The Golden Ratio, Fibonacci Numbers, And Other RecurrencesLeonardo FibonacciInductive Definition or Recurrence Relation for the Fibonacci NumbersSneezwort (Achilleaptarmica)Counting PetalsPineapple whorlsSlide 7Bernoulli Spiral When the growth of the organism is proportional to its sizeSlide 9Slide 10Slide 11Golden Ratio: the divine proportionDefinition of (Euclid)Expanding RecursivelyContinued Fraction RepresentationSlide 17Remember?Slide 191,1,2,3,5,8,13,21,34,55,….Continued fraction representation of a standard fractione.g., 67/29 = 2 with remainder 9/29 = 2 + 1/ (29/9)A Representational CorrespondenceEuclid’s GCD = Continued FractionsSlide 25Sequences That Sum To nSlide 27Slide 28Slide 29Slide 30Slide 31Fibonacci Numbers AgainVisual Representation: TilingSlide 34Slide 35fn+1 = fn + fn-1Slide 37Fibonacci IdentitiesFm+n+1 = Fm+1 Fn+1 + Fm Fn(Fn)2 = Fn-1 Fn+1 + (-1)nSlide 41Slide 42Slide 43Slide 44Slide 45Slide 46(Fn)2 = Fn-1 Fn+1 + (-1)n-1More random factsThe Fibonacci QuarterlySlide 50How to divide polynomials?The Geometric SeriesSlide 53The Infinite Geometric SeriesSlide 55Slide 56Something a bit more complicatedHenceGoing the Other WaySlide 60ThusSlide 62Slide 63Slide 64Linear factors on the bottomGeometric Series (Quadratic Form)Slide 67Power Series Expansion of FSlide 69Slide 70Slide 71Slide 72Slide 73Slide 74Slide 75Slide 76Slide 77Some simple power seriesAl-Karaji’s IdentitiesSlide 80Slide 81Slide 82Slide 83Slide 84Coefficient of Xk in PV = (X2+X)(1-X)-4 is the sum of the first k squares:Polynomials give us closed form expressionsREFERENCESStudy BeeThe Golden Ratio, Fibonacci Numbers, And Other RecurrencesGreat Theoretical Ideas In Computer ScienceJohn Lafferty CS 15-251 Fall 2006Lecture 13 October 10, 2006 Carnegie Mellon UniversityLeonardo FibonacciIn 1202, Fibonacci proposed a problem about the growth of rabbit populations.Inductive Definition or Recurrence Relation for theFibonacci Numbers Stage 0, Initial Condition, or Base Case:Fib(0) = 0; Fib (1) = 1Inductive RuleFor n>1, Fib(n) = Fib(n-1) + Fib(n-2)n 0 1 2 3 4 5 6 7Fib(n) 0 1 1 2 3 5 813Sneezwort (Achilleaptarmica) Each time the plant starts a new shoot it takes two months before it is strong enough to support branching.Counting Petals5 petals: buttercup, wild rose, larkspur,columbine (aquilegia) 8 petals: delphiniums 13 petals: ragwort, corn marigold, cineraria,some daisies 21 petals: aster, black-eyed susan, chicory 34 petals: plantain, pyrethrum 55, 89 petals: michaelmas daisies, theasteraceae family.Pineapple whorlsChurch and Turing were both interested in the number of whorls in each ring of the spiral. The ratio of consecutive ring lengths approaches the Golden Ratio.Bernoulli Spiral When the growth of the organism is proportional to its sizeBernoulli Spiral When the growth of the organism is proportional to its sizeIs there life after andeGolden Ratio: the divine proportion = 1.6180339887498948482045…“Phi” is named after the Greek sculptor Phidias ACABABBCACBCACBCABBCBCBC11 0222Definition of (Euclid)Ratio obtained when you divide a line segment into two