1. 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

2. …..

3. Khufu
B.C.
2,300,000 blocks averaging 2.5 tons each

4. Great Pyramid at Gizeh

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

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

7. Parthenon, Athens (400 B.C.)

8. Pentagon

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

10. 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

11. 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.

12. Profound Truths

13. Expanding Recursively

14. Expanding Recursively

15. Expanding Recursively

16. Expanding Recursively

17. 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

18. 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

19. “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.”

20. 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).

21. Which is the most attractive rectangle?

22. Which is the most attractive rectangle? 1
Golden Rectangle

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

24. 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.

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

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

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

28. 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

29. 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

30. 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

31. 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

32. 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

33. 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

34. 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

35. 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

36. 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?

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

38. Leonhard Euler (1765)

39. Less than .277

41. 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= …
34/21= …
 =

43. 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

44. 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

45. 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/)

46. 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

47. (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

48. 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

49. We have just proved Euler’s result:

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

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

52. 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

53. Lamé 1845
with equality when m is a Fibonacci number

54. 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)

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

56. 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

57.  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

58.  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

59.  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

60. Fibonacci Magic Trick

61. Another Trick!