Great Theoretical Ideas in Computer Science
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1 Great Theoretical Ideas in Computer Science
2 Recurrences, Fibonacci Numbers and Continued Fractions Lecture 9, September 23, 2008
3 Happy Autumnal Equinox
4 Leonardo Fibonacci
5 Leonardo Fibonacci In 1202, Fibonacci proposed a problem about the growth of rabbit populations
6 Rabbit Reproduction
7 Rabbit Reproduction A rabbit lives forever
8 Rabbit Reproduction A rabbit lives forever The population starts as single newborn pair
9 Rabbit Reproduction A rabbit lives forever The population starts as single newborn pair Every month, each productive pair begets a new pair which will become productive after 2 months old
10 Rabbit Reproduction A rabbit lives forever The population starts as single newborn pair Every month, each productive pair begets a new pair which will become productive after 2 months old F n = # of rabbit pairs at the beginning of the n th month
11 month rabbits Rabbit Reproduction A rabbit lives forever The population starts as single newborn pair Every month, each productive pair begets a new pair which will become productive after 2 months old F n = # of rabbit pairs at the beginning of the n th month
12 month rabbits Rabbit Reproduction A rabbit lives forever The population starts as single newborn pair Every month, each productive pair begets a new pair which will become productive after 2 months old F n = # of rabbit pairs at the beginning of the n th month 1
13 month rabbits Rabbit Reproduction A rabbit lives forever The population starts as single newborn pair Every month, each productive pair begets a new pair which will become productive after 2 months old F n = # of rabbit pairs at the beginning of the n th month 1 1
14 month rabbits Rabbit Reproduction A rabbit lives forever The population starts as single newborn pair Every month, each productive pair begets a new pair which will become productive after 2 months old F n = # of rabbit pairs at the beginning of the n th month 1 1 2
15 month rabbits Rabbit Reproduction A rabbit lives forever The population starts as single newborn pair Every month, each productive pair begets a new pair which will become productive after 2 months old F n = # of rabbit pairs at the beginning of the n th month
16 month rabbits Rabbit Reproduction A rabbit lives forever The population starts as single newborn pair Every month, each productive pair begets a new pair which will become productive after 2 months old F n = # of rabbit pairs at the beginning of the n th month
17 month rabbits Rabbit Reproduction A rabbit lives forever The population starts as single newborn pair Every month, each productive pair begets a new pair which will become productive after 2 months old F n = # of rabbit pairs at the beginning of the n th month
18 month rabbits Rabbit Reproduction A rabbit lives forever The population starts as single newborn pair Every month, each productive pair begets a new pair which will become productive after 2 months old F n = # of rabbit pairs at the beginning of the n th month
19 Fibonacci Numbers month rabbits
20 Fibonacci Numbers month rabbits Stage 0, Initial Condition, or Base Case: Fib(1) = 1; Fib (2) = 1
21 Fibonacci Numbers month rabbits Stage 0, Initial Condition, or Base Case: Fib(1) = 1; Fib (2) = 1 Inductive Rule: For n>3, Fib(n) =
22 Fibonacci Numbers month rabbits Stage 0, Initial Condition, or Base Case: Fib(1) = 1; Fib (2) = 1 Inductive Rule: For n>3, Fib(n) = Fib(n-1) + Fib(n-2)
23 Sequences That Sum To n
24 Sequences That Sum To n Let f n+1 be the number of different sequences of 1 s and 2 s that sum to n.
25 Sequences That Sum To n Let f n+1 be the number of different sequences of 1 s and 2 s that sum to n. f 1 = 1
26 Sequences That Sum To n Let f n+1 be the number of different sequences of 1 s and 2 s that sum to n. f 1 = 1 0 = the empty sum
27 Sequences That Sum To n Let f n+1 be the number of different sequences of 1 s and 2 s that sum to n. f 1 = 1 0 = the empty sum f 2 = 1
28 Sequences That Sum To n Let f n+1 be the number of different sequences of 1 s and 2 s that sum to n. f 1 = 1 0 = the empty sum f 2 = 1 1 = 1
29 Sequences That Sum To n Let f n+1 be the number of different sequences of 1 s and 2 s that sum to n. f 1 = 1 0 = the empty sum f 2 = 1 1 = 1 f 3 = 2
30 Sequences That Sum To n Let f n+1 be the number of different sequences of 1 s and 2 s that sum to n. f 1 = 1 0 = the empty sum f 2 = 1 1 = 1 f 3 = 2 2 =
31 Sequences That Sum To n Let f n+1 be the number of different sequences of 1 s and 2 s that sum to n. 4 =
32 Sequences That Sum To n Let f n+1 be the number of different sequences of 1 s and 2 s that sum to n. 4 =
33 Sequences That Sum To n Let f n+1 be the number of different sequences of 1 s and 2 s that sum to n.
34 Sequences That Sum To n Let f n+1 be the number of different sequences of 1 s and 2 s that sum to n. f n+1 = f n + f n-1
35 Sequences That Sum To n Let f n+1 be the number of different sequences of 1 s and 2 s that sum to n. f n+1 = f n + f n-1 # of sequences beginning with a 1 # of sequences beginning with a 2
36 Fibonacci Numbers Again Let f n+1 be the number of different sequences of 1 s and 2 s that sum to n. f n+1 = f n + f n-1 f 1 = 1 f 2 = 1
37 Visual Representation: Tiling Let f n+1 be the number of different ways to tile a 1 n strip with squares and dominoes.