unequal parts such that the ratio of the whole to the larger part is the same as the ratio of the larger to the smaller.A B CExpanding Recursively 111111111111Continued Fraction Representation 1111111111111111111 ....Continued Fraction Representation1111111111111111111 ....1 52= +++++++++++RememberWe already saw the convergents of this CF[1,1,1,1,1,1,1,1,1,1,1,…]are of the formFib(n+1)/Fib(n)n11 5l m2i��-+=f =nnFFHence:Continued Fraction Representation 1111111111111111111 ....1,1,2,3,5,8,13,21,34,55,….2/1 = 23/2 = 1.55/3 = 1.666…8/5 = 1.613/8 = 1.62521/13 = 1.6153846…34/21 = 1.61904… = 1.6180339887498948482045Continued fraction representation of a standard fraction67 121293142= +++e.g., 67/29 = 2 with remainder 9/29= 2 + 1/ (29/9)67 1 1 12 2 229 2 1293 319 942= + = + ++ ++A Representational Correspondence67 1 1 12 2 229 2 1293 319 942= + = + ++ ++Euclid(67,29) 67 div 29 = 2Euclid(29,9) 29 div 9 = 3Euclid(9,2) 9 div 2 = 4Euclid(2,1) 2 div 1 = 2Euclid(1,0)Euclid’s GCD = Continued FractionsEuclid(A,B) = Euclid(B, A mod B)Stop when B=01modA ABB BA B� �= +� �� �Theorem: All fractions have finite continuous fraction expansionsLet us take a slight detour and look at a different representation.Sequences That Sum To nLet fn+1 be the number of different sequences of 1’s and 2’s that sum to n.Example: f5 = 5Sequences That Sum To nLet fn+1 be the number of different sequences of 1’s and 2’s that sum to n.Example: f5 = 54 = 2 + 22 + 1 + 11 + 2 + 11 + 1 + 21 + 1 + 1 + 1Sequences That Sum To nf1f2f3Let fn+1 be the number of different sequences of 1’s and 2’s that sum to n.Sequences That Sum To nf1 = 1 0 = the empty sumf2 = 1 1 = 1f3 = 2 2 = 1 + 1 2Let fn+1 be the number of different sequences of 1’s and 2’s that sum to n.Sequences That Sum To nfn+1 = fn + fn-1Let fn+1 be the number of different sequences of 1’s and 2’s that sum to n.Sequences That Sum To nfn+1 = fn + fn-1Let fn+1 be the number of different sequences of 1’s and 2’s that sum to n.# of sequences beginning with a 2# of sequences beginning with a 1Fibonacci Numbers Againffn+1n+1 = f = fnn + f + fn-1n-1ff11 = 1 f = 1 f22 = 1 = 1Let fn+1 be the number of different sequences of 1’s and 2’s that sum to n.Visual Representation: TilingLet fn+1 be the number of different ways to tile a 1 × n strip with squares and dominoes.Visual Representation: TilingLet fn+1 be the number of different ways to tile a 1 × n strip with squares and dominoes.Visual Representation: Tiling1 way to tile a strip of length 01 way to tile a strip of length 1:2 ways to tile a strip of length 2:fn+1 = fn + fn-1fn+1 is number of ways to tile length n.fn tilings that start with a square.fn-1 tilings that start with a domino.Let’s use this visual representation to prove a couple of Fibonacci identities.Fibonacci IdentitiesSome examples:F2n = F1 + F3 + F5 + … + F2n-1Fm+n+1 = Fm+1 Fn+1 + Fm Fn(Fn)2 = Fn-1 Fn+1 + (-1)nFm+n+1 = Fm+1 Fn+1 + Fm Fnmmnnm-1m-1n-1n-1(Fn)2 = Fn-1 Fn+1 + (-1)n(Fn)2 = Fn-1 Fn+1 + (-1)nn-1n-1Fn tilings of a strip of length n-1(Fn)2 = Fn-1 Fn+1 +
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