GoldenRatio

DivineProportion,GoldenSection,PHI

1/81為何許多國家國旗圖案都喜歡用五角星？中華人民共和國新西蘭朝鮮新加坡2/81美妙五角星——畢達哥斯學派徽章黃金分割，是古希臘畢達哥斯學派從數學原理中發覺出來一個漂亮形式。普通來說，按黃金百分比組成事物都表現出友好和均衡。3/81TimelineofGoldenSection公元前6世紀古希臘畢達哥拉斯學派已研究過正五邊形和正十邊形作圖，所以可推斷他們已知道與此相關黃金分割問題。Phidias(490–430BC)madetheParthenonstatuesthatseemtoembodythegoldenratio.Plato(427–347BC),inhisTimaeus,describesfivepossibleregularsolids(thePlatonicsolids,thetetrahedron,cube,octahedron,dodecahedronandicosahedron),someofwhicharerelatedtothegoldenratio.4/81公元前4世紀，古希臘數學家歐多克索斯第一個系統地研究這個問題，他建立了百分比理論。

Euclid(c.325–c.265BC),inhisElements,gavethefirstrecordeddefinitionofthegoldenratio,whichhecalled,astranslatedintoEnglish,"extremeandmeanratio".

中國古代稱黃金分割為“弦分割”。5/81Fibonacci(1170–1250)mentionedthenumericalseriesnownamedafterhiminhisLiberAbaci;theFibonaccisequenceiscloselyrelatedtothegoldenratio.LucaPacioli(1445–1517)definesthegoldenratioasthe"divineproportion"inhisDivinaProportione.JohannesKepler(1571–1630)describesthegoldenratioasa"preciousjewel":"Geometryhastwogreattreasures:oneistheTheoremofPythagoras,andtheotherthedivisionofalineintoextremeandmeanratio;thefirstwemaycomparetoameasureofgold,thesecondwemaynameapreciousjewel."ThesetwotreasuresarecombinedintheKeplertriangle.6/81CharlesBonnet(1720–1793)pointsoutthatinthespiralphyllotaxisofplantsgoingclockwiseandcounter-clockwisewerefrequentlytwosuccessiveFibonacciseries.MartinOhm(1792–1872)isbelievedtobethefirsttousethetermgoldenerSchnitt(goldensection)todescribethisratio,in1835.EdouardLucas(1842–1891)givesthenumericalsequencenowknownastheFibonaccisequenceitspresentname.MarkBarr(20thcentury)suggeststheGreekletterphi(φ),theinitialletterofGreeksculptorPhidias'sname,asasymbolforthegoldenratio.RogerPenrose(b.1931)discoveredasymmetricalpatternthatusesthegoldenratiointhefieldofaperiodictilings,whichledtonewdiscoveriesaboutquasicrystals.7/811953年，美國數學家J.基弗首先提出優選法optimizationmethod中黃金分割法優選法,是以數學原理為指導，用最可能少試驗次數，盡快找到生產和科學試驗中最優方案一個科學試驗方法。1970-80年代，中國數學家華羅庚在中國推廣，取得很大成績。8/81Goldentriangle,pentagonandpentagram

9/815×88×1313×2121×34①②③④⑤⑥⑦⑧5/8＝0.6258/13≈0.61513/21≈0.61921/34≈0.618以下矩形中,哪些比較勻稱?10/8111/8112/81國旗、明信片、報紙、郵票、書本、桌面、電視屏幕、窗戶、房間等等，都常被設計成靠近于黃金矩形。報幕員站在舞臺寬度0.618處。13/81GoldenRectangle14/81Goldenangle15/81Leafarrangements1/2elm,linden,lime,grasses1/3beech,hazel,grasses,blackberry2/5oak,cherry,apple,holly,plum,commongroundsel3/8poplar,rose,pear,willow5/13pussywillow,almond

許多植物葉片、枝杈或瓣都按黃金分割角度伸展互不重合，有利于光合作用，通風和采光能到達最好效果，這是生物進化結果。16/81LeonardoFibonacci

斐波那契(約1175-約1240)是丟番圖(Diophantos)與費爾馬(PierredeFermat)之間歐洲最出色數論學家，出生在意大利比薩。在著作《算盤書》（LiberAbaci）中，引進了印度阿拉伯數碼（包含0）及其演算法則。數論方面他在丟番圖方程和同余方程方面有主要貢獻。17/81Fibonacci’sSeries《算盤書》“兔子問題”：假設一對兔子每個月能生一對小兔（一雄一雌），而每對小兔在它出生后第三個月，又能開始生小兔，假如沒有死亡，由一對剛出生小兔開始，一年后一共會有多少對兔子？18/81將問題普通化后答案就形成著名斐波那契數列斐波那契數列：1,1,2,3,5,8,13,21,34,55,89,144,233,……

