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1、FibonacciNumbersandtheGoldenRatioSayWhat?What is the Golden Ratio?Well,beforeweanswerthatquestionletsexamineaninterestingsequence(orlist)ofnumbers.Actuallytheseriesstartswith0,1buttomakeiteasierwelljuststartwith:1,1TogetthenextnumberweaddtheprevioustwoTogetthenextnumberweaddtheprevioustwonumberstoge
2、ther.Sonowoursequencebecomesnumberstogether.Sonowoursequencebecomes1,1,2.Thenextnumberwillbe3.Whatdoyou1,1,2.Thenextnumberwillbe3.Whatdoyouthinkthenextnumberinthesequencewillbe?thinkthenextnumberinthesequencewillbe?Remember,weaddtheprevioustwonumberstoRemember,weaddtheprevioustwonumberstogetthenext.
3、Sothenextnumbershouldbe2+3,orgetthenext.Sothenextnumbershouldbe2+3,or5.Hereiswhatoursequenceshouldlooklikeifwe5.Hereiswhatoursequenceshouldlooklikeifwecontinueoninthisfashionforawhile:continueoninthisfashionforawhile:1,1,2,3,5,8,13,21,34,55,89,144,233,377,6101,1,2,3,5,8,13,21,34,55,89,144,233,377,61
4、0Now,Iknowwhatyoumightbethinking:WhatdoesthishavetodowiththeGoldenRatio?Thissequenceofnumberswasfirst“discovered”byamannamedLeonardoFibonacci,andhenceisknownasFibonaccissequence.Math GEEKReallyFamousReallySmartLeonardoFibonacciTherelationshipofthissequencetotheGoldenRatioliesnotintheactualnumbersoft
5、hesequence,butintheratiooftheconsecutivenumbers.Letslookatsomeoftheratiosofthesenumbers:1,1,2,3,5,8,13,21,34,55,89,144,233,377,6101,1,2,3,5,8,13,21,34,55,89,144,233,377,6102/1=2.03/2=1.55/3=1.678/5=1.613/8=1.62521/13=1.61534/21=1.61955/34=1.61889/55=1.618Since a Ratio is basically a fraction (or a d
6、ivision problem) we will find the ratios of these numbers by dividing the larger number by the smaller number that fall consecutively in the series.So, what is the ratio of the 2nd and 3rd numbers?Well, 2 is the 3rd number divided by the 2nd number which is 12 divided by 1 = 2And the ratios continue
7、 like this.Aha!Noticethataswecontinuedownthesequence,theratiosseemtobeconvergingupononenumber(frombothsidesofthenumber)!2/1=2.0(bigger)(bigger)3/2=1.5(smaller)(smaller)5/3=1.67(bigger)(bigger)8/5=1.6(smaller)(smaller)13/8=1.625(bigger)(bigger)21/13=1.615(smaller)(smaller)34/21=1.619(bigger)(bigger)5
8、5/34=1.618(smaller)(smaller)89/55=1.618FibonacciNumbercalculator5/3=1.675/3=1.678/5=1.68/5=1.613/8=1.62513/8=1.62521/13=1.61521/13=1.61534/21=1.61934/21=1.61955/34=1.61855/34=1.61889/55=1.61889/55=1.618 NoticethatIhaveroundedmyratiostothethirddecimalNoticethatIhaveroundedmyratiostothethirddecimalpla
9、ce.Ifweexamine55/34and89/55moreclosely,wewillplace.Ifweexamine55/34and89/55moreclosely,wewillseethattheirdecimalvaluesareactuallynotthesame.Butseethattheirdecimalvaluesareactuallynotthesame.Butwhatdoyouthinkwillhappenifwecontinuetolookatthewhatdoyouthinkwillhappenifwecontinuetolookattheratiosasthenu
10、mbersinthesequencegetlargerandratiosasthenumbersinthesequencegetlargerandlarger?Thatsright:theratiowilleventuallybecomethelarger?Thatsright:theratiowilleventuallybecomethesamenumber,andthatnumberistheGoldenRatio!samenumber,andthatnumberistheGoldenRatio!11231.500000000000000051.666666666666670081.600
11、0000000000000131.6250000000000000211.6153846153846200341.6190476190476200551.6176470588235300891.61818181818182001441.61797752808989002331.61805555555556003771.61802575107296006101.61803713527851009871.61803278688525001,5971.61803444782168002,5841.61803381340013004,1811.61803405572755006,7651.618033
12、963166710010,9461.618033998521800017,7111.618033985017360028,6571.618033990175600046,3681.618033988205320075,0251.6180339889579000TheGoldenRatioiswhatwecallanirrationalnumber:ithasaninfinitenumberofdecimalplacesanditneverrepeatsitself!Generally,weroundtheGoldenRatioto1.618.Here is the decimal value