38 Visual Representation: Tiling 1 way to tile a strip of length 0 1 way to tile a strip of length 1: 2 ways to tile a strip of length 2:
39 f n+1 = f n + f n-1 f n+1 is number of ways to tile length n. f n tilings that start with a square. f n-1 tilings that start with a domino.
40 Fibonacci Identities
41 Fibonacci Identities Some examples:
42 Fibonacci Identities Some examples:
43 Fibonacci Identities Some examples: F 2n = F 1 + F 3 + F F 2n-1
44 Fibonacci Identities Some examples: F 2n = F 1 + F 3 + F F 2n-1
45 Fibonacci Identities Some examples: F 2n = F 1 + F 3 + F F 2n-1 F m+n+1 = F m+1 F n+1 + F m F n
46 Fibonacci Identities Some examples: F 2n = F 1 + F 3 + F F 2n-1 F m+n+1 = F m+1 F n+1 + F m F n
47 Fibonacci Identities Some examples: F 2n = F 1 + F 3 + F F 2n-1 F m+n+1 = F m+1 F n+1 + F m F n (F n ) 2 = F n-1 F n+1 + (-1) n
48 Fm+n+1 = Fm+1 Fn+1 + F m Fn
49 F m+n+1 = F m+1 F n+1 + F m F n m n
50 F m+n+1 = F m+1 F n+1 + F m F n m n m-1 n-1
51 (Fn)2 = Fn-1 Fn+1 + (-1)n
52 (F n ) 2 = F n-1 F n+1 + (-1) n n-1 F n tilings of a strip of length n-1
53 (F n ) 2 = F n-1 F n+1 + (-1) n n (F n ) 2 tilings of two strips of size n-1
54 (F n ) 2 = F n-1 F n+1 + (-1) n n Draw a vertical fault line at the rightmost position (<n) possible without cutting any dominoes
55 (F n ) 2 = F n-1 F n+1 + (-1) n n Swap the tails at the fault line to map to a tiling of 2 (n-1) s to a tiling of an n-2 and an n.
56 (F n ) 2 = F n-1 F n+1 + (-1) n-1 n even n odd
57 Sneezwort (Achilleaptarmica) Each time the plant starts a new shoot it takes two months before it is strong enough to support branching.
58 Counting Petals
59 Counting Petals 5 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, the asteraceae family.
60 The Fibonacci Quarterly
61 Definition of φ (Euclid)
62 Definition of φ (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.
63 Definition of φ (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 C
64 Definition of φ (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. φ = AC AB = AB BC A B C
65 Definition of φ (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. φ = AC AB = AB BC A B C φ 2 =
66 Definition of φ (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. φ = AC AB = AB BC A B C φ 2 = AC BC
67 Definition of φ (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. φ = AC AB = AB BC A B C φ 2 = AC BC φ 2 - φ =
68 Definition of φ (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. φ = AC AB φ 2 = AC BC = φ 2 - φ = AC BC AB BC AB - = BC A B C
69 Definition of φ (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. φ = AC AB φ 2 = AC BC = φ 2 - φ = AC BC AB BC A B C - AB BC = BC BC = 1
70 φ 2 φ 1 = 0
71 φ 2 φ 1 = 0 φ =
72
73 Golden ratio supposed to arise in Parthenon, Athens (400 B.C.) a b The great pyramid at Gizeh Ratio of a person s height to the height of his/her navel
74 Golden ratio supposed to arise in Parthenon, Athens (400 B.C.) a b The great pyramid at Gizeh Ratio of a person s height to the height of his/her navel Mostly circumstantial evidence
