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1、Chapter3SequencesandtreesHowfromtheIChingtotheInterneteraawealthofcodeshavebeendevelopedforexpressinginformationinmanyforms.Throughouthistory,oracleshavebeenconsultedtoobtainadviceonim-portantdecisions,evenineverydaylife.Themostancientanswerswerebinary,limitedtoyes"ando";asin
2、theIChing,theBookofChanges,oneoftheoldestChinesetexts.Answerswererepresentedbyhorizontallines,asolidlineforyesandabrokenlineforno.Itseemsthattheneedforamorenuancedresponseemergedeveninearlytimes,soblocksoftwolineswereintroducedgivingrisetothe22=4ofsolidandbrokenlinescombined.Lat
3、erathirdlinewasinsertedineachblocktodoublethepossiblecombinations,andthe23=8basicsignsoftheIChing,calledtrigrams,wereborn.Theseeightsignswererepresentativeoftheeightelementaryconstituentsofheavenandearth,andhavesurvivedwiththeirwholeexpressivepower.LookattheminFigure3.1,arranged
4、intheirclassicalordering.Thestory,however,isjustattheoutset.Theeightsignsrepresentchangingstates,imagesperpetuallymutating.Thefocusisnotonthethingsintheirbeing,butratherontheirtransitions.Toobtainanevengreatermultiplicitythetrigramswerecombinedinblocksoftwo,toformthecompleteseto
5、fEARTHMOUNTAINWATERWINDTHUNDERFIRELAKEHEAVENFIGURE3.1:TheIChingtrigrams.19©2012byTaylor&FrancisGroup,LLC20MathematicalandAlgorithmicFoundationsoftheInternetthe26=64hexagramscontainedinthebook.Dependingonthepositioninthehexagram,someofthelinesmaybechanging,therebyincreasingevenfu
6、rtherthenumberofpossibleoracularsentences.Thiswealthofcasesisnotsurprising.TheBookofChangesisamonumentofChinesewisdom, ndinginthesignssucientpowertoexpressitself.Incrementingthenumberoflines,orconsideringadoublepossiblemeaningofeachline,wecanincreaseexponentiallythenumberoforac
7、ularsentencestoexpressapracticallyunlimitedsetofsituationswithareasonablysmallsetoflines.Therulingphenomenonisthatofexponentialgrowth,asdiscussedinChapter2,whichactuallyregulatesthe eldofcommunicationmakingitallpossible.Letusspendalittletimeonthispointthatseldomreceivesdueattent
8、ion.3.1TheexpressivenessofsequencesHumancommuni
