A Number You Can Call Your Own Great Theoretical Ideas In Computer Science A Number You Can Call Your Own Is there life after p and e? Lecture 5 CS 15-251

1.6180339887498948482045…..

Khufu 2589-2566 B.C. 2,300,000 blocks averaging 2.5 tons each

Great Pyramid at Gizeh

The ratio of the altitude of a face to half the base

Golden Ratio Divine Proportion  = 1.6180339887498948482045… “Phi” is named after the Greek sculptor Phidias

Parthenon, Athens (400 B.C.)

Pentagon

Ratio of height of the person to height of a person’s navel

Definition 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 C

Definition 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.

Profound Truths

Expanding Recursively

Expanding Recursively

Expanding Recursively

Expanding Recursively

Divina Proportione Luca Pacioli (1509) Pacioli devoted an entire book to the marvelous properties of . The book was illustrated by a friend of his named: Leonardo Da Vinci

Table of contents The first considerable effect The second essential effect The third singular effect The fourth ineffable effect The fifth admirable effect The sixth inexpressible effect The seventh inestimable effect The ninth most excellent effect The twelfth incomparable effect The thirteenth most distinguished effect

“For the sake of our salvation this list of effects must end.” Table of contents “For the sake of our salvation this list of effects must end.”

Aesthetics  plays a central role in renaissance art and architecture. After measuring the dimensions of pictures, cards, books, snuff boxes, writing paper, windows, and such, psychologist Gustav Fechner claimed that the preferred rectangle had sides in the golden ratio (1871).

Which is the most attractive rectangle?

Which is the most attractive rectangle?  1 Golden Rectangle

Bernoulli Spiral When the growth of the organism is proportional to its size

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.

Leonardo Fibonacci In 1202, Fibonacci proposed a problem about the growth of rabbit populations.

Leonardo Fibonacci In 1202, Fibonacci proposed a problem about the growth of rabbit populations.

Leonardo Fibonacci In 1202, Fibonacci proposed a problem about the growth of rabbit populations.

The rabbit reproduction model A rabbit lives forever The population starts as a single newborn pair Every month, each productive pair begets a new pair which will become productive after 2 months old Fn= # of rabbit pairs at the beginning of the nth month month 1 2 3 4 5 6 7 rabbits 8 13

The rabbit reproduction model A rabbit lives forever The population starts as a single newborn pair Every month, each productive pair begets a new pair which will become productive after 2 months old Fn= # of rabbit pairs at the beginning of the nth month month 1 2 3 4 5 6 7 rabbits 8 13

The rabbit reproduction model A rabbit lives forever The population starts as a single newborn pair Every month, each productive pair begets a new pair which will become productive after 2 months old Fn= # of rabbit pairs at the beginning of the nth month month 1 2 3 4 5 6 7 rabbits 8 13

The rabbit reproduction model A rabbit lives forever The population starts as a single newborn pair Every month, each productive pair begets a new pair which will become productive after 2 months old Fn= # of rabbit pairs at the beginning of the nth month month 1 2 3 4 5 6 7 rabbits 8 13

The rabbit reproduction model A rabbit lives forever The population starts as a single newborn pair Every month, each productive pair begets a new pair which will become productive after 2 months old Fn= # of rabbit pairs at the beginning of the nth month month 1 2 3 4 5 6 7 rabbits 8 13

The rabbit reproduction model A rabbit lives forever The population starts as a single newborn pair Every month, each productive pair begets a new pair which will become productive after 2 months old Fn= # of rabbit pairs at the beginning of the nth month month 1 2 3 4 5 6 7 rabbits 8 13

The rabbit reproduction model A rabbit lives forever The population starts as a single newborn pair Every month, each productive pair begets a new pair which will become productive after 2 months old Fn= # of rabbit pairs at the beginning of the nth month month 1 2 3 4 5 6 7 rabbits 8 13

Fn is called the nth Fibonacci number month 1 2 3 4 5 6 7 rabbits 8 13 F1=1, F2=1, and Fn=Fn-1+Fn-2 for n3

Fn is called the nth Fibonacci number month 1 2 3 4 5 6 7 rabbits 8 13 F1=1, F2=1, and Fn=Fn-1+Fn-2 for n3 Fn is defined by a recurrence relation. What is a closed form formula for Fn?

