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應 用 數 學 報 告

應 用 數 學 報 告. 通訊 3A B966C0008 鍾易珩. Q.28 what is the Fibonacci sequence?. 假設 fn 代表一個數列的第 n 項， (n=1.2.3..) n≧3. Divide each number by next unmber. Conversely. Simpler first-order difference equation fn=fn-1 with n≧1 f2=af1=a^2*fo , f3=af2=a^3*fo ingeneral fn=a^n*fo.

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應 用 數 學 報 告

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  1. 應 用 數 學 報 告 通訊3A B966C0008 鍾易珩

  2. Q.28 what is the Fibonacci sequence? 假設fn代表一個數列的第n項，(n=1.2.3..) n≧3

  3. Divide each number by next unmber Conversely

  4. Simpler first-order difference equation fn=fn-1 with n≧1 f2=af1=a^2*fo , f3=af2=a^3*fo ingeneral fn=a^n*fo

  5. Substitude λ^n & fn with roots Note that

  6. Combination two particular solutions A1,A2 are arbitrary constants -->

  7. Choose two term as ”initial conditions” “fo” be placed beginning fo=0,f1=1 following two equations final->

  8. if 14th Fibonacci number is 377 τ^14/√ 5=377.0006 But Fibonacci number are integers

  9. Golden ratio From(28.6) Greek referred If AC=τunits & CB=1units length τ=(1+√ 5)/2 1/ τ= τ-1=1.61803398

  10. A Fibonacci Digression f95=31940434634990099905 ≒3.2*10^19 Insect=10^18 Rubik’s cube=4.3*10^19 1.2.3.4.5=>946383. 946384.946385.. Ex: 792468247 -------3 seconds Ex: 792468247139 ---5 or 6 second

  11. Q.29 so what is the “golden angle”? In a circle line segment ACB =1 Ramain arc=τ arc BA=2π-α to (28.9) Smallest root ≒137.507°

  12. Q.30 why are the angles between leaves ”just so”? “phyllotaxis” ->Related to Fibonacci number,golden ratio Ex: petals lilies (百合花)=3片 buttercups(金鳳花)=5片 delphiniums(飛燕草)=8片 marigolds(金盞菊)=13片 asters(紫菀)=21片 daisies(雛菊)=34片…55….89

  13. H.S.M Coxeter <introduce to geometry> “phyllotaxis is really not a universal law but only a fascinatingly prevalent tendency” Angular intervals =>rational multiples of 360° In practice,plants seem to choose rational approximations to the ”most irrational”number to optimize leaf arrangement.

  14. Depend on plant Leaves are generated approximately 2/5 of a revolution—oak(橡樹)cherry(櫻桃樹) apple(蘋果樹) holly(冬青) plum tree(梅子樹) 1/2—elm(榆樹)grasses(某些青草)lime(酸橙)ilnden(菩提樹) 1/3—beech(山毛櫸)hazel(榛樹) 3/8—poplar(白楊)rose(玫瑰)pear(梨樹)willow(柳樹) 5/13—almond(杏樹)….3/5,8/13,5/13.. Called ”phyllotactic ratios” Ex: 3/8 phyllotactic ratios 8 stems are generated in 3 complete turns.

  15. Distributed two familiess pirals 34 clockwise & 55 counterclockwise Or (55.89) (89.144) Ex: seed of sunflower head

  16. “Brousseau” examine pinecones. Two set of parallel bract spirals. A steep one lower right to upper left Shallower one lower left to upper right. 8 of former ,5 of latter Or (3.5) (8.13) Ex: seed of daisies head

  17. Q:how to these spirals in the first place Ian Stewart <Life’s other secret> “apex” at the center of tip small circular region “primordia” around the apex from small lumps Spirals-the parastichies- Are merely an event of plant’s growth pattern

  18. Golden angles Apex center , primordial also angle ≒137.5° called “divergence angle” Related to “golden ratio τ” If take successtive number ratio of Fibonacci sequence , mutiply by 360°,then substract from answer 360° ==>137.50776

  19. Seed appear gaps While less than 137.5° , and only clockwise Again if angle is more than 137.5° But this time only counterclockwise Seen. ->only angle which seeds pack together without gaps,angel both spirals occur.

  20. End Theodore cook<first book> ”plants aiming Fibonacci angle which will give minimum superposition and maximum exposure..” Ex:given p,q =>integers If (360*p)/q =>radial lines (irrational) Conversely also Follow “Stewart” discussed If a “most irrational” ->Its turn out a “golden number”

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