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:
“So universal and widely related is any transcendent moral greatness, and so nearly identical with greatness everywhere and in every age,—as a pyramid contracts the nearer you approach its apex,—that, when I look over my commonplace-book of poetry, I find that the best of it is oftenest applicable, in part or wholly, to the case of Captain Brown.”
—Henry David Thoreau (1817–1862)
“The mother’s and father’s attitudes toward the child correspond to the child’s own needs.... Mother has the function of making him secure in life, father has the function of teaching him, guiding him to cope with those problems with which the particular society the child has been born into confronts him.”
—Erich Fromm (1900–1980)