A proof of the Kepler conjecture

By T. Hales. 2005

405 citations recorded by the source corpus.

Subfields: Astronomy and Astrophysics, Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/7cd77cf3a7aa730e212a2a07f341ac050eda47ee

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In Astronomy and Astrophysics

In Geometry and Topology

In Statistical and Nonlinear Physics

Citations in the corpus

The sphere packing problem in dimension 8The sphere packing problem in dimension 8 (2016, neutral): “Hales in 1998 [11] and therefore we know that ∆3 = π √ 18 ≈ 0.”
Edwards Statistical Mechanics for Jammed Granular Matter (2016, background): “…conjecture of 1611 stating that the densest arrangement of spheres in three spatial dimensions (3d) is a face-centered-cubic (FCC) crystal with a packing volume fraction φfcc = π/(3 √ 2) ≈ 0.74048... remained an unsolved mathematical problem for almost four centuries (Hales, 2005; Kepler, 1611).”
Ten methods to bound multiple roots of polynomials (2003, methodology): “In our experience approximations of the roots calculated by some standard numerical algorithm are very accurate, with errors of the order of the numerical sensitivity (see Section 3).”
A FORMAL PROOF OF THE KEPLER CONJECTURE (2015, methodology): “In the remainder of this article, we will call the proof of the Kepler conjecture as it appears in [18] the original proof.”
New upper bounds on sphere packings I (2001, background): “e sharp bounds in 8 and 24 dimensions. The sphere packing problem in Rn is trivial for n = 1, and the answer has long been known for n = 2: the standard hexagonal packing is optimal. For n = 3, Hales [Ha] has proved that the obvious packing, known as the “facecentered cubic” packing (equivalently, t”
History of Interactive Theorem Proving (2014, background): “Hales’s proof was indeed eventually published [Hales, 2005], and no significant error has been found in it [Hales et al., 2010].”
The Crystallization Conjecture: A Review (2015, methodology): “A computer-assisted proof was given by Hales in 1999, then published in 2005 in [93].”
Handbook of Granular Computing (2008, background): “Hales, who, in particular, used interval computations to prove that many other placements lead to a smaller density [52].”
Hyperuniform states of matter (2018, background): “74 [157]) and the least dense strictly jammed packings (conjectured to be the ‘tunneled crystals’ with φmin = 2φmax/3 ≈ 0.”
A conceptual breakthrough in sphere packing (2016, background): “The three-dimensional problem was solved by Hales [9] via a lengthy and complex computer-assisted proof, which was extraordinarily difficult to check but has since been completely verified using formal logic [10].”
The sphere packing problem in dimension 24 (2016, background): “Aside from the trivial case of one dimension, the optimal density was previously known only in two [11], three [7], [8], and eight [12] dimensions, with the latter result being a recent breakthrough due to Viazovska; see [1], [9] for expositions.”