Document Open Access Logo

When Ternary Triangulated Disc Packings Are Densest: Examples, Counter-Examples and Techniques

Authors Thomas Fernique, Daria Pchelina

Thumbnail PDF


  • Filesize: 1.37 MB
  • 17 pages

Document Identifiers

Author Details

Thomas Fernique
  • CNRS & LIPN, Univ. Paris Nord, 93430 Villetaneuse, France
Daria Pchelina
  • LIPN, Univ. Paris Nord, 93430 Villetaneuse, France

Cite AsGet BibTex

Thomas Fernique and Daria Pchelina. When Ternary Triangulated Disc Packings Are Densest: Examples, Counter-Examples and Techniques. In 39th International Symposium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 258, pp. 32:1-32:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)


We consider ternary disc packings of the plane, i.e. the packings using discs of three different radii. Packings in which each "hole" is bounded by three pairwise tangent discs are called triangulated. Connelly conjectured that when such packings exist, one of them maximizes the proportion of the covered surface: this holds for unary and binary disc packings. For ternary packings, there are 164 pairs (r, s), 1 > r > s, allowing triangulated packings by discs of radii 1, r and s. In this paper, we enhance existing methods of dealing with maximal-density packings in order to study ternary triangulated packings. We prove that the conjecture holds for 31 triplets of disc radii and disprove it for 40 other triplets. Finally, we classify the remaining cases where our methods are not applicable. Our approach is based on the ideas present in the Hales' proof of the Kepler conjecture. Notably, our proof features local density redistribution based on computer search and interval arithmetic.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
  • Disc packing
  • density
  • interval arithmetic


  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    PDF Downloads


  1. N. Bedaride and T. Fernique. Density of Binary Disc Packings: The Nine Compact Packings. Discrete and Computational Geometry, 67:1-24, 2022. URL:
  2. R. Connelly, S. Gortler, E. Solomonides, and M. Yampolskaya. Circle packings, triangulations, and rigidity. Oral presentation at the conference for the 60th birthday of Thomas C. Hales, 2018. Google Scholar
  3. R. Connelly and S. J. Gortler. Packing Disks by Flipping and Flowing. Discret. Comput. Geom., 66:1262-1285, 2021. Google Scholar
  4. J. Conway and N.J.A. Sloane. Sphere Packings, Lattices and Groups. Grundlehren der mathematischen Wissenschaften. Springer New York, 1998. Google Scholar
  5. S. L. Devadoss and J. O'Rourke. Discrete and Computational Geometry. Princeton University Press, 2011. Google Scholar
  6. The Sage Developers. Sage mathematics software (version 9.0)., 2020.
  7. E. Fayen, A. Jagannathan, G. Foffi, and F. Smallenburg. Infinite-pressure phase diagram of binary mixtures of (non)additive hard disks. The Journal of Chemical Physics, 152(20):204901, 2020. Google Scholar
  8. L. Fejes Tóth. Über die dichteste Kugellagerung. Math. Z., 48:676-684, 1943. URL:
  9. L. Fejes Tóth. Compact Packing of Circles. Studia Sci. Math. Hungar., 19:103-107, 1984. Google Scholar
  10. L. Fejes Tóth and J. Molnár. Unterdeckung und Überdeckung der Ebene durch Kreise. Mathematische Nachrichten, 18:235-243, 1958. Google Scholar
  11. S. P. Fekete, P Keldenich, and C. Scheffer. Packing disks into disks with optimal worst-case density. Discrete and Computational Geometry, 2022. URL:
  12. S. P. Fekete, S. Morr, and C. Scheffer. Split packing: Packing circles into triangles with optimal worst-case density. In Algorithms and Data Structures, pages 373-384. Springer International Publishing, 2017. Google Scholar
  13. T. Fernique. A densest ternary circle packing in the plane., 2019.
  14. T. Fernique. Density of binary disc packings: Lower and upper bounds. Experimental Mathematics, pages 1-12, 2022. Google Scholar
  15. T. Fernique, A. Hashemi, and O. Sizova. Compact packings of the plane with three sizes of discs. Discret. Comput. Geom., 66(2):613-635, 2021. Google Scholar
  16. T. Fernique and D. Pchelina. Compact packings are not always the densest., 2021.
  17. T. Fernique and D. Pchelina. Density of triangulated ternary disc packings., 2022.
  18. A. Florian. Ausfüllung der Ebene durch Kreise. Rendiconti del Circolo Matematico di Palermo, 9:300-312, 1960. Google Scholar
  19. C. F. Gauss. Untersuchungen über die Eigenschaften der positiven ternären quadratischen Formen von Ludwig August Seber. Göttingische gelehrte Anzeigen, 1831. Google Scholar
  20. T. C. Hales. A proof of the Kepler conjecture. Annals of Mathematics, 162(3):1065-1185, 2005. Google Scholar
  21. T. C. Hales, M. Adams, G. Bauer, D. T. Dang, J. Harrison, T. L. Hoang, C. Kaliszyk, V. Magron, S. McLaughlin, T. T. Nguyen, T. Q. Nguyen, T. Nipkow, S. Obua, J. Pleso, J. Rute, A. Solovyev, A. H. T. Ta, T. N. Tran, D. T. Trieu, J. Urban, K. K. Vu, and R. Zumkeller. A formal proof of the Kepler conjecture. Forum of Mathematics, Pi, 5:e2, 2017. URL:
  22. T. C. Hales and S. P. Ferguson. The Kepler conjecture. Discrete Comput. Geom., 36(1):1-269, 2006. Google Scholar
  23. T. C. Hales and S. P. Ferguson. A Formulation of the Kepler Conjecture, pages 83-133. Springer New York, New York, NY, 2011. Google Scholar
  24. A. Heppes. On the densest packing of discs of radius 1 and √2-1. Studia Scientiarum Mathematicarum Hungarica, 36:433-454, 2000. Google Scholar
  25. A. Heppes. Some densest two-size disc packings in the plane. Discrete and Computational Geometry, 30:241-262, 2003. Google Scholar
  26. A. B. Hopkins, F. H. Stillinger, and S. Torquato. Densest binary sphere packings. Phys. Rev. E, 85:021130, 2012. Google Scholar
  27. T. Kennedy. A densest compact planar packing with two sizes of discs., 2005.
  28. T. Kennedy. Compact packings of the plane with two sizes of discs. Discret. Comput. Geom., 35(2):255-267, 2006. URL:
  29. J. Lagarias. Bounds for local density of sphere packings and the Kepler conjecture. Discrete and Computational Geometry, 27:165-193, 2002. URL:
  30. M. Messerschmidt. The number of configurations of radii that can occur in compact packings of the plane with discs of n sizes is finite., 2021. URL:
  31. P. I. O’Toole and T. S. Hudson. New high-density packings of similarly sized binary spheres. The Journal of Physical Chemistry C, 115(39):19037-19040, 2011. Google Scholar
  32. T. Paik, B. T. Diroll, C. R. Kagan, and C. B. Murray. Binary and Ternary Superlattices Self-Assembled from Colloidal Nanodisks and Nanorods. Journal of the American Chemical Society, 137(20):6662-6669, 2015. Google Scholar
  33. A. Thue. Über die dichteste Zusammenstellung von kongruenten Kreisen in einer Ebene. Christiania Videnskabs-Selskabets Skrifter, I. Math.-Naturv. Klasse, 1:1-9, 1910. Google Scholar
Questions / Remarks / Feedback

Feedback for Dagstuhl Publishing

Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail