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.

Read more about this topic:  Kepler Conjecture

Famous quotes containing the words related and/or problems:

    Women stand related to beautiful nature around us, and the enamoured youth mixes their form with moon and stars, with woods and waters, and the pomp of summer. They heal us of awkwardness by their words and looks. We observe their intellectual influence on the most serious student. They refine and clear his mind: teach him to put a pleasing method into what is dry and difficult.
    Ralph Waldo Emerson (1803–1882)

    I have said many times, and it is literally true, that there is absolutely nothing that could keep me in business, if my job were simply business to me. The human problems which I deal with every day—concerning employees as well as customers—are the problems that fascinate me, that seem important to me.
    Hortense Odlum (1892–?)