FibonacciNumbersandtheGoldenRatioSayWhat?What is the Golden Ratio?Well,beforeweanswerthatquestionletsexamineaninterestingsequence(orlist)ofnumbers.Actuallytheseriesstartswith0,1buttomakeiteasierwelljuststartwith:1,1TogetthenextnumberweaddtheprevioustwoTogetthenextnumberweaddtheprevioustwonumberstogether.Sonowoursequencebecomesnumberstogether.Sonowoursequencebecomes1,1,2.Thenextnumberwillbe3.Whatdoyou1,1,2.Thenextnumberwillbe3.Whatdoyouthinkthenextnumberinthesequencewillbe?thinkthenextnumberinthesequencewillbe?Remember,weaddtheprevioustwonumberstoRemember,weaddtheprevioustwonumberstogetthenext.Sothenextnumbershouldbe2+3,orgetthenext.Sothenextnumbershouldbe2+3,or5.Hereiswhatoursequenceshouldlooklikeifwe5.Hereiswhatoursequenceshouldlooklikeifwecontinueoninthisfashionforawhile:continueoninthisfashionforawhile:1,1,2,3,5,8,13,21,34,55,89,144,233,377,6101,1,2,3,5,8,13,21,34,55,89,144,233,377,610Now,Iknowwhatyoumightbethinking:WhatdoesthishavetodowiththeGoldenRatio?Thissequenceofnumberswasfirst“discovered”byamannamedLeonardoFibonacci,andhenceisknownasFibonaccissequence.Math GEEKReallyFamousReallySmartLeonardoFibonacciTherelationshipofthissequencetotheGoldenRatioliesnotintheactualnumbersofthesequence,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 division 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 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)55/34=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 NoticethatIhaveroundedmyratiostothethirddecimalNoticethatIhaveroundedmyratiostothethirddecimalplace.Ifweexamine55/34and89/55moreclosely,wewillplace.Ifweexamine55/34and89/55moreclosely,wewillseethattheirdecimalvaluesareactuallynotthesame.Butseethattheirdecimalvaluesareactuallynotthesame.Butwhatdoyouthinkwillhappenifwecontinuetolookatthewhatdoyouthinkwillhappenifwecontinuetolookattheratiosasthenumbersinthesequencegetlargerandratiosasthenumbersinthesequencegetlargerandlarger?Thatsright:theratiowilleventuallybecomethelarger?Thatsright:theratiowilleventuallybecomethesamenumber,andthatnumberistheGoldenRatio!samenumber,andthatnumberistheGoldenRatio!11231.500000000000000051.666666666666670081.6000000000000000131.6250000000000000211.6153846153846200341.6190476190476200551.6176470588235300891.61818181818182001441.61797752808989002331.61805555555556003771.61802575107296006101.61803713527851009871.61803278688525001,5971.61803444782168002,5841.61803381340013004,1811.61803405572755006,7651.618033963166710010,9461.618033998521800017,7111.618033985017360028,6571.618033990175600046,3681.618033988205320075,0251.6180339889579000TheGoldenRatioiswhatwecallanirrationalnumber:ithasaninfinitenumberofdecimalplacesanditneverrepeatsitself!Generally,weroundtheGoldenRatioto1.618.Here is the decimal value 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 ordinary text, in lines across, so Phi is 161803398874.)Read this as ordinary text, in lines across, so Phi is 161803398874.) DpsDps: :16180339887498948482045868343656381177203091798057616180339887498948482045868343656381177203091798057650502862135448622705260462818286213544862270526046281890244970720720418939113749024497072072041893911374100100847540880753868917521266338622235369317931800607667263584754088075386891752126633862223536931793180060766726354433389086595939582905638322661319928290267884433389086595939582905638322661319928290267882002000675208766892501711696207032221043206752087668925017116962070322210432162695486262963136144381497587012203408058879544547492461856953641626954862629631361443814975870122034080588795445474924618569536430030086444924108644492410443207713449470495658467885098743394422125448770664780915884607499887124007652174432077134494704956584678850987433944221254487706647809158846074998871240076521705751797880575179788400400341662562494075890697040002812104276217711177780531531714101170466659934166256249407589069704000281210427621771117778053153171410117046665991466979873176135600670874807101466979873176135600670874807105005001317952368942752194843530567830022878569978291317952368942752194843530567830022878569978297783478458782289110976250030269615617002504643382437764861028383126833037242926777834784587822891109762500302696156170025046433824377648610283831268330372429267526311653392473167111211588186385133162038400522216579128667529465490681131715995263116533924731671112115881863851331620384005222165791286675294654906811317159934323597349498509040947621322298101726107059611645629909816290555208524790352406343235973494985090409476213222981017261070596116456299098162905552085247903524060201727997471753427775927786256194320827505131218156285512224809394712341451702202017279974717534277759277862561943208275051312181562855122248093947123414517022373580577278616008688382952304592647878017889921990270776903895321968198615143783735805772786160086883829523045926478780178899219902707769038953219681986151437803149974110692608867429622675756052317277752035361393620314997411069260886742962267575605231727775203536139362100010001076738937645560606010767389376455606060592165894667595519004005559089502295309423124823555921658946675955190040055590895022953094231248235521221212212415444006470340565734797241544400647034056573479766397239494994658457887303962309037503399385621024236902513868041457799569812244663972394949946584578873039623090375033993856210242369025138680414577995698122445747178034173126453220416397232134044449487302315417676893752103068737880344170057471780341731264532204163972321340444494873023154176768937521030687378803441700939544096279558986787232095124268935573097045095956844017555198819218020640529059395440962795589867872320951242689355730970450959568440175551988192180206405290551893494759260073485228210108819464454422231889131929468962200230144377026992300518934947592600734852282101088194644544222318891319294689622002301443770269923007803085261180754519288770502109684249362713592518760777884665836150238913493333178030852611807545192887705021096842493627135925187607778846658361502389134933331223105339232136243192637289106705033992822652635562090297986424727597725655086152231053392321362431926372891067050339928226526355620902979864247275977256550861548754357482647181414512700060238901620777322449943530889990950168032811219432048487543574826471814145127000602389016207773224499435308899909501680328112194320481964387675863314798571911397815397807476150772211750826945863932045652098969855519643876758633147985719113978153978074761507722117508269458639320456520989698555678141069683728840587461033781054443909436835835813811311689938555769754841491446781410696837288405874610337810544439094368358358138113116899385557697548414914453415091295407005019477548616307542264172939468036731980586183391832859913039607534150912954070050194775486163075422641729394680367319805861833918328599130396072014455950449779212076124785645916160837059498786006970189409886400764436170933420144559504497792120761247856459161608370594987860069701894098864007644361709334172709191433650137151727091914336501371520002000 