Techniques we have seen so far: Guess and verify Guess the form and solve for the coefficients Decorate the tree

Leonhard Euler (1765)

Less than .277

1,1,2,3,4,8,13,21,34,55,…. 2/1 = 2 3/2 = 1.5 5/3 = 1.666… 8/5 = 1.6 13/8 = 1.625 21/13= 1.6153846… 34/21= 1.61904…  = 1.6180339887498948482045

A technique to derive the formula for the Fibonacci numbers Fn is defined by two conditions: Base condition: F0=0, F1=1 Inductive condition: Fn=Fn-1+Fn-2

Consider solutions of the form: Fn= cn for some complex constant c Forget the base condition and concentrate on satisfying the inductive condition Inductive condition: Fn=Fn-1+Fn-2 Consider solutions of the form: Fn= cn for some complex constant c C must satisfy: cn - cn-1 - cn-2 = 0

cn - cn-1 - cn-2 = 0 iff cn-2(c2 - c1 - 1) = 0 iff c=0 or c2 - c1 - 1 = 0 Iff c = 0, c = , or c = -(1/)

ROTTEN LUCK c = 0, c = , or c = -(1/) So for all these values of c the inductive condition is satisfied: cn - cn-1 - cn-2 = 0 Do any of them happen to satisfy the base condition as well? c0=0 and c1=1? ROTTEN LUCK

(a g(n) + b h(n)) + (a g(n-1) + b h(n-1)) + (a g(n-2) + b h(n-2)) = 0 Insight: if 2 functions g(n) and h(n) satisfy the inductive condition then so does a g(n) + b h(n) for all complex a and b g(n)-g(n-1)-g(n-2)=0 ag(n)-ag(n-1)-ag(n-2)=0 h(n)-h(n-1)-h(n-2)=0 bh(n)-bh(n-1)-bh(n-2)=0 (a g(n) + b h(n)) + (a g(n-1) + b h(n-1)) + (a g(n-2) + b h(n-2)) = 0

a,b a n + b (-1/ )n satisfies the inductive condition Adjust a and b to fit the base conditions. n=0: a+b = 0 n=1: a 1 + b (-1/ )1 = 1 a= 1/5 b= -1/5

We have just proved Euler’s result:

Sneezwort (Achilleaptarmica) Each time the plant starts a new shoot it takes two months before it is stong enough to support branching.

Euclid’s GCD algorithm EUCLID(A,B) *\ requires AB0 If B=0 then Return A else Return Euclid( B, A mod B)

Euclid’s GCD algorithm EUCLID(A,B) *\ requires AB0 If B=0 then Return A else Return Euclid(B, A mod B) T(m) = the largest number of recursive calls that Euclid makes on any input pair with B=m

Lamé 1845 with equality when m is a Fibonacci number

Lemma: Euclid(Fk+1,Fk) makes k recursive calls Euclid(Fk+1,Fk) will call … Euclid(Fk,Fk-1) will call … Euclid(Fk-1,Fk-2) will call … . Euclid(F2,F1) will call … Euclid(F1,F0)

Lemma: Euclid(Fk+1,Fk) makes k recursive calls Corollary: T(Fk)  k Corollary: when m is Fibonacci

Lemma: Euclid(Fk+1,Fk) makes k recursive calls We now want to prove the converse: Euclid(A,B) makes k calls  A  Fk+1 and B  Fk

 k  2, Euclid(A,B) makes k calls  A  Fk+1 and B  Fk Base: k=2 B > 0 since EUCLID doesn’t halt right away. B  1 and A  2 B  F2 A  F3

 k  2, Euclid(A,B) makes k calls  A  Fk+1 and B  Fk Assume we have proved the hypothesis up to k-1 K>2 means EUCLID(A, B) will call EUCLID(B, A mod B) which will make k-1 recursive calls By induction: B  Fk and A mod B  Fk-1

 k  2, Euclid(A,B) makes k calls  A  Fk+1 and B  Fk By induction: B  Fk and A mod B  Fk-1 B + (A mod B)  Fk + Fk-1  Fk+1 A  B + (A mod B) Thus: A  Fk+1

Fibonacci Magic Trick

Another Trick!