黄金比例市公开课获奖课件省名师优质课赛课一等奖课件

GoldenRatio

DivineProportion,GoldenSection,PHI

1/81为何许多国家国旗图案都喜欢用五角星？中华人民共和国新西兰朝鲜新加坡2/81美妙五角星——毕达哥斯学派徽章黄金分割，是古希腊毕达哥斯学派从数学原理中发觉出来一个漂亮形式。普通来说，按黄金百分比组成事物都表现出友好和均衡。3/81TimelineofGoldenSection公元前6世纪古希腊毕达哥拉斯学派已研究过正五边形和正十边形作图，所以可推断他们已知道与此相关黄金分割问题。Phidias(490–430BC)madetheParthenonstatuesthatseemtoembodythegoldenratio.Plato(427–347BC),inhisTimaeus,describesfivepossibleregularsolids(thePlatonicsolids,thetetrahedron,cube,octahedron,dodecahedronandicosahedron),someofwhicharerelatedtothegoldenratio.4/81公元前4世纪，古希腊数学家欧多克索斯第一个系统地研究这个问题，他建立了百分比理论。

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

中国古代称黄金分割为“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，美国数学家J.基弗首先提出优选法optimizationmethod中黄金分割法优选法,是以数学原理为指导，用最可能少试验次数，尽快找到生产和科学试验中最优方案一个科学试验方法。1970-80年代，中国数学家华罗庚在中国推广，取得很大成绩。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,哪些比较匀称?10/8111/8112/81国旗、明信片、报纸、邮票、书本、桌面、电视屏幕、窗户、房间等等，都常被设计成靠近于黄金矩形。报幕员站在舞台宽度0.618处。13/81GoldenRectangle14/81Goldenangle15/81Leafarrangements1/2elm,linden,lime,grasses1/3beech,hazel,grasses,blackberry2/5oak,cherry,apple,holly,plum,commongroundsel3/8poplar,rose,pear,willow5/13pussywillow,almond

许多植物叶片、枝杈或瓣都按黄金分割角度伸展互不重合，有利于光合作用，通风和采光能到达最好效果，这是生物进化结果。16/81LeonardoFibonacci

斐波那契(约1175-约1240)是丢番图(Diophantos)与费尔马(PierredeFermat)之间欧洲最出色数论学家，出生在意大利比萨。在著作《算盘书》（LiberAbaci）中，引进了印度阿拉伯数码（包含0）及其演算法则。数论方面他在丢番图方程和同余方程方面有主要贡献。17/81Fibonacci’sSeries《算盘书》“兔子问题”：假设一对兔子每个月能生一对小兔（一雄一雌），而每对小兔在它出生后第三个月，又能开始生小兔，假如没有死亡，由一对刚出生小兔开始，一年后一共会有多少对兔子？18/81将问题普通化后答案就形成著名斐波那契数列斐波那契数列：1,1,2,3,5,8,13,21,34,55,89,144,233,……

从第三项开始每一项都是数列中前两项之和。第n个月时兔子数就是斐波那契数列第n项。19/81FibonaccinumbersandtheGoldenNumber20/81FibonacciNumbers,theGoldenSectionandTrees

著名“鲁德维格定律”是F数列在植物学中应用。数学家泽林斯基在一次国际数学会上指出，树年分枝数目就是F数列，即枝数增加遵照F数列规律．21/81英国T·W·汤姆森爵士指出．假如一棵树一直保持幼时长高和长粗百分比，那它终将会因自己“细高个子”而翻倒；所以它选择了长高和长粗最正确百分比：0.618．禾本植物(如小麦、水稻)茎节，可看到其相邻两节之比为1∶1.618或1∶2.472(依品种不一样而异)．血管粗细比：1∶1.618。22/81Pinecones许多植物叶片、花瓣、果粒数与F数列相吻合．松果上鳞片分布都与F数列相关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菲氏数过月季花，为21瓣。达尔文数过波斯菊恰好144瓣，其中55瓣和89。米切尔马斯花，157瓣，真中13瓣与另外144瓣相比，尤其长且弯曲向内，他认为157为F数列中13和144合成。向日葵外缘花瓣分为55和89瓣两种不一样形态。瓣在形态上有显著差异：一个长丝卷曲向内，一个平展舒放向外。25/81FibonacciRectangles26/81FibonacciSpiralsAlogarithmicspiral,equiangularspiralorgrowthspiralisaspecialkindofspiralcurvewhichoftenappearsinnature.ThelogarithmicspiralwasfirstdescribedbyDescartesandlaterextensivelyinvestigatedbyJakobBernoulli,whocalleditSpiramirabilis,"themarvelousspiral".上帝之眼27/81Cutawayofanautilusshellshowingthechambersarrangedinanapproximatelylogarithmicspiral海洋鹦鹉螺、蜗牛，一些动物角质体上，有甲壳软体动物身上，都有黄金螺线28/81AlowpressureareaoverIcelandshowsanapproximatelylogarithmicspiralpatternThearmsofspiralgalaxiesoftenhavetheshapeofalogarithmicspiral,heretheWhirlpoolGalaxyRomanescobroccoli,showingfractalforms蕨类植物琴状梢头，其螺线为黄金螺线29/81FibonacciPhyllotaxis30/81Flowers,VegetablesandFruit31/81Seedheads向日葵不但葵盘上有一左一右黄金螺线，而且每朵小花或果花上也有两条黄金螺线；更奇异是，每套螺线总数都符合F数列：如有21条左旋，则必有13条石旋，其总数必为34条．32/81FibonacciSpirals33/8134/81ManandGoldenSection菲波那契大量调查后，人体肚脐以下长度与身高之比比值0.618，被视为“标准美人”。芭蕾舞蹈员身形合黄金百分比，在人体绘画和雕塑等应用，如古希腊神话中太阳神阿波罗形象，女神维纳斯塑像。肚脐以上部分黄金分割点在咽喉，肚脐以下部分黄金点在膝关节，上肢部分黄金点在肘关节．人体肚脐还是胎儿营养供给，同时也是医疗效果黄金点。35/8136/81普通人腰与脚底距离占身高0.58，而下肢较长人显得身材颀长，更有美感。踮起脚尖能够增加腰与脚底距离，使得这一距离与身高比值更靠近0.618。给人以更为优美艺术形象.37/81人体最感舒适温度约23℃(气温)精神愉快时，人脑电波频率下限(8赫兹)与上限(12.9赫兹)之比，恰为黄金数。38/81FibonacciFingers?2handseachofwhichhas...5fingers,eachofwhichhas...3partsseparatedby...2knuckles39/81GoldenSectioninProductionandScienceResearch1953年美国基弗在首先提出来，1970年以后在中国进行了推广。为了到达优质、高产、低耗等目标，逐步发展起来优选法中0.618法(黄金分割法)，在生产实践和科学试验中有广泛应用。40/81GoldenSectioninArchitectandArt世界上许多美好建筑物都是按黄金分割百分比建造．古希腊雅典女神庙；法国埃菲尔铁塔。意大利著名画家达·芬奇在他作品中经常选择0.618∶1百分比关系。从声学角度来看，管弦乐器在黄金分割点上奏出声音最悦耳，许多著名音乐作品，其中高潮出现地方大多和黄金分割点靠近。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