從第三項開始每一項都是數列中前兩項之和。第n個月時兔子數就是斐波那契數列第n項。19/81FibonaccinumbersandtheGoldenNumber20/81FibonacciNumbers,theGoldenSectionandTrees

著名“魯德維格定律”是F數列在植物學中應用。數學家澤林斯基在一次國際數學會上指出，樹年分枝數目就是F數列，即枝數增加遵照F數列規律．21/81英國T·W·湯姆森爵士指出．假如一棵樹一直保持幼時長高和長粗百分比，那它終將會因自己“細高個子”而翻倒；所以它選擇了長高和長粗最正確百分比：0.618．禾本植物(如小麥、水稻)莖節，可看到其相鄰兩節之比為1∶1.618或1∶2.472(依品種不一樣而異)．血管粗細比：1∶1.618。22/81Pinecones許多植物葉片、花瓣、果粒數與F數列相吻合．松果上鱗片分布都與F數列相關23/81Petalsonflowers3petals:lily,iris

5petals:buttercup,wildrose,larkspur,columbine(aquilegia)

8petals:delphiniums

13petals:ragwort,cornmarigold,cineraria,somedaisies

21petals:aster,black-eyedsusan,chicory

34petals:plantain,pyrethrum

55,89petals:michaelmasdaisies,theasteraceaefamily.24/81Petalsonflowers菲氏數過月季花，為21瓣。達爾文數過波斯菊恰好144瓣，其中55瓣和89。米切爾馬斯花，157瓣，真中13瓣與另外144瓣相比，尤其長且彎曲向內，他認為157為F數列中13和144合成。向日葵外緣花瓣分為55和89瓣兩種不一樣形態。瓣在形態上有顯著差異：一個長絲卷曲向內，一個平展舒放向外。25/81FibonacciRectangles26/81FibonacciSpiralsAlogarithmicspiral,equiangularspiralorgrowthspiralisaspecialkindofspiralcurvewhichoftenappearsinnature.ThelogarithmicspiralwasfirstdescribedbyDescartesandlaterextensivelyinvestigatedbyJakobBernoulli,whocalleditSpiramirabilis,"themarvelousspiral".上帝之眼27/81Cutawayofanautilusshellshowingthechambersarrangedinanapproximatelylogarithmicspiral海洋鸚鵡螺、蝸牛，一些動物角質體上，有甲殼軟體動物身上，都有黃金螺線28/81AlowpressureareaoverIcelandshowsanapproximatelylogarithmicspiralpatternThearmsofspiralgalaxiesoftenhavetheshapeofalogarithmicspiral,heretheWhirlpoolGalaxyRomanescobroccoli,showingfractalforms蕨類植物琴狀梢頭，其螺線為黃金螺線29/81FibonacciPhyllotaxis30/81Flowers,VegetablesandFruit31/81Seedheads向日葵不但葵盤上有一左一右黃金螺線，而且每朵小花或果花上也有兩條黃金螺線；更奇異是，每套螺線總數都符合F數列：如有21條左旋，則必有13條石旋，其總數必為34條．32/81FibonacciSpirals33/8134/81ManandGoldenSection菲波那契大量調查后，人體肚臍以下長度與身高之比比值0.618，被視為“標準美人”。芭蕾舞蹈員身形合黃金百分比，在人體繪畫和雕塑等應用，如古希臘神話中太陽神阿波羅形象，女神維納斯塑像。肚臍以上部分黃金分割點在咽喉，肚臍以下部分黃金點在膝關節，上肢部分黃金點在肘關節．人體肚臍還是胎兒營養供給，同時也是醫療效果黃金點。35/8136/81普通人腰與腳底距離占身高0.58，而下肢較長人顯得身材頎長，更有美感。踮起腳尖能夠增加腰與腳底距離，使得這一距離與身高比值更靠近0.618。給人以更為優美藝術形象.37/81人體最感舒適溫度約23℃(氣溫)精神愉快時，人腦電波頻率下限(8赫茲)與上限(12.9赫茲)之比，恰為黃金數。38/81FibonacciFingers?2handseachofwhichhas...5fingers,eachofwhichhas...3partsseparatedby...2knuckles39/81GoldenSectioninProductionandScienceResearch1953年美國基弗在首先提出來，1970年以后在中國進行了推廣。為了到達優質、高產、低耗等目標，逐步發展起來優選法中0.618法(黃金分割法)，在生產實踐和科學試驗中有廣泛應用。40/81GoldenSectioninArchitectandArt世界上許多美好建筑物都是按黃金分割百分比建造．古希臘雅典女神廟；法國埃菲爾鐵塔。意大利著名畫家達·芬奇在他作品中經常選擇0.618∶1百分比關系。從聲學角度來看，管弦樂器在黃金分割點上奏出聲音最悅耳，許多著名音樂作品，其中高潮出現地方大多和黃金分割點靠近。41/8142/81TheancientEgyptianswerethefirsttousemathematicsinart.Itseemsalmostcertainthattheyascribedmagicalpropertiestothegoldensection(goldenratio,divineproportion,phi)andusedinthedesignoftheirgreatpyramids.

