of sizes 1 and r. We prove here that there are exactly 164 pairs (rs) allowing a compact packing by discs of sizes 1, r and s.


Circle packing Compact packing Triangulated packing 



We thank Tom Kennedy for pointing us the reference [17], fortunately after we completed our proof so that our approach has not been influenced. We thank Thierry Monteil for answering various questions about SageMath, as well as Bruno Salvy for discussions on Gröbner basis. We thank the referees of a short conference version of this paper [9], as well as the referees of this long version.

Supplementary material

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Authors and Affiliations

  1. 1.Université Paris 13, CNRS, Sorbonne Paris Cité, UMR 7030VilletaneuseFrance
  2. 2.Department of Mathematical SciencesIsfahan University of TechnologyIsfahanIran
  3. 3.Faculty of MathematicsHigher School of EconomicsMoscowRussia
  4. 4.Semenov Institute of Chemical PhysicsMoscowRussia

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