(6.1)--Chapter6_OnRatio.pdf

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1、How Does Ballet Show The NewUnderstanding of Human Form?On RatioLife on MathematicsThe confusion about the Wedding dressThe ratio is used to compare certain measures between things,which is a commonmeans for people to feel the world.People use percentage to describe the growth of economy compared to

2、 the previousyear.A farmer has improved his planting method and the price of oranges reached threetimes as much as that in last year.A kid in the kindergarten complained to the teacher,“Ijust had half my bowl of dumplings,and the Bully took them all!”It is easy for people to receive a momentary perc

3、eption by hearing the expression ofratio.In many cases,the meaning of the value is not important,but when people use ratioto depict spirit or feelings in Chinese,it can be vivid and hit the spot,such as“multipliedvalue”,“100 times the spirit”and“pay multiplied attention to your safety”.Ratios can so

4、metimes be specific and sometimes be vague and mysterious.They hidein peoples consciousness,providing a quick cognition of experiencing the world at anytime,and in many cases,quietly controlling peoples decisions.Hesitation at The Wedding DressSilians weddingSilian on high heels?The cheongsamThe che

5、ongsam,qipao in Chinese,evolved from the robes worn by thewomen of the Manchu Eight Banners in the Qing Dynasty.Theoriginal robes were not open on both sides,and the upper and lowermeasurements were relatively close.The cheongsamNowadays,the cheongsam absorbs the characteristics ofwestern fashion,so

6、 the both sides of the robe are openand the waist is tightened,which can highlight thefeminine waist curve and the beauty of oriental women.Therefore,since the 1920s,the cheongsam has graduallybecome a representative clothing for Chinese women,especially in ceremonial activities.At the opening cerem

7、ony of the 2008 Beijing OlympicGames,the Chinese cheongsam was selected as the attireof Swedish female athletes.Assorting in pairsAccording to legend,the Manchu warriors had to cross a mud pond in a battle.They imitated the white crane and tied the tree shrubs on the shoes.Finally,they passed throug

8、h the mud pond.Wearing high heels is generally a common match for cheongsam,which makes women look elegant and proud.In the Qing Dynasty,noble women wore cheongsam with special high-heeled shoes called flowerpot shoes.The shoes bottom is thick and its size is relatively small,therefore,anyone wearin

9、g this kind of shoes must be slow and cautious,which invisibly adds to the calm and composed taste of royal women.Secret of Barbie DollBarbie Doll on high heels The fact:the ratio of leg length to bodylength nearly reached 0.618.The golden sectionWhy did the first crisis in the history of mathematic

10、s arise?How is the irrationality of irrational numbers embedded in the Western culture and by what way is it reasonably recognized?The golden sectionThe golden secti on:it is to divide a line into two parts so that the whole length divided by the longer part is also equal to the longer part divided

11、by the shorter part.+=abaab+=+=tt11=atb=t20 61851.The golden ratioabOrigin of Golden Ratio Most people believe that the golden ratiooriginated from the Pythagorean school.The Pythagorean school once studied themapping of regular pentagons and regulardecagons.Mapping of regular pentagons and regular

12、decagonsPythagoras(circa 580-500 B.C.)was a mathematician and philosopherof ancient Greece.The school he founded took“number”as the tenet.Heused number to study the development of a wide range of objectivethings,including the temperament.He proposed a theory of harmony,that is,simultaneous sounding

13、of two or more tones.In addition,theGougu Theorem called in China is known as the Pythagorean Theorem,because it is discovered by Pythagorean in the west.Origin of Golden Ratio Later,Eudoxus,a mathematician in ancient Greece,systematicallystudied the golden ratio.Eudoxus(circa400-347B.C.)wasanancien

14、tGreekmathematicianandastronomer.Hemadeoutstandingcontributions in geometry,astronomy and other fields.Hisgreatestachievementisthecreationofthetheoryofproportions.Is a number？Do the irrationals exist?The dictum of the Pythagorean school:all is number.251Is Rational？2CBA211Right triangle why did Hipp

15、asus believe is not a rational number?Rationals:reasonable,Irrationals:unreasonable,2proportional.disproportional.The First Crisis of Mathematics must not be rational.definitely problematic,and the previous assumption is assumption.Therefore,theis contradictory to.It must also be even qwhich means a