43/81IfwetakeacrosssectionoftheGreatPyramid,wegetarighttriangle,theso-calledEgyptianTriangle.Theratiooftheslantheightofthepyramid(hypotenuseofthetriangle)tothedistancefromgroundcenter(halfthebasedimension)is1.61804...whichdiffersfromphibyonlyoneunitinthefifthdecimalplace.Ifweletthebasedimensionbe2units,thenthesidesoftherighttriangleareintheproportion1:sqrt(phi):phiandthepyramidhasaheightofsqrt(phi).44/81TheMedievalbuildersofchurchesandcathedralsapproachedthedesignoftheirbuildingsinmuchthesamewayastheGreeks.Agoodgeometricstructurewastheiraim.Insideandout,theirbuildingswereintricateconstructionsbasedonthegoldensection.

45/8146/81Pythagoras(560-480BC),theGreekgeometer,wasespeciallyinterestedinthegoldensection,andprovedthatitwasthebasisfortheproportionsofthehumanfigure.Heshowedthatthehumanbodyisbuiltwitheachpartinadefinitegoldenproportiontoalltheotherparts.

47/81Pythagoras'discoveriesoftheproportionsofthehumanfigurehadatremendouseffectonGreekart.Everypartoftheirmajorbuildings,downtothesmallestdetailofdecoration,wasconstructeduponthisproportion.48/81TheParthenonwasperhapsthebestexampleofamathematicalapproachtoart.

49/81Onceitsruinedtriangularpedimentisrestored,...50/81theancienttemplefitsalmostpreciselyintoagoldenrectangle.51/81Furtherclassicsubdivisionsoftherectanglealignperfectlywithmajorarchitecturalfeaturesofthestructure.52/8153/81MathematicianshadthecontributionoftheGreeksinmindwhentheychristenedtheratio"phi"intributetothegreatPhidias,whousedtheproportionfrequentlyinhissculpture.

54/8155/8156/81Butwhilstinarchitecturetherewasthisverygreatinterestingeometry,artistsseemedtohavelostallinterestinthegoldensectionandinmathematicsasawhole.Inthe16thCentury,LucaPacioli(1445-1514),geometerandfriendofthegreatRenaissancepainters,rediscoveredthe"goldensecret".Hispublicationdevotedtothenumberphi,DivinaProportione,wasillustratedbynolessanartistthan...57/81Hehadearlier,likePythagoras,madeaclosestudyofthehumanfigureandhadshownhowallitsdifferentpartswererelatedbythegoldensection.

58/8159/8160/81Leonardo'sunfinishedcanvasSaintJeromeshowsthegreatscholarwithalionlyingathisfeet.Agoldenrectanglefitssoneatlyaroundthecentralfigurethatitisoftensaidtheartistdeliberatelypaintedthefiguretoconformtothoseproportions.KnowingLeonardo'sloveof"geometricalrecreations"ashedescribedthem,thisisquitelikely.61/81LeonardodaVinci(1451-1519).Leonardohadforalongtimedisplayedanardentinterestinthemathematicsofartandnature.62/81NoticehowtheclassicsubdivisionoftherectanglelinesupwithSt.Jerome'sextendedarm.63/81ThegoldenrectanglesinDaVinci'sMonaLisaabound.VisitthewebpageMonaLisaApplettoaddgoldenrectanglesinteractivelytohisfamousmasterpiece.64/8165/81Michelangelo'sHolyFamily...isnotableforitspositioningoftheprincipalfiguresinalignmentwithapentagramorgoldenstar.66/81Hackingbacktoclassicalthemesandtechniquesfortheirinspiration,artistsoftheRenaissancelikeMichelangelo(1475-1564)andRaphael(1483-1530)oncemorebegantoconstructtheircompositionsonthegoldenratio.TheproportionsofMichelangelo'sDavidconformtothegoldenratiofromthelocationofthenavelwithrespecttotheheighttotheplacementofthejointsinthefingers.67/81Raphael'sCrucifixion...isanotherwell-knownexample.Theprincipalfiguresoutlineagoldentriangle...68/81whichcanbeusedtolocateoneofitsunderlyingpentagrams.69/81Thisself-portraitbyRembrandt(1606-1669)...isanexampleoftriangularcomposition-holdingtogetheranintricatesubjectwithinthreestraightlines.Thedifferentlengthsofthesidesaddalittlevariety.Aperpendicularlinefromtheapexofthe