Editing Kepler conjecture (section)

==Related problems==
;[[Axel Thue|Thue]]'s theorem: The regular hexagonal packing is the densest [[circle packing]] in the plane (1890). The density is {{frac|{{pi}}|{{sqrt|12}}}}.
:The 2-dimensional analog of the Kepler conjecture; the proof is elementary. Henk and Ziegler attribute this result to Lagrange, in 1773 (see references, pag. 770).
:A simple proof by Chau and Chung from 2010 uses the [[Delaunay triangulation]] for the set of points that are centers of circles in a saturated circle packing.<ref>{{cite arXiv|last1=Chang|first1=Hai-Chau|last2=Wang|first2=Lih-Chung|title=A Simple Proof of Thue's Theorem on Circle Packing|eprint=1009.4322|date=22 September 2010|class=math.MG}}</ref>

;The hexagonal [[honeycomb theorem]]: The most efficient partition of the plane into equal areas is the regular hexagonal tiling.<ref>{{cite journal|last1=Hales|first1=Thomas C.|title=The Honeycomb Conjecture|url=https://doi.org/10.1007/s004540010071|journal=[[Discrete & Computational Geometry]]|volume=25|issue=|pages=1–22|arxiv=math/9906042|date=20 May 2002|doi=10.1007/s004540010071 |s2cid=14849112 }}</ref>
:Related to Thue's theorem.

;[[Dodecahedral conjecture]]: The volume of the [[Voronoi diagram|Voronoi polyhedron]] of a sphere in a packing of equal spheres is at least the volume of a regular dodecahedron with inradius 1. McLaughlin's proof,<ref>{{cite journal |last1=Hales |first1=Thomas C. |last2=McLaughlin |first2=Sean |year=2010 |title=The Dodecahedral Conjecture |journal=[[Journal of the American Mathematical Society]] |volume=23 |issue=2 |pages=299–344 |doi=10.1090/S0894-0347-09-00647-X |arxiv=math.MG/9811079 |bibcode=2010JAMS...23..299H }}</ref> for which he received the 1999 [[Frank and Brennie Morgan Prize for Outstanding Research in Mathematics by an Undergraduate Student|Morgan Prize]].
:A related problem, whose proof uses similar techniques to Hales' proof of the Kepler conjecture. Conjecture by L. Fejes Tóth in the 1950s.

;The [[Weaire–Phelan structure#The Kelvin conjecture|Kelvin problem]]: What is the most efficient [[foam]] in 3 dimensions? This was conjectured to be solved by the [[Kelvin structure]], and this was widely believed for over 100 years, until disproved in 1993 by the discovery of the [[Weaire–Phelan structure]]. The surprising discovery of the Weaire–Phelan structure and disproof of the Kelvin conjecture is one reason for the caution in accepting Hales' proof of the Kepler conjecture.

;[[Sphere packing]] in higher dimensions: In 2016, [[Maryna Viazovska]] announced proof of the optimal sphere packing in dimension 8, which quickly led to a solution in dimension 24.<ref>{{citation |last1=Klarreich |first1=Erica |author-link1=Erica Klarreich |title=Sphere Packing Solved in Higher Dimensions |url=https://www.quantamagazine.org/sphere-packing-solved-in-higher-dimensions-20160330/ |magazine=Quanta Magazine |date=March 30, 2016}}</ref> However, the optimal sphere packing question in dimensions other than 1, 2, 3, 8, and 24 is still open.

;[[Ulam's packing conjecture]]: It is unknown whether there is a convex solid whose optimal [[packing density]] is lower than that of the sphere.