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The Golden Ratio: A Divine Proportion in Computer Science

Explore the profound truths and applications of the Golden Ratio in computer science, art, architecture, and nature. Learn about its historical significance, mathematical properties, and its influence on Fibonacci numbers.

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The Golden Ratio: A Divine Proportion in Computer Science

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  1. Great Theoretical Ideas In Computer Science A Number You Can Call Your Own Is there life afterp ande? Lecture 5 CS 15-251

  2. 1.6180339887498948482045…..

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

  4. Great Pyramid at Gizeh

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

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

  7. Pentagon

  8. 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

  9. 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.

  10. Profound Truths

  11. Expanding Recursively

  12. Expanding Recursively

  13. Expanding Recursively

  14. Expanding Recursively

  15. Divina ProportioneLuca 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

  16. 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

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

  18. 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).

  19. Which is the most attractive rectangle?

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

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

  22. 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.

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

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

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

  26. 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 • Fn= # of rabbit pairs at the beginning of the nth month

  27. 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 at 2 months old • Fn= # of rabbit pairs at the beginning of the nth month

  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 at 2 months old • Fn= # of rabbit pairs at the beginning of the nth month

  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 at 2 months old • Fn= # of rabbit pairs at the beginning of the nth month

  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 at 2 months old • Fn= # of rabbit pairs at the beginning of the nth month

  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 at 2 months old • Fn= # of rabbit pairs at the beginning of the nth month

  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 at 2 months old • Fn= # of rabbit pairs at the beginning of the nth month

  33. Fn is called the nth Fibonacci number F1=1, F2=1, and Fn=Fn-1+Fn-2 for n3

  34. Fn is called the nth Fibonacci number 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?

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

  36. Leonhard Euler (1765)

  37. Less than .277

  38. 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

  39. 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

  40. 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

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

  42. 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

  43. 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 • (a g(n) + b h(n)) + (a g(n-1) + b h(n-1)) + (a g(n-2) + b h(n-2)) = 0

  44. a,b a n + b (-1/ )nsatisfies 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

  45. We have just proved Euler’s result:

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

  47. 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

  48. Lamé 1845 • with equality when m is a Fibonacci number

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