16、nd get,So let must be even.2,which means by can be divided，Then prime numbers.are twoq and pwhich,in can be written as is a rational number,and it Assume that:Proof22pq=222pq=2p2p2pm=2242mq=222qm=2The First Crisis in History of MathematicsHippasuss fatewhat is the relation between irrationals and ra

17、tionals?Fibonacci sequenceFibonacci(1 1 7 5-1 2 5 0)was an Italianmathematician.He popularized the HinduArabic numeral system in the Western Worldprimarily through his composition in 1202of Liber Abaci.He also introduced Europe tothe sequence of Fibonacci numbersFibonacci posed a most famous questio

18、n:How many pairs of rabbitswill be produced in a year,beginning with a single pair,if in everymonth each pair bears a new pair which becomes productive from thesecond month on?Fibonacci sequence:thesum of the preceding twois exactly the value of thelatter item.1，1，2，3，5，8，13，21，Fibonacci sequenceThe

19、 ratio of two adjacent numbers can be made into a new sequence:2 3 5 8 13,3 5 8 13 21The general term of the Fibonacci sequence is:the limit of it is .510.618211515()()225nnnu+=1111515()()5122limlim0.61821515()()22+=+nnnnnnnnuuThe development trend of the above numbers:Rational Sequences Approach an

20、 IrrationalThe sequence:We can understand irrational numbers through limit ofrationals.3 5 8 13,2 3 58.that makes sequence,there is definitely a rational For any irrational number 251I nulimnnuI=Irrational Sequences Approach a RationalThe relationship between the rational numbers and the irrationaln

21、umbers is very close.For any rational number R,there is definitely an irrational sequence 123,1,2,3,nnv nv v vv=that makes limnnvR=Why does everyone like pentagrams?What is the optimum seeking method?Why does the five-pointed star become a symbol respected and admired by many nations and countries?H

22、ow does the beauty of ratio quietly find her way into peoples subconscious?Optimum Seeking MethodsMr.Hua Luogeng:the founder of modern Chinese mathematics.He was the founder and pioneer of Chinese analytic number theory,matrix geometry,classical group,automorphic function,the functions of several co

23、mplex variables and other areas and theories.He was praised as the Father of Modern Chinese Mathematics.Optimum Seeking Method How to define the carbon content of steel?The higher the carbon content is,the higher the strengthof steel will be,but the toughness decreases.Conversely,when the carbon con

24、tent of steel is low,the strength ofsteel is low,but its toughness is high.Problem:How much the amount of carbon should be from the range of 1000 grams to 2000 grams?Experiment Method of exhaustion:Adding carbon from 1000g,1001g,1002g,till 2000g.It might work out in textbooks,but it is obviously not

25、 possible in reality.So,the amount of carbon depends entirely on the use of steel.Optimum seeking method in experimentationFind a paper strip to express 1000-2000g of carbon，and then,find two golden pointsof this strip:C point and D point,corresponding to the carbon content of 1,382g and1,618g.First

26、ly,two experiments will be carried out on these two carbon contents.If the testresult based on 1,382g is close to the requirement,you can tear off the end of 1,618g;conversely,if 1,618g is better,you can tear off the left end marking 1,382g.Next,you need to find the two golden points in the remainin

27、g strip,and then keep continuing to experiment.A（1000）C（1382）D（1618）B（2000）A very interesting nature of the golden sectionInteresting nature:Assume that point C and D are two golden points of line segment AB.When you remove the line AC in the line AB,the remaining golden point D happens to be the go

28、lden point of the remaining line CB.215=CBDB253=DB215=CBThe point D happens to be the golden point of CB.1ACBD,Golden ratio of rifle In 1918,an American corporal in the Expeditionarymade transformations to the rifle.The length of the gunwas increased and it was convenient to aim and shoot.Later,it w

29、as discovered that the ratio of the body to the handle of theimproved rifle was exactly the golden ratio.Natural beauty brought by the golden ratioThe magical effect of the golden ratioThere is always an irresistible beauty by using this ratio in their design.A typical examplefive-pointed star The g

