The symmetry beauty in Mathematics数学中的对称美.docx
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The symmetry beauty in Mathematics数学中的对称美.docx
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ThesymmetrybeautyinMathematics数学中的对称美
ThesymmetrybeautyinMathematics(数学中的对称美)
ThesymmetrybeautyinMathematics(数学中的对称美)
ThesymmetrybeautyinMathematics
Abstract:
symmetryusuallyreferstothegraphicorobjectonacertainpoint,lineorplane,thesize,shapeandarrangementthathasacorrespondingrelation,inmathematics,theconceptofsymmetryisslightlyextendedtosomerelatedconceptshaveoftenregardedassymmetricoropposing,suchsymmetryhasbecomeanimportantcomponentpartofmathematics,symmetryisavastsubjectintwoaspectsofartandnatureareofgreatsignificance,itisthefundamentalmathematics,beautyandsymmetryarecloselylinked.
Keywords:
symmetricalgraphs,mathematics,symmetry,beauty
Symmetryinnaturehasmanythings,suchasmapleleaves,snowflakes,etc.,symmetryitselfisakindofharmony,akindofbeauty.Theapplicationofmathematicsisalsoverybroad,suchas:
everyoneisveryfamiliarwiththeaxialsymmetrygraphics,etc.,infact,accordingtotheprincipleofsymmetry,inprimaryschoolmathematicsinthevariousknowledgeareas,canbefoundintheapplicationofthisrule.Howtoletthestudentsgraspthebasicprincipleofsymmetrytosolvesomepracticalproblems,findtheintrinsicunitybetweenthings,gowithinthissimplemathematicalthinking,principleandcontainprofoundphilosophyessence,whichrequiresustounderstandthehiddendeepinthebackproblem,accordingtosomecasestheauthorfoundinteachingthatexplainshowtofindsymmetryinmathematics.
First,getinspirationfromapalindrome,solvingarithmetic
sequence
Palindromenumberhasmany,suchas0:
2002yearsisapalindromenumber,thenextpalindromenumbertowaituntil2112,integermultiplication,themostinterestingpalindromenumberis:
1x1=1,11x11=121111*111=12321.Accordingtothislawcanbecalculatedby:
111111111*111111111=12345678987654321,thestudentsforthisspecialpalindromeresults,allfeelverysurprised,ithasstronginterest,lamentedthesymmetrynumber.Asakindofsymmetricalbeauty,becomeaneternaltheoremintheuniverse,likeYinthereisYang,blackiswhite,that'smorelikesomeiffy,modernphysicstheoryinferenceasamatterwithantimatter,asweseeinlife,allthingsfeelisthesamematerialintheuniverse,thereweseenotseeantimatterenergyandmatterareequal,sotheuniversewasbalanced,youliketheuniverse,therearealso"antiyou",ifonedayyou"ahandshake,thenyouandyou"suddenlydisappear,like5+(-5)=0,somewhatabsurd,butthisassumptioninanswersomeproblems,butitiscleverandeasytounderstand.
Asintheprimaryschoolofgooddegreeofstudentsonthesumofarithmeticseries,mostlyusingtheformula:
(a+end)*/2itemsofteaching,forstudentstomasterandisdifficulttounderstand.Asa"badwomanweaving"theancientarithmetic:
awomanatherweaving,wovenBudoubieverydaytoreducesomeday,reductionisequaltothenumberofthefirstday,shemadefivefeet,thelastdayofthetextureofaruler,atotalofthirtydaysforatotaloffabric,fabrichowmanyfeetofcloth?
Thedifficultyofthisquestionisthat,apartfromthefirstandlastday,theclothwoveninthemiddleisnotaninteger,anditisnoteasytoweavelessclotheverydaythanthe
previousday.Thesymmetryoftheideaisthisanswer:
supposethatthereisanothergirllikeweavingandthewoman,butsheandthewomanWeaver'ssituationisjusttheopposite:
thegirleverydayoftheBudoubifabrictoincreasethenumberandincreaseisequaltothenumberofthefirstday,sheweavesaruler,thelastdaythefabricisfivefeet,alsomadethirtydays,thetotallengthoftheweavinggirlandwomenareequal,thenumberofwomenclothclotheverydaytoreducethenumberandincreaseeverygirlweavingisequal,sothetwodaytotalofclothissixfeet,thirtydaysatotalof6fabrics*30=180feet,each90feetoffabric.
ThisproblemliesinthesubtletiesofasetofarithmeticsequenceAbstractsumintotheimageapalindromenumbersummationmethodisvivid,wonderfulanddifferentapproachesbutequallysatisfactoryresultsinphysicsthatmatterandantimatter.
Infact,asthesumofanarithmeticseriescanbeusedinthissolution,usingthesymmetricalthinkingtounderstandarithmeticsequencethanthesimpleformulatoemphasizeandvividmulti.
