Finitely many radius sets for compact sphere packings
For a packing of balls with pairwise disjoint interiors in $\mathbb{R}^d$, take the centres as vertices, and as simplices the sets $E$ of at most $d + 1$ centres whose balls are pairwise tangent and whose convex hull $\operatorname{conv}(E)$ contains no other centre. The packing is compact when these simplices form a geometric simplicial complex whose underlying space is all of $\mathbb{R}^d$ and in which every simplex lies in a $d$-simplex. For integers $d, n \ge 2$ let $\Pi_{d,n}$ be the set of tuples $0 < r_0 < r_1 < \cdots < r_{n-1} = 1$ that occur as the exact set of radii of a compact packing of $\mathbb{R}^d$. In the plane, Kennedy showed that $\Pi_{2,2}$ has nine elements (2006), Fernique, Hashemi and Sizova that $\Pi_{2,3}$ has 164 (2021), and Messerschmidt that $\Pi_{2,n}$ is finite for every $n$ (2024). In space, Fernique showed that $\Pi_{3,2}$ has one element (2021) and, in a preprint, that $\Pi_{3,3}$ has four (2019). Messerschmidt and Kikianty proved, for all $d, n \ge 2$, that the radius tuples are finite in number among compact packings in which the triangulation of the unit sphere associated with every ball belongs to a fixed set with a rigidity property they call heteroperturbative; the set of all triangulations of the circle has it, the set of all triangulations of the $2$-sphere does not.
Problem. Is $\Pi_{d,n}$ finite for every $d \ge 3$ and every $n \ge 2$? All $n$ radii must occur, the packing need not be periodic, and radius tuples are counted, not packings.
Source: Messerschmidt and Kikianty (2025), Conjecture 1.1, for all d, n ≥ 2; the case d = 2 is Messerschmidt's theorem (2024). Stated for d ≥ 3 by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- MG — Metric Geometry
- Secondary
- CO
- Keywords
- AIM Open Applied Problems, compact packings, sphere packings
References
@article{messerschmidt2025,
author = {Messerschmidt, Miek and Kikianty, Eder},
title = {On compact packings of {E}uclidean space with spheres of finitely many sizes},
journal = {Discrete Comput. Geom.},
volume = {74},
number = {3},
year = {2025},
pages = {719--742},
doi = {10.1007/s00454-024-00628-y},
eprint = {2305.00758},
archivePrefix = {arXiv},
}
@article{messerschmidt2024,
author = {Messerschmidt, Miek},
title = {The number of configurations of radii that can occur in compact packings of the plane with discs of $n$ sizes is finite},
journal = {Discrete Comput. Geom.},
volume = {71},
number = {2},
year = {2024},
pages = {667--682},
doi = {10.1007/s00454-022-00471-z},
eprint = {2110.15831},
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},
}
@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},
}
@article{fernique2021space,
author = {Fernique, Thomas},
title = {Compact packings of space with two sizes of spheres},
journal = {Discrete Comput. Geom.},
volume = {65},
number = {4},
year = {2021},
pages = {1287--1295},
doi = {10.1007/s00454-019-00140-8},
eprint = {1812.03850},
archivePrefix = {arXiv},
}
@misc{fernique2019space,
author = {Fernique, Thomas},
title = {Compact packings of space with three sizes of spheres},
year = {2019},
eprint = {1912.02293},
archivePrefix = {arXiv},
}
@misc{aim-compact-packing-radius-finiteness,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Finitely many radius sets for compact sphere packings}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/634-compact-packing-radius-finiteness.md},
note = {Entry 634 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-00580, "Finitely many radius sets for compact sphere packings", clopen.solutions.