13、of Phi to 2000 places grouped in blocks of 5 Here is the decimal value of Phi to 2000 places grouped in blocks of 5 decimal digits. The value of phi is the same but begins with 06. decimal digits. The value of phi is the same but begins with 06. instead of 16. . instead of 16. . Read this as ordinar
14、y text, in lines across, so Phi is 161803398874.)Read this as ordinary text, in lines across, so Phi is 161803398874.) DpsDps: :1618033988749894848204586834365638117720309179805761618033988749894848204586834365638117720309179805765050286213544862270526046281828621354486227052604628189024497072072041
15、893911374902449707207204189391137410010084754088075386891752126633862223536931793180060766726358475408807538689175212663386222353693179318006076672635443338908659593958290563832266131992829026788443338908659593958290563832266131992829026788200200067520876689250171169620703222104320675208766892501711
16、696207032221043216269548626296313614438149758701220340805887954454749246185695364162695486262963136144381497587012203408058879544547492461856953643003008644492410864449241044320771344947049565846788509874339442212544877066478091588460749988712400765217443207713449470495658467885098743394422125448770
17、664780915884607499887124007652170575179788057517978840040034166256249407589069704000281210427621771117778053153171410117046665993416625624940758906970400028121042762177111777805315317141011704666599146697987317613560067087480710146697987317613560067087480710500500131795236894275219484353056783002287
18、856997829131795236894275219484353056783002287856997829778347845878228911097625003026961561700250464338243776486102838312683303724292677783478458782289110976250030269615617002504643382437764861028383126833037242926752631165339247316711121158818638513316203840052221657912866752946549068113171599526311
19、653392473167111211588186385133162038400522216579128667529465490681131715993432359734949850904094762132229810172610705961164562990981629055520852479035240634323597349498509040947621322298101726107059611645629909816290555208524790352406020172799747175342777592778625619432082750513121815628551222480939
20、471234145170220201727997471753427775927786256194320827505131218156285512224809394712341451702237358057727861600868838295230459264787801788992199027077690389532196819861514378373580577278616008688382952304592647878017889921990270776903895321968198615143780314997411069260886742962267575605231727775203
21、536139362031499741106926088674296226757560523172777520353613936210001000107673893764556060601076738937645560606059216589466759551900400555908950229530942312482355592165894667595519004005559089502295309423124823552122121221241544400647034056573479724154440064703405657347976639723949499465845788730396
22、230903750339938562102423690251386804145779956981224466397239494994658457887303962309037503399385621024236902513868041457799569812244574717803417312645322041639723213404444948730231541767689375210306873788034417005747178034173126453220416397232134044449487302315417676893752103068737880344170093954409
23、627955898678723209512426893557309704509595684401755519881921802064052905939544096279558986787232095124268935573097045095956844017555198819218020640529055189349475926007348522821010881946445442223188913192946896220023014437702699230051893494759260073485228210108819464454422231889131929468962200230144
24、377026992300780308526118075451928877050210968424936271359251876077788466583615023891349333317803085261180754519288770502109684249362713592518760777884665836150238913493333122310533923213624319263728910670503399282265263556209029798642472759772565508615223105339232136243192637289106705033992822652635
25、562090297986424727597725655086154875435748264718141451270006023890162077732244994353088999095016803281121943204848754357482647181414512700060238901620777322449943530889990950168032811219432048196438767586331479857191139781539780747615077221175082694586393204565209896985551964387675863314798571911397