WeworkwithanotherimportantirrationalnumberinGeometry:pi,whichisapproximately3.14.SincewedontwanttomaketheGoldenRatiofeelleftout,wewillgiveititsownGreekletter:phi. Phiwhich is equal to:OnemoreinterestingthingaboutPhiisitsOnemoreinterestingthingaboutPhiisitsreciprocal.Ifyoutaketheratioofanynumberinreciprocal.IfyoutaketheratioofanynumberintheFibonaccisequencetothenextnumber(thisistheFibonaccisequencetothenextnumber(thisisthereverseofwhatwedidbefore),theratiowillthereverseofwhatwedidbefore),theratiowillapproachtheapproximation0.618.Thisistheapproachtheapproximation0.618.ThisisthereciprocalofPhi:1/1.618=0.618.ItishighlyreciprocalofPhi:1/1.618=0.618.Itishighlyunusualforthedecimalintegersofanumberandunusualforthedecimalintegersofanumberanditsreciprocaltobeexactlythesame.Infact,Iitsreciprocaltobeexactlythesame.Infact,Icannotnameanothernumberthathasthiscannotnameanothernumberthathasthisproperty!Thisonlyaddstothemystiqueoftheproperty!ThisonlyaddstothemystiqueoftheGoldenRatioandleadsustoask:WhatmakesitGoldenRatioandleadsustoask:Whatmakesitsospecial?sospecial?TheGoldenRatioisnotjustsomenumberthatmathTheGoldenRatioisnotjustsomenumberthatmathteachersthinkiscool.Theinterestingthingisthatitkeepsteachersthinkiscool.Theinterestingthingisthatitkeepspoppingupinstrangeplaces-placesthatwemaynotpoppingupinstrangeplaces-placesthatwemaynotordinarilyhavethoughttolookforit.Itisimportanttonoteordinarilyhavethoughttolookforit.ItisimportanttonotethatFibonaccididnotinventtheGoldenRatio;hejustthatFibonaccididnotinventtheGoldenRatio;hejustdiscoveredoneinstanceofwhereitappearednaturally.Indiscoveredoneinstanceofwhereitappearednaturally.InfactcivilizationsasfarbackandasfarapartastheAncientfactcivilizationsasfarbackandasfarapartastheAncientEgyptians,theMayans,aswellastheGreeksdiscoveredEgyptians,theMayans,aswellastheGreeksdiscoveredtheGoldenRatioandincorporateditintotheirownart,theGoldenRatioandincorporateditintotheirownart,architecture,anddesigns.TheydiscoveredthattheGoldenarchitecture,anddesigns.TheydiscoveredthattheGoldenRatioseemstobeNaturesperfectnumber.ForsomeRatioseemstobeNaturesperfectnumber.Forsomereason,itjustseemstoappealtoournaturalinstincts.Thereason,itjustseemstoappealtoournaturalinstincts.Themostbasicexampleisinrectangularobjects.mostbasicexampleisinrectangularobjects.Lookatthefollowingrectangles:Lookatthefollowingrectangles: Nowaskyourself,whichofthemseemstobethemostNowaskyourself,whichofthemseemstobethemostnaturallyattractiverectangle?Ifyousaidthefirstone,thennaturallyattractiverectangle?Ifyousaidthefirstone,thenyouareprobablythetypeofpersonwholikeseverythingtoyouareprobablythetypeofpersonwholikeseverythingtobesymmetrical.Mostpeopletendtothinkthatthethirdbesymmetrical.Mostpeopletendtothinkthatthethirdrectangleisthemostappealing.rectangleisthemostappealing.IfyouweretomeasureeachrectangleslengthIfyouweretomeasureeachrectangleslengthandwidth,andcomparetheratiooflengthtowidthandwidth,andcomparetheratiooflengthtowidthforeachrectangleyouwouldseethefollowing:foreachrectangleyouwouldseethefollowing:Rectangleone:Ratio1:1Rectangleone:Ratio1:1Rectangletwo:Ratio2:1Rectangletwo:Ratio2:1RectangleThree:Ratio1.618:1RectangleThree:Ratio1.618:1HaveyoufiguredoutwhythethirdrectangleistheHaveyoufiguredoutwhythethirdrectangleisthemostappealing?Thatsright-becausetheratioofmostappealing?Thatsright-becausetheratioofitslengthtoitswidthistheGoldenRatio!ForitslengthtoitswidthistheGoldenRatio!Forcenturies,designersofartandarchitecturehavecenturies,designersofartandarchitecturehaverecognizedthesignificanceoftheGoldenRatioinrecognizedthesignificanceoftheGoldenRatiointheirwork.theirwork.LetsseeifwecandiscoverwheretheGoldenRatioappearsineverydayobjects.Useyourmeasuringtooltocomparethelengthandthewidthofrectangularobjectsintheclassroomorinyourhouse(dependingonwhereyouarerightnow).Trytochooseobjectsthataremeanttobevisuallyappealing.WereyousurprisedtofindtheGoldenRatioinsomanyWereyousurprisedtofindtheGoldenRatioinsomanyplaces?Itshardtobelievethatwehavetakenitforgrantedplaces?Itshardtobelievethatwehavetakenitforgrantedforsolong,isntit?forsolong,isntit?ObjectObjectLengthLengthWidthWidthRatioRatioindex cardindex cardphotographphotographpicture framepicture frametextbooktextbookdoor framedoor framecomputer computer screenscreenTV screenTV screenHowaboutinmusic?LetstakealookatthepianokeyboarddoyouseeAnythingfamiliar?Countthenumberofkeys(notes)ineachofthebracketsCountthenumberofkeys(notes)ineachofthebracketsYouwillseethenumbers2,3,5,8,13.coincidence?Youwillseethenumbers2,3,5,8,13.coincidence?DoesitlookliketheFibonaccisequenceitshouldDoesitlookliketheFibonaccisequenceitshouldbecauseitis!becauseitis!How about Architecture?FindtheGoldenRatiointheParthenon.FindtheGoldenRatiointheParthenon.1.LetsstartbydrawingarectanglearoundtheParthenon,1.LetsstartbydrawingarectanglearoundtheParthenon,fromtheleftmostpillartotherightandfromthebaseofthefromtheleftmostpillartotherightandfromthebaseofthepillarstothehighestpoint.pillarstothehighestpoint.2.Measurethelengthandthewidthofthisrectangle.Now2.Measurethelengthandthewidthofthisrectangle.Nowfindtheratioofthelengthtothewidth.Isthenumberfairlyfindtheratioofthelengthtothewidth.IsthenumberfairlyclosetotheGoldenRatio?closetotheGoldenRatio?3.Nowlookabovethepillars.Youshouldnoticesome3.Nowlookabovethepillars.YoushouldnoticesomerectanglesonthefaceoftheParthenon.FindtheratioofrectanglesonthefaceoftheParthenon.Findtheratioofthelengthtothewidthofoneoftheserectangles.Noticethelengthtothewidthofoneoftheserectangles.Noticeanything?anything?TherearemanyotherplaceswheretheGoldenRatioTherearemanyotherplaceswheretheGoldenRatioappearsintheParthenon,allofwhichwecannotseeappearsintheParthenon,allofwhichwecannotseebecauseweonlyhaveafrontalviewofthestructure.Thebecauseweonlyhaveafrontalviewofthestructure.Thebuildingisbuiltonarectangularplotoflandwhichhappensbuildingisbuiltonarectangularplotoflandwhichhappenstobe.youguessedit-aGoldenRectangle!tobe.youguessedit-aGoldenRectangle!Once its ruined triangular pediment is restored, .the ancient temple fits almost precisely into a golden rectangle.Further classic subdivisions of the rectangle align perfectly with major architectural features of the structure. Furtherclassicsubdivisionsoftherectanglealignperfectlywithmajorarchitecturalfeaturesofthestructure.The Golden Ratio in ArtNowletsgobackandtrytodiscovertheGoldenRatioinart.WewillconcentrateontheworksofLeonardodaVinci,ashewasnotonlyagreatartistbutalsoageniuswhenitcametomathematicsandinvention.The