Largest radius ratio in nonuniform triangulated circle packings
A triangulated packing of the plane is a packing of closed disks with pairwise disjoint interiors whose contact graph, drawn with straight segments between the centres of tangent disks, triangulates the whole plane. For such a packing whose smallest and largest radii are attained, $0 < r_{\min} \le r_{\max} < \infty$, let $q = r_{\min}/r_{\max}$. Let $q_* = 0.6510501858\ldots$ be the root near $0.65105$ of $$89x^8 + 1344x^7 + 4008x^6 - 464x^5 - 2410x^4 + 176x^3 + 296x^2 - 96x + 1 = 0.$$ Kennedy showed that exactly nine ratios occur in triangulated packings with two sizes of disk (2006), and Fernique, Hashemi and Sizova that exactly 164 pairs of ratios occur with three sizes (2021). One of these pairs, with radii $1$, $r \approx 0.834$ and $q_*$, admits a unique triangulated packing in which all three sizes occur; it is periodic, and Fernique proved it is the densest packing with these three sizes (2019). Connelly and Zhang proved that in a triangulated packing with all radii in $[0.62, 1]$ every disk is tangent to five, six or seven others (2025).
Problem. Does every triangulated packing with at least two distinct radii, whose smallest and largest radii are attained, satisfy $q \le q_*$? The packing need not be periodic, and the number of distinct radii is not bounded.
Source: Connelly and Zhang (2025), Conjecture 1, who conjecture that the answer is yes; the extremal packing is the three-radius packing of Fernique, Hashemi and Sizova (2021), studied by Fernique (2019).
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- MG — Metric Geometry
- Secondary
- CO
- Keywords
- AIM Open Applied Problems, circle packings, triangulations
References
@article{connelly2025,
author = {Connelly, Robert and Zhang, Zhen},
title = {Rigidity of circle packings with flexible radii},
journal = {Discrete Comput. Geom.},
volume = {74},
number = {3},
year = {2025},
pages = {585--618},
doi = {10.1007/s00454-025-00776-9},
eprint = {2206.07165},
archivePrefix = {arXiv},
}
@article{fernique2021plane,
author = {Fernique, Thomas and Hashemi, Amir and Sizova, Olga},
title = {Compact packings of the plane with three sizes of discs},
journal = {Discrete Comput. Geom.},
volume = {66},
number = {2},
year = {2021},
pages = {613--635},
doi = {10.1007/s00454-019-00166-y},
eprint = {1810.02231},
archivePrefix = {arXiv},
}
@misc{fernique2019,
author = {Fernique, Thomas},
title = {A densest ternary circle packing in the plane},
year = {2019},
eprint = {1912.02297},
archivePrefix = {arXiv},
}
@article{kennedy2006,
author = {Kennedy, Tom},
title = {Compact packings of the plane with two sizes of discs},
journal = {Discrete Comput. Geom.},
volume = {35},
number = {2},
year = {2006},
pages = {255--267},
doi = {10.1007/s00454-005-1172-4},
eprint = {math/0407145},
archivePrefix = {arXiv},
}
@misc{aim-triangulated-packing-radius-ratio,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Largest radius ratio in nonuniform triangulated circle packings}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/631-triangulated-packing-radius-ratio.md},
note = {Entry 631 at commit 24edaad, 2026-10-15},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00651, "Largest radius ratio in nonuniform triangulated circle packings", clopen.solutions.