Kepler Conjecture - Related Problems

Related Problems

Thue's theorem
The regular hexagonal packing is the densest sphere packing in the plane. (1890)
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).
The hexagonal honeycomb conjecture
The most efficient partition of the plane into equal areas is the regular hexagonal tiling. Hales' proof (1999).
Related to Thue's theorem.
The dodecahedron conjecture
The volume of the 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, for which he received the 1999 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 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 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
The optimal sphere packing question in dimensions bigger than 3 is still open.

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