Annunciation - The Annunciation - UsingtheleftsideofthepaintingasaUsingtheleftsideofthepaintingasaside,createasquareontheleftofthepaintingbyinsertingside,createasquareontheleftofthepaintingbyinsertingaverticalline.Noticethatyouhavecreatedasquareandaaverticalline.Noticethatyouhavecreatedasquareandarectangle.TherectangleturnsouttobeaGoldenrectangle.TherectangleturnsouttobeaGoldenRectangle,ofcourse.Also,drawinahorizontallinethatisRectangle,ofcourse.Also,drawinahorizontallinethatis61.8%ofthewaydownthepainting(.618-theinverseof61.8%ofthewaydownthepainting(.618-theinverseoftheGoldenRatio).Drawanotherlinethatis61.8%ofthetheGoldenRatio).Drawanotherlinethatis61.8%ofthewayupthepainting.Tryagainwithverticallinesthatarewayupthepainting.Tryagainwithverticallinesthatare61.8%ofthewayacrossbothfromlefttorightandfrom61.8%ofthewayacrossbothfromlefttorightandfromrighttoleft.Youshouldnowhavefourlinesdrawnacrossrighttoleft.Youshouldnowhavefourlinesdrawnacrossthepainting.Noticethattheselinesintersectimportantthepainting.Noticethattheselinesintersectimportantpartsofthepainting,suchastheangel,thewoman,etc.partsofthepainting,suchastheangel,thewoman,etc.Coincidence?Ithinknot!Coincidence?Ithinknot!The Mona Lisa - Measurethelengthandthewidthofthepaintingitself.Theratiois,ofcourse,Golden.DrawarectanglearoundMonasface(fromthetopoftheforeheadtothebaseofthechin,andfromleftcheektorightcheek)andnoticethatthis,too,isaGoldenrectangle.LeonardoLeonardodadaVincistalentasanartistmaywellVincistalentasanartistmaywellhavebeenoutweighedbyhistalentsasahavebeenoutweighedbyhistalentsasamathematician.Heincorporatedgeometryintomathematician.Heincorporatedgeometryintomanyofhispaintings,withtheGoldenRatiobeingmanyofhispaintings,withtheGoldenRatiobeingjustoneofhismanymathematicaltools.Whydojustoneofhismanymathematicaltools.Whydoyouthinkheuseditsomuch?Expertsagreethatyouthinkheuseditsomuch?ExpertsagreethatheprobablythoughtthatGoldenmeasurementsheprobablythoughtthatGoldenmeasurementsmadehispaintingsmoreattractive.Maybehewasmadehispaintingsmoreattractive.Maybehewasjustalittletooobsessedwithperfection.However,justalittletooobsessedwithperfection.However,hewasnottheonlyonetouseGoldenpropertieshewasnottheonlyonetouseGoldenpropertiesinhiswork.inhiswork.Constructing A Golden Rectangle Constructing A Golden Rectangle IsntitstrangethattheGoldenRatiocameupinsuchIsntitstrangethattheGoldenRatiocameupinsuchunexpectedplaces?Wellletsseeifwecanfindoutwhy.unexpectedplaces?Wellletsseeifwecanfindoutwhy.TheGreekswerethefirsttocallphitheGoldenRatio.TheyTheGreekswerethefirsttocallphitheGoldenRatio.Theyassociatedthenumberwithperfection.Itseemstobepartassociatedthenumberwithperfection.ItseemstobepartofhumannatureorinstinctforustofindthingsthatcontainofhumannatureorinstinctforustofindthingsthatcontaintheGoldenRationaturallyattractive-suchastheperfecttheGoldenRationaturallyattractive-suchastheperfectrectangle.Realizingthis,designershavetriedtorectangle.Realizingthis,designershavetriedtoincorporatetheGoldenRatiointotheirdesignssoastoincorporatetheGoldenRatiointotheirdesignssoastomakethemmorepleasingtotheeye.Doors,notebookmakethemmorepleasingtotheeye.Doors,notebookpaper,textbooks,etc.allseemmoreattractiveiftheirsidespaper,textbooks,etc.allseemmoreattractiveiftheirsideshavearatioclosetophi.Now,letsseeifwecanconstructhavearatioclosetophi.Now,letsseeifwecanconstructourownperfectrectangle.ourownperfectrectangle.Method