30、olden ratio in five-pointed stardividing points36 degrees0618There are two dividing points in one side of a five-pointed star cut by the other two sides.These two dividing points happen to be the golden section points of the side.This ratio is 2sin18.The Golden Ratio in the ArchitecturesThe ratio：81

31、30.6154 Where is the mystery of golden sections?Laozis doctrine and The beauty of RatioWhat kind of cornerstone of philosophy did Laozis doctrine provide for the beauty of ratio?What is the mathematical connotation of the“degree”we often say?How does ballet show peoples new understanding of human fo

32、rm?Symmetrical Beauty Based on The NatureTwo symmetrical legsTrees grow symmetricallyInvariance in SymmetryEmmy Noether(1882-1935):German female mathematician,whose research fields are abstract algebra and theoretical physics.She has made outstanding contributions to the theory of rings,fields and a

33、lgebras.One of the key ideas she introduced is now knownas Noethers theorem:every symmetry in the laws of nature corresponds to a conservation.isosceles trianglesbilaterally symmetricalSpatial Translation Symmetry Isnt the spatial translational symmetry eternal?play billiardsAsymmetry Two famous phy

34、sicists Yang Zhenning and Li Zhengdao pointed out thatin the microscopic world,for two identical particles in a weaklyinteracting environment,their motion laws are not necessarily identical.Later,the famous scientist Wu Jianxiong affirmed this view withexperiments.In the dynamic world of biology,“as

35、ymmetry”is absolute,due to the asymmetrical molecularstructure of protoplasm within the cell.Asymmetry The asymmetry imposes great effects on thevitality and continuation of our lives and hasan important significance for the evolution oflife.Further studies have shown that the asymmetrical molecular

36、 structure ismore active than a symmetric molecular structure,and the metabolismof the asymmetrical one is more active.Symmetry and AsymmetryScenery of photography:the subject of the photography is placed in the middle,occupying the majority of the photo;the background is occupied in the secondary p

37、lace,occupying a relatively small part.changes in aesthetic orientation.Symmetry:often based on the need for balance;Asymmetry:to lead to the development and change;symmetry makes things present a static beautyasymmetry gives things a dynamic sense of return.The Rule of Thirds in photographyScenery

38、in the middleThe Rule of ThirdsLaozis PhilosophyTao Te ChingThe change between the strong and the weakoccurs，which results in movement.yin and yangThe Relation between Strongness and WeaknessUsing the male,being female,Using strength,being weak,Using the light,being dark,Dialectic of strength and we

39、akness Laozi mentioned that,“Fill a cup to its brim and it is easily spilled;Temper a sword to its hardest and it is easily broken;Amass the greatesttreasure and it is easily stolen;”Laozi lived in the Spring and AutumnPeriod.At that time,some vassal stateswere engaged in wars.Laozi said that“So whe

40、n a large country submits to a small country,itwill adopt the small country;when a small country submits to a largecountry,it will be adopted by the large country.”Which is bigger:ab0ba1The line segment:a larger part and a smaller part.The Big and The SmallLaozis philosophical point of view:the big

41、should be modest,the small should not be afraid of the power of the big.，b a abbab=+According to Laozis point of view:being the big,keeping the small.Moderation:to ensure the internal balance in things development and change.The golden section is still a balance,which is an asymmetricalbalance,a bal

42、ance in dynamic development.Golden RatioGolden RatioAnother BalanceBalletAs a most elegant dance,ballet was born in the Italian Renaissance,later developed in France and popularized in the world.Ballet has a remarkable dance movement,that is,the actress has tostand en pointe.Thus,ballet is also call

43、ed toe dance.The ballet Swan Lake is about the beautiful love betweenthe prince and the princess.When we talk about ballet,wecan still remember that the four little swans jumped enpointe along with the bright and compact melody composedby Tchaikovsky.？Questions：You can never see anyone walking on th

44、eir toes in real life.Infact,standing en pointe has subverted the walking habits of human beings.The Beauty of Human Body Ratio Perfect human bodyStanding en pointe is a kind of salvation of body formwithout the help of any external force,a pursuit and yearningof beauty,based on ratio.Wear high-heeled shoesStanding en pointeGolden Ratio