Editing Kepler conjecture (section)

{{Short description|Math theorem about sphere packing}}
The '''Kepler conjecture''', named after the 17th-century mathematician and astronomer [[Johannes Kepler]], is a [[mathematics|mathematical]] [[theorem]] about [[sphere packing]] in three-dimensional [[Euclidean space]]. It states that no arrangement of equally sized [[sphere]]s filling space has a greater [[packing density|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 Callister Hales|Thomas Hales]], following an approach suggested by {{harvtxt|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 (proof assistant)|Isabelle]] and [[HOL Light]] [[proof assistant]]s.  In 2017, the formal proof was accepted by the journal ''[[Forum of Mathematics|Forum of Mathematics, Pi]]''.<ref name="formalproof">{{cite journal
 |last1=Hales|first1=Thomas|author-link1=Thomas Callister Hales
 |last2=Adams|first2=Mark |last3=Bauer|first3=Gertrud |last4=Dang|first4=Tat Dat |last5=Harrison|first5=John
 |last6=Hoang|first6=Le Truong |last7=Kaliszyk|first7=Cezary |last8=Magron|first8=Victor |last9=McLaughlin|first9=Sean
 |last10=Nguyen|first10=Tat Thang |last11=Nguyen|first11=Quang Truong |last12=Nipkow|first12=Tobias |last13=Obua|first13=Steven
 |last14=Pleso|first14=Joseph |last15=Rute|first15=Jason |last16=Solovyev|first16=Alexey |last17=Ta|first17=Thi Hoai An
 |last18=Tran|first18=Nam Trung |last19=Trieu|first19=Thi Diep |last20=Urban|first20=Josef |last21=Vu|first21=Ky
 |last22=Zumkeller|first22=Roland
 |title=A Formal Proof of the Kepler Conjecture
 |journal=Forum of Mathematics, Pi
 |date=29 May 2017 |volume=5 |page=e2  |doi=10.1017/fmp.2017.1 |doi-access=free
 |hdl=2066/176365|hdl-access=free}}</ref>