26、815397807476150772211750826945863932045652098969855567814106968372884058746103378105444390943683583581381131168993855576975484149144678141069683728840587461033781054443909436835835813811311689938555769754841491445341509129540700501947754861630754226417293946803673198058618339183285991303960753415091
27、2954070050194775486163075422641729394680367319805861833918328599130396072014455950449779212076124785645916160837059498786006970189409886400764436170933420144559504497792120761247856459161608370594987860069701894098864007644361709334172709191433650137151727091914336501371520002000 Weworkwithanotherim
28、portantirrationalnumberinGeometry:pi,whichisapproximately3.14.SincewedontwanttomaketheGoldenRatiofeelleftout,wewillgiveititsownGreekletter:phi. Phiwhich is equal to:OnemoreinterestingthingaboutPhiisitsOnemoreinterestingthingaboutPhiisitsreciprocal.Ifyoutaketheratioofanynumberinreciprocal.Ifyoutaketh
29、eratioofanynumberintheFibonaccisequencetothenextnumber(thisistheFibonaccisequencetothenextnumber(thisisthereverseofwhatwedidbefore),theratiowillthereverseofwhatwedidbefore),theratiowillapproachtheapproximation0.618.Thisistheapproachtheapproximation0.618.ThisisthereciprocalofPhi:1/1.618=0.618.Itishig
30、hlyreciprocalofPhi:1/1.618=0.618.Itishighlyunusualforthedecimalintegersofanumberandunusualforthedecimalintegersofanumberanditsreciprocaltobeexactlythesame.Infact,Iitsreciprocaltobeexactlythesame.Infact,Icannotnameanothernumberthathasthiscannotnameanothernumberthathasthisproperty!Thisonlyaddstothemys
31、tiqueoftheproperty!ThisonlyaddstothemystiqueoftheGoldenRatioandleadsustoask:WhatmakesitGoldenRatioandleadsustoask:Whatmakesitsospecial?sospecial?TheGoldenRatioisnotjustsomenumberthatmathTheGoldenRatioisnotjustsomenumberthatmathteachersthinkiscool.Theinterestingthingisthatitkeepsteachersthinkiscool.T
32、heinterestingthingisthatitkeepspoppingupinstrangeplaces-placesthatwemaynotpoppingupinstrangeplaces-placesthatwemaynotordinarilyhavethoughttolookforit.Itisimportanttonoteordinarilyhavethoughttolookforit.ItisimportanttonotethatFibonaccididnotinventtheGoldenRatio;hejustthatFibonaccididnotinventtheGolde
33、nRatio;hejustdiscoveredoneinstanceofwhereitappearednaturally.Indiscoveredoneinstanceofwhereitappearednaturally.InfactcivilizationsasfarbackandasfarapartastheAncientfactcivilizationsasfarbackandasfarapartastheAncientEgyptians,theMayans,aswellastheGreeksdiscoveredEgyptians,theMayans,aswellastheGreeksd
34、iscoveredtheGoldenRatioandincorporateditintotheirownart,theGoldenRatioandincorporateditintotheirownart,architecture,anddesigns.TheydiscoveredthattheGoldenarchitecture,anddesigns.TheydiscoveredthattheGoldenRatioseemstobeNaturesperfectnumber.ForsomeRatioseemstobeNaturesperfectnumber.Forsomereason,itju
35、stseemstoappealtoournaturalinstincts.Thereason,itjustseemstoappealtoournaturalinstincts.Themostbasicexampleisinrectangularobjects.mostbasicexampleisinrectangularobjects.Lookatthefollowingrectangles:Lookatthefollowingrectangles: Nowaskyourself,whichofthemseemstobethemostNowaskyourself,whichofthemseem
36、stobethemostnaturallyattractiverectangle?Ifyousaidthefirstone,thennaturallyattractiverectangle?Ifyousaidthefirstone,thenyouareprobablythetypeofpersonwholikeseverythingtoyouareprobablythetypeofpersonwholikeseverythingtobesymmetrical.Mostpeopletendtothinkthatthethirdbesymmetrical.Mostpeopletendtothink
37、thatthethirdrectangleisthemostappealing.rectangleisthemostappealing.IfyouweretomeasureeachrectangleslengthIfyouweretomeasureeachrectangleslengthandwidth,andcomparetheratiooflengthtowidthandwidth,andcomparetheratiooflengthtowidthforeachrectangleyouwouldseethefollowing:foreachrectangleyouwouldseethefo
