The Kepler conjecture, named after the 17th-century mathematician and astronomer Johannes Kepler, is a mathematical theorem about sphere packing in three-dimensional Euclidean space. It states that no arrangement of equally sized spheres filling space has a greater average density than that of the cubic close packing (face-centered cubic) and hexagonal close packing arrangements. The density of these arrangements is around 74.05%.
In 1998, Thomas Hales, following an approach suggested by Fejes Tóth (1953), announced that he had a proof of the Kepler conjecture. Hales' proof is a proof by exhaustion involving the checking of many individual cases using complex computer calculations. Referees said that they were "99% certain" of the correctness of Hales' proof, and the Kepler conjecture was accepted as a theorem. In 2014, the Flyspeck project team, headed by Hales, announced the completion of a formal proof of the Kepler conjecture using a combination of the Isabelle and HOL Light proof assistants. In 2017, the formal proof was accepted by the journal Forum of Mathematics, Pi.[1]
^Hales, Thomas; Adams, Mark; Bauer, Gertrud; Dang, Tat Dat; Harrison, John; Hoang, Le Truong; Kaliszyk, Cezary; Magron, Victor; McLaughlin, Sean; Nguyen, Tat Thang; Nguyen, Quang Truong; Nipkow, Tobias; Obua, Steven; Pleso, Joseph; Rute, Jason; Solovyev, Alexey; Ta, Thi Hoai An; Tran, Nam Trung; Trieu, Thi Diep; Urban, Josef; Vu, Ky; Zumkeller, Roland (29 May 2017). "A Formal Proof of the Kepler Conjecture". Forum of Mathematics, Pi. 5: e2. doi:10.1017/fmp.2017.1. hdl:2066/176365.
The Keplerconjecture, named after the 17th-century mathematician and astronomer Johannes Kepler, is a mathematical theorem about sphere packing in three-dimensional...
discrete geometry, he settled the Keplerconjecture on the density of sphere packings and the honeycomb conjecture. In 2014, he announced the completion...
later became known as the Keplerconjecture, a statement about the most efficient arrangement for packing spheres. Kepler wrote the influential mathematical...
lengthy arguments, proved this central conjecture, and remarked that this theorem is equivalent to Keplerconjecture for regular arrangements. In two papers...
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Johannes Keplerconjectured that this is the maximum possible density amongst both regular and irregular arrangements—this became known as the Kepler conjecture...
Callister Hales (almost certainly) proves the Keplerconjecture. 1999 – the full Taniyama–Shimura conjecture is proven. 2000 – the Clay Mathematics Institute...
the most dense arrangement of atoms has an APF of about 0.74 (see Keplerconjecture), obtained by the close-packed structures. For multiple-component...
The proof takes about 500 pages spread over about 20 papers. 2005 Keplerconjecture. Hales's proof of this involves several hundred pages of published...
algorithms. Thomas C. Hales and Samuel P. Ferguson, for proving the Keplerconjecture on the densest possible sphere packings. 2012: Sanjeev Arora, Satish...
down. The most prominent examples are the four color theorem and the Keplerconjecture. Both of these theorems are only known to be true by reducing them...
many spheres has a longer history of investigation, from which the Keplerconjecture is most well-known. Atoms in crystal structures can be simplistically...
packing constant of K. The Keplerconjecture is concerned with the packing constant of 3-balls. Ulam's packing conjecture states that 3-balls have the...
densest possible packing of equal spheres in ordinary space (see Keplerconjecture). It consists of copies of a single cell, the rhombic dodecahedron...
Sean McLaughlin proved the conjecture in 1998, following the same strategy that led Hales to his proof of the Keplerconjecture. The proofs rely on extensive...
fewer dimensions, and the proof of the three-dimensional version (the Keplerconjecture) involved long computer calculations. In contrast, Viazovska's proof...
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