One1. Well start by making a square, any square (just remember 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 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 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 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 the 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 - Method TwoNow,letstryadifferentmethodthatwillrelatetheNow,letstryadifferentmethodthatwillrelatetherectangletotheFibonacciserieswelookedat.WellstartrectangletotheFibonacciserieswelookedat.Wellstartwithasquare.Thesizedoesnotmatter,aslongasallwithasquare.Thesizedoesnotmatter,aslongasallsidesarecongruent.Welluseasmallsquaretoconservesidesarecongruent.Welluseasmallsquaretoconservespace,becausewearegoingtobuildourgoldenrectanglespace,becausewearegoingtobuildourgoldenrectanglearoundthissquare.Pleasenotethatthegoldenareaisaroundthissquare.Pleasenotethatthegoldenareaiswhatyourrectanglewilleventuallylooklike.whatyourrectanglewilleventuallylooklike.Now,letsbuildanother,congruentsquarerightnexttotheNow,letsbuildanother,congruentsquarerightnexttothefirstone:firstone:Nowwehavearectanglewithawidth1andlength2units.Nowwehavearectanglewithawidth1andlength2units.Letsbuildasquareontopofthisrectangle,sothatthenewLetsbuildasquareontopofthisrectangle,sothatthenewsquarewillhaveasideof2units:squarewillhaveasideof2units:Noticethatwehaveanewrectanglewithwidth2andNoticethatwehaveanewrectanglewithwidth2andlength3.length3.Letscontinuetheprocess,buildinganothersquareontheLetscontinuetheprocess,buildinganothersquareontherightofourrectangle.Thissquarewillhaveasideof3:rightofourrectangle.Thissquarewillhaveasideof3:Nowwehavearectangleofwidth3andlength5.Nowwehavearectangleofwidth3andlength5.Again,letsbuilduponthisrectangleandconstructasquareunderneath,withasideof5:Thenewrectanglehasawidthof5andalengthof8.Letscontinuetotheleftwithasquarewithside8:Haveyounoticedthepatternyet?Thenewrectanglehasawidthof8andalengthof13.Letscontinuewithonefinalsquareontop,withasideof13:Ourfinalrectanglehasawidthof13andalengthof21.Ourfinalrectanglehasawidthof13andalengthof21.NoticethatwehaveconstructedourgoldenrectangleusingNoticethatwehaveconstructedourgoldenrectangleusingsquarethathadsuccessivesidelengthsfromtheFibonaccisquarethathadsuccessivesidelengthsfromtheFibonaccisequence(1,1,2,3,5,8,13,.)!Nowonderourrectanglesequence(1,1,2,3,5,8,13,.)!Nowonderourrectangleisgolden!Eachsuccessiverectanglethatweconstructedisgolden!EachsuccessiverectanglethatweconstructedhadawidthandlengththatwereconsecutivetermsinthehadawidthandlengththatwereconsecutivetermsintheFibonaccisequence.SoifwedividethelengthbytheFibonaccisequence.Soifwedividethelengthbythewidth,wewillarriveattheGoldenRatio!Ofcourse,ourwidth,wewillarriveattheGoldenRatio!Ofcourse,ourrectangleisnotperfectlygolden.Wecouldkeeptherectangleisnotperfectlygolden.Wecouldkeeptheprocessgoinguntilthesidesapproximatedtheratiobetter,processgoinguntilthesidesapproximatedtheratiobetter,butforourpurposesalengthof21andawidthof13arebutforourpurposesalengthof21andawidthof13aresufficient.sufficient.3421DotheMath!34dividedby21=1.61904761904Rememberthatthefartherintothesequencewegotheclosertheratiogetstobeingperfect!Thisrectangleshouldseemverywellproportionedtoyou,i.e.itshouldbepleasingtotheeye.Ifitisnt,maybeyouneedyoureyeschecked!Constructing