38、llowing:Rectangleone:Ratio1:1Rectangleone:Ratio1:1Rectangletwo:Ratio2:1Rectangletwo:Ratio2:1RectangleThree:Ratio1.618:1RectangleThree:Ratio1.618:1HaveyoufiguredoutwhythethirdrectangleistheHaveyoufiguredoutwhythethirdrectangleisthemostappealing?Thatsright-becausetheratioofmostappealing?Thatsright-bec
39、ausetheratioofitslengthtoitswidthistheGoldenRatio!ForitslengthtoitswidthistheGoldenRatio!Forcenturies,designersofartandarchitecturehavecenturies,designersofartandarchitecturehaverecognizedthesignificanceoftheGoldenRatioinrecognizedthesignificanceoftheGoldenRatiointheirwork.theirwork.Letsseeifwecandi
40、scoverwheretheGoldenRatioappearsineverydayobjects.Useyourmeasuringtooltocomparethelengthandthewidthofrectangularobjectsintheclassroomorinyourhouse(dependingonwhereyouarerightnow).Trytochooseobjectsthataremeanttobevisuallyappealing.WereyousurprisedtofindtheGoldenRatioinsomanyWereyousurprisedtofindthe
41、GoldenRatioinsomanyplaces?Itshardtobelievethatwehavetakenitforgrantedplaces?Itshardtobelievethatwehavetakenitforgrantedforsolong,isntit?forsolong,isntit?ObjectObjectLengthLengthWidthWidthRatioRatioindex cardindex cardphotographphotographpicture framepicture frametextbooktextbookdoor framedoor framec
42、omputer computer screenscreenTV screenTV screenHowaboutinmusic?LetstakealookatthepianokeyboarddoyouseeAnythingfamiliar?Countthenumberofkeys(notes)ineachofthebracketsCountthenumberofkeys(notes)ineachofthebracketsYouwillseethenumbers2,3,5,8,13.coincidence?Youwillseethenumbers2,3,5,8,13.coincidence?Doe
43、sitlookliketheFibonaccisequenceitshouldDoesitlookliketheFibonaccisequenceitshouldbecauseitis!becauseitis!How about Architecture?FindtheGoldenRatiointheParthenon.FindtheGoldenRatiointheParthenon.1.LetsstartbydrawingarectanglearoundtheParthenon,1.LetsstartbydrawingarectanglearoundtheParthenon,fromthel
44、eftmostpillartotherightandfromthebaseofthefromtheleftmostpillartotherightandfromthebaseofthepillarstothehighestpoint.pillarstothehighestpoint.2.Measurethelengthandthewidthofthisrectangle.Now2.Measurethelengthandthewidthofthisrectangle.Nowfindtheratioofthelengthtothewidth.Isthenumberfairlyfindtherati
45、oofthelengthtothewidth.IsthenumberfairlyclosetotheGoldenRatio?closetotheGoldenRatio?3.Nowlookabovethepillars.Youshouldnoticesome3.Nowlookabovethepillars.YoushouldnoticesomerectanglesonthefaceoftheParthenon.FindtheratioofrectanglesonthefaceoftheParthenon.Findtheratioofthelengthtothewidthofoneoftheser
46、ectangles.Noticethelengthtothewidthofoneoftheserectangles.Noticeanything?anything?TherearemanyotherplaceswheretheGoldenRatioTherearemanyotherplaceswheretheGoldenRatioappearsintheParthenon,allofwhichwecannotseeappearsintheParthenon,allofwhichwecannotseebecauseweonlyhaveafrontalviewofthestructure.Theb
47、ecauseweonlyhaveafrontalviewofthestructure.Thebuildingisbuiltonarectangularplotoflandwhichhappensbuildingisbuiltonarectangularplotoflandwhichhappenstobe.youguessedit-aGoldenRectangle!tobe.youguessedit-aGoldenRectangle!Once its ruined triangular pediment is restored, .the ancient temple fits almost p
48、recisely into a golden rectangle.Further classic subdivisions of the rectangle align perfectly with major architectural features of the structure. Furtherclassicsubdivisionsoftherectanglealignperfectlywithmajorarchitecturalfeaturesofthestructure.The Golden Ratio in ArtNowletsgobackandtrytodiscoverth
49、eGoldenRatioinart.WewillconcentrateontheworksofLeonardodaVinci,ashewasnotonlyagreatartistbutalsoageniuswhenitcametomathematicsandinvention.The Annunciation - The Annunciation - UsingtheleftsideofthepaintingasaUsingtheleftsideofthepaintingasaside,createasquareontheleftofthepaintingbyinsertingside,cre
50、ateasquareontheleftofthepaintingbyinsertingaverticalline.Noticethatyouhavecreatedasquareandaaverticalline.Noticethatyouhavecreatedasquareandarectangle.TherectangleturnsouttobeaGoldenrectangle.TherectangleturnsouttobeaGoldenRectangle,ofcourse.Also,drawinahorizontallinethatisRectangle,ofcourse.Also,dr
51、awinahorizontallinethatis61.8%ofthewaydownthepainting(.618-theinverseof61.8%ofthewaydownthepainting(.618-theinverseoftheGoldenRatio).Drawanotherlinethatis61.8%ofthetheGoldenRatio).Drawanotherlinethatis61.8%ofthewayupthepainting.Tryagainwithverticallinesthatarewayupthepainting.Tryagainwithverticallin
52、esthatare61.8%ofthewayacrossbothfromlefttorightandfrom61.8%ofthewayacrossbothfromlefttorightandfromrighttoleft.Youshouldnowhavefourlinesdrawnacrossrighttoleft.Youshouldnowhavefourlinesdrawnacrossthepainting.Noticethattheselinesintersectimportantthepainting.Noticethattheselinesintersectimportantparts
53、ofthepainting,suchastheangel,thewoman,etc.partsofthepainting,suchastheangel,thewoman,etc.Coincidence?Ithinknot!Coincidence?Ithinknot!The Mona Lisa - Measurethelengthandthewidthofthepaintingitself.Theratiois,ofcourse,Golden.DrawarectanglearoundMonasface(fromthetopoftheforeheadtothebaseofthechin,andfr
54、omleftcheektorightcheek)andnoticethatthis,too,isaGoldenrectangle.LeonardoLeonardodadaVincistalentasanartistmaywellVincistalentasanartistmaywellhavebeenoutweighedbyhistalentsasahavebeenoutweighedbyhistalentsasamathematician.Heincorporatedgeometryintomathematician.Heincorporatedgeometryintomanyofhispa
55、intings,withtheGoldenRatiobeingmanyofhispaintings,withtheGoldenRatiobeingjustoneofhismanymathematicaltools.Whydojustoneofhismanymathematicaltools.Whydoyouthinkheuseditsomuch?Expertsagreethatyouthinkheuseditsomuch?ExpertsagreethatheprobablythoughtthatGoldenmeasurementsheprobablythoughtthatGoldenmeasu
56、rementsmadehispaintingsmoreattractive.Maybehewasmadehispaintingsmoreattractive.Maybehewasjustalittletooobsessedwithperfection.However,justalittletooobsessedwithperfection.However,hewasnottheonlyonetouseGoldenpropertieshewasnottheonlyonetouseGoldenpropertiesinhiswork.inhiswork.Constructing A Golden R
57、ectangle Constructing A Golden Rectangle IsntitstrangethattheGoldenRatiocameupinsuchIsntitstrangethattheGoldenRatiocameupinsuchunexpectedplaces?Wellletsseeifwecanfindoutwhy.unexpectedplaces?Wellletsseeifwecanfindoutwhy.TheGreekswerethefirsttocallphitheGoldenRatio.TheyTheGreekswerethefirsttocallphith
58、eGoldenRatio.Theyassociatedthenumberwithperfection.Itseemstobepartassociatedthenumberwithperfection.ItseemstobepartofhumannatureorinstinctforustofindthingsthatcontainofhumannatureorinstinctforustofindthingsthatcontaintheGoldenRationaturallyattractive-suchastheperfecttheGoldenRationaturallyattractive
59、-suchastheperfectrectangle.Realizingthis,designershavetriedtorectangle.Realizingthis,designershavetriedtoincorporatetheGoldenRatiointotheirdesignssoastoincorporatetheGoldenRatiointotheirdesignssoastomakethemmorepleasingtotheeye.Doors,notebookmakethemmorepleasingtotheeye.Doors,notebookpaper,textbooks
60、,etc.allseemmoreattractiveiftheirsidespaper,textbooks,etc.allseemmoreattractiveiftheirsideshavearatioclosetophi.Now,letsseeifwecanconstructhavearatioclosetophi.Now,letsseeifwecanconstructourownperfectrectangle.ourownperfectrectangle.Method One1. Well start by making a square, any square (just rememb
61、er that all sides 1. Well start by making a square, any square (just remember that all sides have to have the same length, and all angles have to measure 90 degrees!):have to have the same length, and all angles have to measure 90 degrees!):2.Now, lets divide the square in half (bisect it). Be sure
62、to use your protractor 2.Now, lets divide the square in half (bisect it). Be sure to use your protractor to divide the base and to form another 90 degree angle:to divide the base and to form another 90 degree angle:Measure the length of the diagonal and make a note Measure the length of the diagonal