75 Expanding Recursively
76 Expanding Recursively
77 Continued Fraction Representation
78 A (Simple) Continued Fraction Is Any Expression Of The Form: where a, b, c, are whole numbers.
79 A Continued Fraction can have a finite or infinite number of terms. We also denote this fraction by [a,b,c,d,e,f, ]
80 A Finite Continued Fraction Denoted by [2,3,4,2,0,0,0, ]
81 An Infinite Continued Fraction Denoted by [1,2,2,2, ]
82 Recursively Defined Form For CF
83 Continued fraction representation of a standard fraction
84 e.g., 67/29 = 2 with remainder 9/29 = 2 + 1/ (29/9)
85 Ancient Greek Representation: Continued Fraction Representation
86 Ancient Greek Representation: Continued Fraction Representation = [1,1,1,1,0,0,0, ]
87 Ancient Greek Representation: Continued Fraction Representation
88 Ancient Greek Representation: Continued Fraction Representation = [1,1,1,1,1,0,0,0, ]
89 Ancient Greek Representation: Continued Fraction Representation = [1,1,1,1,1,1,0,0,0, ]
90 A Pattern? Let r 1 = [1,0,0,0, ] = 1 r 2 = [1,1,0,0,0, ] = 2/1 r 3 = [1,1,1,0,0,0 ] = 3/2 r 4 = [1,1,1,1,0,0,0 ] = 5/3 and so on. Theorem: r n = Fib(n+1)/Fib(n)
91 1,1,2,3,5,8,13,21,34,55,.
92 1,1,2,3,5,8,13,21,34,55,. 2/1 = 2
93 1,1,2,3,5,8,13,21,34,55,. 2/1 = 2 3/2 = 1.5
94 1,1,2,3,5,8,13,21,34,55,. 2/1 = 2 3/2 = 1.5 5/3 = 1.666
95 1,1,2,3,5,8,13,21,34,55,. 2/1 = 2 3/2 = 1.5 5/3 = /5 = 1.6
96 1,1,2,3,5,8,13,21,34,55,. 2/1 = 2 3/2 = 1.5 5/3 = /5 = /8 = 1.625
97 1,1,2,3,5,8,13,21,34,55,. 2/1 = 2 3/2 = 1.5 5/3 = /5 = /8 = /13 =
98 1,1,2,3,5,8,13,21,34,55,. 2/1 = 2 3/2 = 1.5 5/3 = /5 = /8 = /13 = /21 =
99 1,1,2,3,5,8,13,21,34,55,. 2/1 = 2 3/2 = 1.5 5/3 = /5 = /8 = /13 = /21 = φ =
100 Pineapple whorls Church 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.
101
102 Proposition: Any finite continued fraction evaluates to a rational. Theorem Any rational has a finite continued fraction representation.
103 Hmm. Finite CFs = Rationals. Then what do infinite continued fractions represent?
104 An infinite continued fraction
105 Quadratic Equations X 2 3x 1 = 0 X 2 = 3X + 1 X = 3 + 1/X X = 3 + 1/X = 3 + 1/[3 + 1/X] =
106 A Periodic CF
107 Theorem: Any solution to a quadratic equation has a periodic continued fraction. Converse: Any periodic continued fraction is the solution of a quadratic equation. (try to prove this!)
108 So they express more than just the rationals What about those non-recurring infinite continued fractions?
109 Non-periodic CFs
110 What is the pattern?
111 No one knows! What is the pattern?
112 What a cool representation! Finite CF: Rationals Periodic CF: Quadratic roots And some numbers reveal hidden regularity.
113 More good news: Convergents Let α = [a 1, a 2, a 3,...] be a CF. Define: C 1 = [a 1,0,0,0,0..] C 2 = [a 1,a 2,0,0,0,...] C 3 = [a 1,a 2,a 3,0,0,...] and so on. C k is called the k-th convergent of α α is the limit of the sequence C 1, C 2, C 3,
114 Best Approximator Theorem A rational p/q is the best approximator to a real α if no rational number of denominator smaller than q comes closer to α.
115 Best Approximator Theorem A rational p/q is the best approximator to a real α if no rational number of denominator smaller than q comes closer to α. BEST APPROXIMATOR THEOREM: Given any CF representation of α, each convergent of the CF is a best approximator for α!
116 Best Approximators of π C 1 = 3 C 2 = 22/7 C 3 = 333/106 C 4 = 355/113 C 5 = /33102 C 6 =104348/33215
117 Continued Fraction Representation
118 Continued Fraction Representation
119 Remember? We already saw the convergents of this CF [1,1,1,1,1,1,1,1,1,1,1, ] are of the form Fib(n+1)/Fib(n) Hence:
120 1,1,2,3,5,8,13,21,34,55,. 2/1 = 2 3/2 = 1.5 5/3 = /5 = /8 = /13 = /21 = φ =
121 As we ve seen...
122 Going the Other Way
123 What is the Power Series Expansion of z/(1-z-z 2 )? What does this look like when we expand it as an infinite sum?
124 Since the bottom is quadratic we can factor it:
125 Write as a sum of two terms where Solving:
126 Now use the geometric series
127
128 Leonhard Euler (1765) J. P. M. Binet (1843) A de Moivre (1730) The n th Fibonacci number is:
129
130
131 Recurrences and generating functions Golden ratio Continued fractions Convergents Here s What You Need to Know Closed form for Fibonaccis 74
132
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