a Golden SpiralNoticehowwebuiltourrectangleinacounterclockwisedirection.ThisleadsusintoanotherinterestingcharacteristicoftheGoldenRatio.Letslookattherectanglewithallofourconstructionlinesdrawnin:Wearegoingtoconcentrateonthesquaresthatwedrew,Wearegoingtoconcentrateonthesquaresthatwedrew,startingwiththetwosmallestones.Letsstartwiththeonestartingwiththetwosmallestones.Letsstartwiththeoneontheright.Connecttheupperrightcornertothelowerleftontheright.ConnecttheupperrightcornertothelowerleftcornerwithanarcthatisonefourthofacirclecornerwithanarcthatisonefourthofacircleWearegoingtoconcentrateonthesquaresthatWearegoingtoconcentrateonthesquaresthatwedrew,startingwiththetwosmallestones.Letswedrew,startingwiththetwosmallestones.Letsstartwiththeoneontheright.Connecttheupperstartwiththeoneontheright.Connecttheupperrightcornertothelowerleftcornerwithanarcthatrightcornertothelowerleftcornerwithanarcthatisonefourthofacircle:isonefourthofacircle:ThencontinueyourlineintothesecondsquareonThencontinueyourlineintothesecondsquareontheleft,againwithanarcthatisonefourthofatheleft,againwithanarcthatisonefourthofacircle:circle:WewillcontinuethisprocessuntileachsquareWewillcontinuethisprocessuntileachsquarehasanarcinsideofit,withallofthemconnectedhasanarcinsideofit,withallofthemconnectedasacontinuousline.Thelineshouldlooklikeaasacontinuousline.Thelineshouldlooklikeaspiralwhenwearedone.Hereisanexampleofspiralwhenwearedone.Hereisanexampleofwhatyourspiralshouldlooklike:whatyourspiralshouldlooklike:GoldenSpiralmanipulativeNowwhatwasthepointofthat?Thepointisthatthisgoldenspiraloccursfrequentlyinnature.Ifyoulookcloselyenough,youmightfindagoldenspiralintheheadofadaisy,inapinecone,insunflowers,orinanautilusshellthatyoumightfindonabeachoreveninyourear!Herearesomeexamples:So,whydoshapesthatexhibittheGoldenRatioseemmoreappealingtothehumaneye?Noonereallyknowsforsure.ButwedohaveevidencethattheGoldenRatioseemstobeNaturesperfectnumber.SomebodywithalotoftimeontheirhandsdiscoveredthatSomebodywithalotoftimeontheirhandsdiscoveredthattheindividualfloretsofthedaisy(andofasunflowerastheindividualfloretsofthedaisy(andofasunfloweraswell)growintwospiralsextendingoutfromthecenter.Thewell)growintwospiralsextendingoutfromthecenter.Thefirstspiralhas21arms,whiletheotherhas34.Dothesefirstspiralhas21arms,whiletheotherhas34.Dothesenumberssoundfamiliar?numberssoundfamiliar?Theyshould-theyareFibonaccinumbers!Andtheirratio,ofcourse,istheGoldenRatio.Wecansaythesamethingaboutthespiralsofapinecone,wherespiralsfromthecenterhave5and8arms,respectively(orof8and13,dependingonthesize)-again,twoFibonaccinumbers:Apineapplehasthreearmsof5,8,and13-evenmoreevidencethatthisisnotacoincidence.NowisNatureplayingsomekindofcruelgamewithus?Nooneknowsforsure,butscientistsspeculatethatplantsthatgrowinspiralformationdosoinFibonaccinumbersbecausethisarrangementmakesfortheperfectspacingforgrowth.Soforsomereason,thesenumbersprovidetheperfectarrangementformaximumgrowthpotentialandsurvivaloftheplant.Dothesefacesseemattractivetoyou?Dothesefacesseemattractivetoyou?Manypeopleseemtothinkso.Butwhy?IsManypeopleseemtothinkso.Butwhy?Istheresomethingspecificineachoftheirtheresomethingspecificineachoftheirfacesthatattractsustothem,orisourfacesthatattractsustothem,orisourattractiongovernedbyoneofNaturesattractiongovernedbyoneofNaturesrules?Doesthishaveanythingtodowithrules?DoesthishaveanythingtodowiththeGoldenRatio?IthinkyoualreadyknowtheGoldenRatio?Ithinkyoualreadyknowtheanswertothatquestion.Letstrytotheanswertothatquestion.LetstrytoanalyzethesefacestoseeiftheGoldenanalyzethesefacestoseeiftheGoldenRatioispresentornot.HereshowweareRatioispresentornot.HereshowwearegoingtoconductoursearchfortheGoldengoingtoconductoursearchfortheGoldenRatio:wewillmeasurecertainaspectsofRatio:wewillmeasurecertainaspectsofeachpersonsface.Thenwewillcompareeachpersonsface.Thenwewillcomparetheirratios.Letsbegin.Wewillneedthetheirratios.Letsbegin.Wewillneedthefollowingmeasurements,tothenearestfollowingmeasurements,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=Outsidedistancebetweeneyes=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: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 the 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 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 dimensionsEvenwhenviewedEvenwhenviewedfromtheside,thefromtheside,thehumanheadhumanheadillustratestheillustratestheDivineProportion.DivineProportion.ThefirstgoldensectionThefirstgoldensection( (blueblue)fromthefrontofthe)fromthefrontoftheheaddefinesthepositionofheaddefinesthepositionoftheearopening.Thetheearopening.Thesuccessivegoldensectionssuccessivegoldensectionsdefinetheneck(definetheneck(yellowyellow),),thebackoftheeye(thebackoftheeye(greengreen)andthefrontoftheeyeandandthefrontoftheeyeandbackofthenoseandmouthbackofthenoseandmouth( (magentamagenta).Thedimensions).ThedimensionsofthefacefromtoptoofthefacefromtoptobottomalsoexhibitthebottomalsoexhibittheDivineProportion,intheDivineProportion,inthepositionsoftheeyebrowpositionsoftheeyebrow( (blueblue),nose(),nose(yellowyellow)and)andmouth(mouth(greengreenandandmagentamagenta).). TheearreflectstheshapeofTheearreflectstheshapeofaFibonaccispiral.aFibonaccispiral.ThefronttwoincisorteethformaThefronttwoincisorteethformagoldenrectangle,withaphiratiointhegoldenrectangle,withaphiratiointheheighthheighthtothewidth.tothewidth.TheratioofthewidthofthefirsttoothTheratioofthewidthofthefirsttoothtothesecondtoothfromthecenteristothesecondtoothfromthecenterisalsophi.alsophi.TheratioofthewidthofthesmiletotheTheratioofthewidthofthesmiletothethirdtoothfromthecenterisphiasthirdtoothfromthecenterisphiaswell.well.VisitthesiteofDr.EddyLevinformoreVisitthesiteofDr.EddyLevinformoreontheontheGoldenSectionandDentistryGoldenSectionandDentistry. .YourhandshowsPhiandtheFibonacciSeriesyourindexfinger.TheratioofyourforearmtohandisPhiThe Human BodyThehumanbodyisbasedonPhiand5ThehumanbodyillustratestheGoldenSection.Wellusethesamebuildingblocksagain:TheProportionsintheBodyThewhitelineisthebodysheight.Theblueline,agoldensectionofthewhiteline,definesthedistancefromtheheadtothefingertipsTheyellowline,agoldensectionoftheblueline,definesthedistancefromtheheadtothenavelandtheelbows.Thegreenline,agoldensectionoftheyellowline,definesthedistancefromtheheadtothepectoralsandinsidetopofthearms,thewidthoftheshoulders,thelengthoftheforearmandtheshinbone.Themagentaline,agoldensectionofthegreenline,definesthedistancefromtheheadtothebaseoftheskullandthewidthoftheabdomen.Thesectionedportionsofthemagentalinedeterminethepositionofthenoseandthehairline.Althoughnotshown,thegoldensectionofthemagentaline(alsotheshortsectionofthegreenline)definesthewidthoftheheadandhalfthewidthofthechestandthehips.