63、 and make a note of it. of it. Now, draw in one of the diagonals of one of the rectangles Now extend the base of the square from the midpoint Now extend the base of the square from the midpoint of the base by a distance equal to the length of the of the base by a distance equal to the length of the
64、diagonaldiagonalConstruct a new line perpendicular to the base at the end of Construct a new line perpendicular to the base at the end of our new line, and then connect to form a rectangle:our new line, and then connect to form a rectangle:Measure the length and the width of your rectangle.Measure t
65、he length and the width of your rectangle. Now, find the ratio of the length to the width.Are you surprised by the result? The rectangle you have made is called a Golden Rectangle because it is perfectly proportional.Constructing a Golden Rectangle - Method TwoConstructing a Golden Rectangle - Metho
66、d TwoNow,letstryadifferentmethodthatwillrelatetheNow,letstryadifferentmethodthatwillrelatetherectangletotheFibonacciserieswelookedat.WellstartrectangletotheFibonacciserieswelookedat.Wellstartwithasquare.Thesizedoesnotmatter,aslongasallwithasquare.Thesizedoesnotmatter,aslongasallsidesarecongruent.Wel
67、luseasmallsquaretoconservesidesarecongruent.Welluseasmallsquaretoconservespace,becausewearegoingtobuildourgoldenrectanglespace,becausewearegoingtobuildourgoldenrectanglearoundthissquare.Pleasenotethatthegoldenareaisaroundthissquare.Pleasenotethatthegoldenareaiswhatyourrectanglewilleventuallylooklike
68、.whatyourrectanglewilleventuallylooklike.Now,letsbuildanother,congruentsquarerightnexttotheNow,letsbuildanother,congruentsquarerightnexttothefirstone:firstone:Nowwehavearectanglewithawidth1andlength2units.Nowwehavearectanglewithawidth1andlength2units.Letsbuildasquareontopofthisrectangle,sothatthenew
69、Letsbuildasquareontopofthisrectangle,sothatthenewsquarewillhaveasideof2units:squarewillhaveasideof2units:Noticethatwehaveanewrectanglewithwidth2andNoticethatwehaveanewrectanglewithwidth2andlength3.length3.Letscontinuetheprocess,buildinganothersquareontheLetscontinuetheprocess,buildinganothersquareon
70、therightofourrectangle.Thissquarewillhaveasideof3:rightofourrectangle.Thissquarewillhaveasideof3:Nowwehavearectangleofwidth3andlength5.Nowwehavearectangleofwidth3andlength5.Again,letsbuilduponthisrectangleandconstructasquareunderneath,withasideof5:Thenewrectanglehasawidthof5andalengthof8.Letscontinu
71、etotheleftwithasquarewithside8:Haveyounoticedthepatternyet?Thenewrectanglehasawidthof8andalengthof13.Letscontinuewithonefinalsquareontop,withasideof13:Ourfinalrectanglehasawidthof13andalengthof21.Ourfinalrectanglehasawidthof13andalengthof21.NoticethatwehaveconstructedourgoldenrectangleusingNoticetha
72、twehaveconstructedourgoldenrectangleusingsquarethathadsuccessivesidelengthsfromtheFibonaccisquarethathadsuccessivesidelengthsfromtheFibonaccisequence(1,1,2,3,5,8,13,.)!Nowonderourrectanglesequence(1,1,2,3,5,8,13,.)!Nowonderourrectangleisgolden!Eachsuccessiverectanglethatweconstructedisgolden!Eachsuc
73、cessiverectanglethatweconstructedhadawidthandlengththatwereconsecutivetermsinthehadawidthandlengththatwereconsecutivetermsintheFibonaccisequence.SoifwedividethelengthbytheFibonaccisequence.Soifwedividethelengthbythewidth,wewillarriveattheGoldenRatio!Ofcourse,ourwidth,wewillarriveattheGoldenRatio!Ofc
74、ourse,ourrectangleisnotperfectlygolden.Wecouldkeeptherectangleisnotperfectlygolden.Wecouldkeeptheprocessgoinguntilthesidesapproximatedtheratiobetter,processgoinguntilthesidesapproximatedtheratiobetter,butforourpurposesalengthof21andawidthof13arebutforourpurposesalengthof21andawidthof13aresufficient.