Two.Theapplicationofthesymmetryprinciplefromtheaxialsymmetryfigure
Accordingtothehalfoftheaxialsymmetryfigureandthesymmetricalaxis,theotherhalffigureoftheaxialsymmetryfigurecanbeaccuratelydrawn,whichisacommonproblemaftertheteachingoftheaxialsymmetryfigure.Inmathematics,axialsymmetryalsoprovidessomeinsightsintopeople's
researchintomathematics,forexample,itisoftenusedingametheory.Suchas:
thereare21piecesonthetable,arrangedinarow,youcangetoneatatime,youcanalsotaketwopieces,andevencantakethreepieces.WheretogetpawnOK,notinordertotake,butwithtwoorthreepiecemustbeadjacenttothatnospaceorotherpieces,twopeopletaketurnstoask:
"whogotthelastonewhowins,ifyoutaketoensurethewin?
"
Thisproblemseemscomplicated,accordingtothepermutationandcombinationmethodinordertogetmany11analysisclearlytoolaborious,infactbyusingsymmetryprincipleisverysimple,aslongasthefirstpersontotakeawaythemiddlegrain,eleventhgrainofchess,sotheleftandrightsidesoftheremainingtengrains,sothattheotherpartytogetthepieces,youtakeontherightsideofthepiece,andthenumberandpositionandheissymmetrical,iftheotherpartywiththerighttopawn,youwillgetthepiecesaccordingtohim,anywayaslongaskeeptheleftandrightsides.Therestofthenumberandlocationofthesame,aslongashetooksome,youalsohavetotake,sothelastgrainboundtofallintoyourhandstherefore,totakethewin,ifthepiecesare20grains(evennumber),youwilltakethemiddletwo,letbothsidesoftheremaining9chesspieces,soyouwin.
Therearesimilartopicssuchas:
numberoneyuancointwopeopletaketurnstoputitinalargedisk,thecoincannotoverlapbetweenrequirements,whocanputwholoses,isthefirstplacetowinorwinback?
Apparently,accordingtotheprincipleofsymmetry,thefirstputpeopleaslongasthefirsttooccupythecenter,afterwhichtheotherputyouasheputontheoppositesideofsymmetry,aslongashehasthespace
pendulum,thenknownasthelocalrelativealsomusthaveaspacependulum,untiltheothersidecanputsofar,eachothertolose.Infact,basiccharacteristicsofthinkingmethodofthesetwoquestionsarefromtheaxisofsymmetry,theteacherintheteachingcontentoftheendofaxissymmetrycanpenetratethisknowledgeproperly,studentswillingtolearn,andenhancetheuseofaxialsymmetryknowledgeanddeepunderstanding,findsymmetricalbeauty,feelthecharmofmathematics.
Representationofdualitybysymmetricgraphs.Thesymmetricalbeautyofmathematicsisnaturallyexpressedinthedualityofmathematicalelementsandthedualityofmathematicalpropositions.Butdualityhasnogeometricintuition.Projectivegeometryisanexcellentwaytomakethe"dual"visualsymmetryinmathematicsandtouseitformathematicalproblems.
Example1inEuclideanplanegeometry,theproposition"overtwopoints"canmakeastraightline".Wechangethe"point"into"straightline","straightline"to"point",andthenchangetherelativewordsappropriately,thenthepropositionbecomes"twostraightline"and"onepoint"".Thispropositionisclearlywrong,becausethereisnointersectionpointwhenthetwolinesareparallel.Thisshowsthattherelationbetweenpointandlineisnotsymmetricalinthisproposition.Assumethattwoparallellinesintersectatinfinity,whenthepointandthelineformasymmetricrelationship.ItisherethatDishagestablishedtheprojectivegeometrytheory
preliminarily.Weshowthedualityrelationofexample1intuitively,usingthefollowinggraph:
theuplinkrepresents
twopoints,andastraightlinecanbedetermined;themeaningofthearrowis"todetermine"".Thetwoarrowsontheleftandrightindicatethepositionoftheswitchingpointandthestraightline.Thedownlinkmeans"twolinestoonepoint",andthemeaningofthearrowis"intersection",whichisbasedontheassumptionthatthegraph
(1)becomestheperfectrectangle,andthedualrelationbetweenthepointandthelineistransformedinto
Geometricalsymmetry.Thesymmetrybeautyinmathematicsprovidesauniquemethodformathematicalresearch,thatis,symmetry.Inasimpleway,symmetryisthemethodofthinkingthatusessymmetry.
The"assumption"usedinexample1istheuseofsymmetry.Mathematiciansusethismethodtorevealanddiscoveralotofmathematicalmysteriesandgetveryusefultheoriesandconclusions.Itisalsothesymmetrymethodthatenlightensustotranslatedualityintosymmetry".Inadditiontoprojectivegeometry,suchapplicationsarealsopresentinmodernalgebra.
Example2[1]setsKasadomain,andAisak-space.AiscalledaKalgebra,iftheelementsinAhavemultiplicationandunitelements,andthemultiplicationsatisfiestheunionlaw.Weuse:
A,A,mappingwithAmultiplication,withA,Ksaid:
epsilonmappingunitofA,isacombinationoflawandunitrespectivelycanbeexpressedasthefollowingcommutativediagram:
allthearrowsandreverse,replace,substitute,andexchange,andmap:
thedeltaiscalledthecomultiplication,iscalledmorethanunitexchangegraph(3)calledthecomultiplicationcoassociativelaw.Here,multiplicationand
multiplication,thecombininglawandthere
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