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1、Chapter1AnunconventionalintroductiontotheInternetHowEuleransweredaperplexingquestionaboutajourneyinthehistoriccityofK•onigsberg,andhowthatjourneycanbere-shapedformoreinterestingpurposesusingInternetmathematics.ThesketchmapinFigure1.1isoneofthemostsacredimagesinthehistoryofmat
2、hematics.DrawnbySwissscientistLeonhardPaulEulerforanarticlepublishedin1736,itwasintendedtodepicttheconnectionsamongthedi erentdistrictsofthecityofK•onigsbergviasevenbridgescrossingtheriverPregel.WhyK•onigsberg;whyEuler;and,aboveall,whyarewediscussingthishere?FormanycenturiesK
3、•onigsberg,intheheartofEasternPrussia,wasoneofthemostimportantcitiesintheworld.Itdeservedabetterfate.Conqueredbymanifoldrulers,invadedbyvariousarmies,thecity nallybecamepartoftheSovietUnionunderthenameofKaliningradattheendofWorldWarII,duringwhichbrutalaerialbombardmenthaddest
4、royedalmostallofitshistoricmonuments.EventhecelebrateduniversityAlbertina,foundedinthesixteenthcenturybyAlbertofBrandenburg,fellvictimtothebombs.ItwasherethatImmanuelKantspenthisentirelife;and,accordingtoaslightlydu-bioustradition,itwasalsotheoriginofatantalizingproblem, nall
5、ybroughttotheattentionofEulerin1735.Atthattime,EulerheldapositionintheImperialRussianAcademyofSciencesfoundedbyczarPetertheGreatinSt.Petersburg.Theproblemposedtohimwasverysimple,andscarcelyinterestingat rstglance.K•onigsberghadsevenbridgesconnectingfourdistrictsofthecitysepar
6、atedbythebendsoftheriver.InEuler'sdrawingthedistrictsareindicatedbyA;B;C;Dandthebridgesbya;b;c;d;e;f;g.ThehistoriccenterofthecitywasinA,theKneiphofisland,siteoftheUniversityAlbertinaand(onecenturylater)ofKant'sgrave.Thequestionraisedbythecuriouscitizenswaswhetheritwaspossible
7、totakeawalkstartingfromanarbitrarypointofthecity,crossingeachbridgeexactlyonce,andreturningtothestartingpoint.Ifyouhavenotseenthisproblembefore,considerspendingafewmomentsonit.Itshould1©2012byTaylor&FrancisGroup,LLC2MathematicalandAlgorithmicFoundationsoftheInternetFIGURE1.1:
8、ThecityofK•onigsbergintheEuler'sdrawing.notbediculttoconvinceyourse