75、sufficient.3421DotheMath!34dividedby21=1.61904761904Rememberthatthefartherintothesequencewegotheclosertheratiogetstobeingperfect!Thisrectangleshouldseemverywellproportionedtoyou,i.e.itshouldbepleasingtotheeye.Ifitisnt,maybeyouneedyoureyeschecked!Constructing a Golden SpiralNoticehowwebuiltourrectang
76、leinacounterclockwisedirection.ThisleadsusintoanotherinterestingcharacteristicoftheGoldenRatio.Letslookattherectanglewithallofourconstructionlinesdrawnin:Wearegoingtoconcentrateonthesquaresthatwedrew,Wearegoingtoconcentrateonthesquaresthatwedrew,startingwiththetwosmallestones.Letsstartwiththeonestar
77、tingwiththetwosmallestones.Letsstartwiththeoneontheright.Connecttheupperrightcornertothelowerleftontheright.ConnecttheupperrightcornertothelowerleftcornerwithanarcthatisonefourthofacirclecornerwithanarcthatisonefourthofacircleWearegoingtoconcentrateonthesquaresthatWearegoingtoconcentrateonthesquares
78、thatwedrew,startingwiththetwosmallestones.Letswedrew,startingwiththetwosmallestones.Letsstartwiththeoneontheright.Connecttheupperstartwiththeoneontheright.Connecttheupperrightcornertothelowerleftcornerwithanarcthatrightcornertothelowerleftcornerwithanarcthatisonefourthofacircle:isonefourthofacircle:
79、ThencontinueyourlineintothesecondsquareonThencontinueyourlineintothesecondsquareontheleft,againwithanarcthatisonefourthofatheleft,againwithanarcthatisonefourthofacircle:circle:WewillcontinuethisprocessuntileachsquareWewillcontinuethisprocessuntileachsquarehasanarcinsideofit,withallofthemconnectedhas
80、anarcinsideofit,withallofthemconnectedasacontinuousline.Thelineshouldlooklikeaasacontinuousline.Thelineshouldlooklikeaspiralwhenwearedone.Hereisanexampleofspiralwhenwearedone.Hereisanexampleofwhatyourspiralshouldlooklike:whatyourspiralshouldlooklike:GoldenSpiralmanipulativeNowwhatwasthepointofthat?T
81、hepointisthatthisgoldenspiraloccursfrequentlyinnature.Ifyoulookcloselyenough,youmightfindagoldenspiralintheheadofadaisy,inapinecone,insunflowers,orinanautilusshellthatyoumightfindonabeachoreveninyourear!Herearesomeexamples:So,whydoshapesthatexhibittheGoldenRatioseemmoreappealingtothehumaneye?Noonere
82、allyknowsforsure.ButwedohaveevidencethattheGoldenRatioseemstobeNaturesperfectnumber.SomebodywithalotoftimeontheirhandsdiscoveredthatSomebodywithalotoftimeontheirhandsdiscoveredthattheindividualfloretsofthedaisy(andofasunflowerastheindividualfloretsofthedaisy(andofasunfloweraswell)growintwospiralsext
83、endingoutfromthecenter.Thewell)growintwospiralsextendingoutfromthecenter.Thefirstspiralhas21arms,whiletheotherhas34.Dothesefirstspiralhas21arms,whiletheotherhas34.Dothesenumberssoundfamiliar?numberssoundfamiliar?Theyshould-theyareFibonaccinumbers!Andtheirratio,ofcourse,istheGoldenRatio.Wecansaythesa
84、methingaboutthespiralsofapinecone,wherespiralsfromthecenterhave5and8arms,respectively(orof8and13,dependingonthesize)-again,twoFibonaccinumbers:Apineapplehasthreearmsof5,8,and13-evenmoreevidencethatthisisnotacoincidence.NowisNatureplayingsomekindofcruelgamewithus?Nooneknowsforsure,butscientistsspecul
85、atethatplantsthatgrowinspiralformationdosoinFibonaccinumbersbecausethisarrangementmakesfortheperfectspacingforgrowth.Soforsomereason,thesenumbersprovidetheperfectarrangementformaximumgrowthpotentialandsurvivaloftheplant.Dothesefacesseemattractivetoyou?Dothesefacesseemattractivetoyou?Manypeopleseemto
86、thinkso.Butwhy?IsManypeopleseemtothinkso.Butwhy?Istheresomethingspecificineachoftheirtheresomethingspecificineachoftheirfacesthatattractsustothem,orisourfacesthatattractsustothem,orisourattractiongovernedbyoneofNaturesattractiongovernedbyoneofNaturesrules?Doesthishaveanythingtodowithrules?Doesthisha
87、veanythingtodowiththeGoldenRatio?IthinkyoualreadyknowtheGoldenRatio?Ithinkyoualreadyknowtheanswertothatquestion.Letstrytotheanswertothatquestion.LetstrytoanalyzethesefacestoseeiftheGoldenanalyzethesefacestoseeiftheGoldenRatioispresentornot.HereshowweareRatioispresentornot.Hereshowwearegoingtoconduct
