Kepler conjecture
In mathematics,Kepler conjecture isconjecture about sphere packingthree dimensional Euclidean space. It says that no arrangementequal spheres filling space hasgreater average density than that ofcubic close packing (face centred cubic)hexagonal close packing arrangements. The densitythese arrangements islittle over 74%.In 1998 Thomas Hales, presently Andrew Mellon Professor atUniversityPittsburgh, announced that he hadproof ofKepler conjecture. Hales' proof isproof by exhaustion involving checkingmany individual cases using complex computer calculations. Referees have said that they"99% certain" ofcorrectnessHales' proof. SoKepler conjecturenow very closebecomingtheorem.
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2 Origins 3 Nineteenth century 4 Twentieth century 5 Hales' proof 6 A formal proof 7 References 8 External links |
Background
Imagine fillinglarge containersmall equal-sized spheres. The density ofarrangement isproportion ofvolume ofcontainer thattaken up byspheres. In ordermaximisenumberspheres incontainer, you needfind an arrangement withhighest possible density, so thatspherespacked together as closely as possible.Experiment shows that droppingspheresrandomly will achievedensityaround 65%. However,higher density can be achieved by carefully arrangingspheres as follows. Start withlayerspheres inhexagonal lattice, then putnext layerspheres inlowest points you can find abovefirst layer,so on - thisjustway you see oranges stacked inshop. This natural methodstackingspheres creates onetwo similar patterns called cubic close packinghexagonal close packing. Eachthese two arrangements has an average density of
Origins
The conjecturenamed after Johannes Kepler, who statedconjecture1661Strena sue de nive sexangula. Kepler had startedstudy arrangementsspheres asresulthis correspondence withEnglish mathematicianastronomer Thomas Harriot1606. Harriot wasfriendassistantSir Walter Raleigh, who had set Harriotproblemdetermining how beststack cannon balls ondeckshis ships. Harriot publishedstudyvarious stacking patterns1591,went ondevelop an early versionatomic theory.Nineteenth century
Kepler did not haveproof ofconjecture, andnext step was taken by German mathematician Carl Friedrich Gauss, who publishedpartial solution1831. Gauss proved thatKepler conjecturetrue ifspheres havebe arranged inregular lattice.This meant that any packing arrangement that disprovedKepler conjecture would havebe an irregular one. But eliminating all possible irregular arrangementsvery difficult,thiswhat madeKepler conjecture so hardprove. In fact, thereirregular arrangements thatdenser thancubic close packing arrangement oversmall enough volume, but any attemptextend these arrangementsfilllarger volume always reduces their density.
After Gauss, no further progress was made towards provingKepler conjecture innineteenth century. In 1900 David Hilbert includedin his listtwenty three unsolved problemsmathematics -forms partHilbert's eighteenth problem.
Twentieth century
The next step towardssolution was taken by Hungarian mathematician László Fejes Tóth. In 1953 Fejes Tóth showed thatproblemdeterminingmaximum densityall arrangements (regularirregular) could be reduced tofinite (but very large) numbercalculations. This meant thatproof by exhaustion was,principle, possible. As Fejes Tóth realised,fast enough computer could turn this theoretical result intopractical approach toproblem.Meanwhile, attempts were madefind an upper bound formaximum densityany possible arrangementspheres. English mathematician Claude Ambrose Rogers established an upper bound valueabout 78%1958,subsequent efforts by other mathematicians reduced this value slightly, but this was stilllong way abovecubic close packing density74%.
There were also some failed proofs. American architectgeometer Buckminster Fuller claimedhaveproof1975, but this was soon foundbe incorrect. In 1993 Wu-Yi-Hsang atUniversityCalifornia, Berkeley publishedpaperwhich he claimedproveKepler conjecture using geometric methods. This was also foundbe incorrect.
Hales' proof
Followingapproach suggested by Fejes Tóth, Thomas Hales, then atUniversityMichigan, determined thatmaximum densityall arrangements could be found by minimisingfunction150 variables. In 1992, assisted by his graduate student Samuel Ferguson, he embarked onresearch programmesystematically apply linear programming methodsfindlower bound onvaluethis functioneach one ofsetover 5,000 different configurationsspheres. If an lower bound could be foundevery onethese configurations that was greater thanvalue forcubic close packing arrangement, thenKepler conjecture would be proved. To find lower boundsall cases involved solving around 100,000 linear programming problems.When presentingprogresshis project1996, Hales said thatend wassight, butmight take "a year or two"complete. In August 1998 Hales announced thatproof was complete. At that stageconsisted250 pagesnotes3 gigabytescomputer programs, dataresults.
Despiteunusual nature ofproof,editors ofAnnalsMathematics agreedpublish it, providedwas accepted bypaneltwelve referees. In 2003, after four yearswork,head ofreferee's panel Gábor Fejes Tóth (sonLászló Fejes Tóth) reported thatpanel were "99% certain" ofcorrectness ofproof, butcould not certifycorrectnessall ofcomputer calculations.
In February 2003 Hales published100-page paper describingnon-computer parthis proofdetail.
A formal proof
In January 2003 Hales announcedstart ofcollaborative projectproducecomplete formal proof ofKepler conjecture. The aim isremove any remaining uncertainty aboutvalidity ofproof by creatingformal proof that can be verified by automated theorem proving software such as HOL. This projectcalled Project FlysPecK -F, PK standingFormal ProofKepler. Hales estimates that producingcomplete formal proof will take around 20 yearswork !References
- L.G. Szpiro (2003) Kepler's Conjecture Wiley, John & Sons Inc. ISBN 0471086010
- Thomas C. Hales (2003) A Proof ofKepler Conjecture