88、oursearchfortheGoldengoingtoconductoursearchfortheGoldenRatio:wewillmeasurecertainaspectsofRatio:wewillmeasurecertainaspectsofeachpersonsface.Thenwewillcompareeachpersonsface.Thenwewillcomparetheirratios.Letsbegin.Wewillneedthetheirratios.Letsbegin.Wewillneedthefollowingmeasurements,tothenearestfoll
89、owingmeasurements,tothenearesttenthofacentimeter:tenthofacentimeter:a=Top-of-headtochin=cma=Top-of-headtochin=cmb=Top-of-headtopupil=cmb=Top-of-headtopupil=cmc=Pupiltoc=Pupiltonosetipnosetip=cm=cmd=Pupiltolip=cmd=Pupiltolip=cme=Widthofnose=cme=Widthofnose=cmf=Outsidedistancebetweeneyes=cmf=Outsidedi
90、stancebetweeneyes=cmg=Widthofhead=cmg=Widthofhead=cmh=Hairlinetopupil=cmh=Hairlinetopupil=cmi=i=NosetipNosetiptochin=cmtochin=cmj=Lipstochin=cmj=Lipstochin=cmk=Lengthoflips=cmk=Lengthoflips=cml=l=NosetipNosetiptolips=cmtolips=cma/g=cmb/d=cmi/j=cmi/c=cme/l=cmf/h=cmk/e=cmNow find the following ratios:
91、faceappletThe blue line defines a perfect square of the pupils and outside corners of the mouth. The golden section of these four blue lines defines the nose, the tip of the nose, the inside of the nostrils, the two rises of the upper lip and the inner points of the ear. The blue line also defines t
92、he distance from the upper lip to the bottom of the chin.The yellow line, a golden section of the blue line, defines the width of the nose, the distance between the eyes and eye brows and the distance from the pupils to the tip of the nose.The green line, a golden section of the yellow line defines
93、the width of the eye, the distance at the pupil from the eye lash to the eye brow and the distance between the nostrils.The magenta line, a golden section of the green line, defines the distance from the upper lip to the bottom of the nose and several dimensionsEvenwhenviewedEvenwhenviewedfromthesid
94、e,thefromtheside,thehumanheadhumanheadillustratestheillustratestheDivineProportion.DivineProportion.ThefirstgoldensectionThefirstgoldensection( (blueblue)fromthefrontofthe)fromthefrontoftheheaddefinesthepositionofheaddefinesthepositionoftheearopening.Thetheearopening.Thesuccessivegoldensectionssucce
95、ssivegoldensectionsdefinetheneck(definetheneck(yellowyellow),),thebackoftheeye(thebackoftheeye(greengreen)andthefrontoftheeyeandandthefrontoftheeyeandbackofthenoseandmouthbackofthenoseandmouth( (magentamagenta).Thedimensions).Thedimensionsofthefacefromtoptoofthefacefromtoptobottomalsoexhibitthebotto
96、malsoexhibittheDivineProportion,intheDivineProportion,inthepositionsoftheeyebrowpositionsoftheeyebrow( (blueblue),nose(),nose(yellowyellow)and)andmouth(mouth(greengreenandandmagentamagenta).). TheearreflectstheshapeofTheearreflectstheshapeofaFibonaccispiral.aFibonaccispiral.Thefronttwoincisorteethfo
97、rmaThefronttwoincisorteethformagoldenrectangle,withaphiratiointhegoldenrectangle,withaphiratiointheheighthheighthtothewidth.tothewidth.TheratioofthewidthofthefirsttoothTheratioofthewidthofthefirsttoothtothesecondtoothfromthecenteristothesecondtoothfromthecenterisalsophi.alsophi.Theratioofthewidthoft
98、hesmiletotheTheratioofthewidthofthesmiletothethirdtoothfromthecenterisphiasthirdtoothfromthecenterisphiaswell.well.VisitthesiteofDr.EddyLevinformoreVisitthesiteofDr.EddyLevinformoreontheontheGoldenSectionandDentistryGoldenSectionandDentistry. .YourhandshowsPhiandtheFibonacciSeriesyourindexfinger.The
99、ratioofyourforearmtohandisPhiThe Human BodyThehumanbodyisbasedonPhiand5ThehumanbodyillustratestheGoldenSection.Wellusethesamebuildingblocksagain:TheProportionsintheBodyThewhitelineisthebodysheight.Theblueline,agoldensectionofthewhiteline,definesthedistancefromtheheadtothefingertipsTheyellowline,agol
100、densectionoftheblueline,definesthedistancefromtheheadtothenavelandtheelbows.Thegreenline,agoldensectionoftheyellowline,definesthedistancefromtheheadtothepectoralsandinsidetopofthearms,thewidthoftheshoulders,thelengthoftheforearmandtheshinbone.Themagentaline,agoldensectionofthegreenline,definesthedistancefromtheheadtothebaseoftheskullandthewidthoftheabdomen.Thesectionedportionsofthemagentalinedeterminethepositionofthenoseandthehairline.Althoughnotshown,thegoldensectionofthemagentaline(alsotheshortsectionofthegreenline)definesthewidthoftheheadandhalfthewidthofthechestandthehips.
