1. WELCOME Seena.V Assistant Professor Department of Mathematics

2. Fibonacci Numbers And Golden Ratio

3. Who Was Fibonacci?

5. About the Origin of Fibonacci Sequence

6. Fibonacci Sequence was discovered after an investigation on the reproduction of rabbits.

7. Fibonacci’s Rabbits Problem: Suppose a newly-born pair of rabbits (one male, one female) are put in a field. Rabbits are able to mate at the age of one month so that at the end of its second month, a female can produce another pair of rabbits. Suppose that the rabbits never die and that the female always produces one new pair (one male, one female) every month from the second month on. How many pairs will there be in one year?

8. Pairs 1 pair At the end of the first month there is still only one pair

9. Pairs 1 pair 1 pair 2 pairs End first month… only one pair At the end of the second month the female produces a new pair, so now there are 2 pairs of rabbits

10. Pairs 1 pair 1 pair 2 pairs 3 pairs End second month… 2 pairs of rabbits At the end of the third month, the original female produces a second pair, making 3 pairs in all in the field. End first month… only one pair

11. Pairs 1 pair 1 pair 2 pairs 3 pairs End third month… 3 pairs 5 pairs End first month… only one pair End second month… 2 pairs of rabbits At the end of the fourth month, the first pair produces yet another new pair, and the female born two months ago produces her first pair of rabbits also, making 5 pairs.

13. So 144 Pairs will be there at the end of One Year….

14. Fibonacci sequence in Nature

18. Spirals seen in the arrangement of seeds in the head of this sunflower number 34 in a counterclockwise direction

19. and 55 in a clockwise direction

20. Note that 34 and 55 are the ninth and tenth Fibonacci numbers respectively .

21. Also note that The flower itself has 34 petals.

26. Note that 8 and 13 are Consecutive Fibonacci numbers

30. The Fibonacci numbers can be found in pineapples and bananas. Bananas have 3 or 5 flat sides, Pineapple scales have Fibonacci spirals in sets of 8, 13, 21

32. The Golden Ratio

33. The golden ratio is an irrational mathematical constant, approximately equals to 1.6180339887

34. The golden ratio is often denoted by the Greek letter φ (Phi) So φ = 1.6180339887

37. a b a+b a+b a = a b = φ

38. φ = 1+ √5 2 = 1.618

39. One interesting thing about Phi is its reciprocal 1/ φ = 1/1.618 = 0.618 . It is highly unusual for the decimal integers of a number and its reciprocal to be exactly the same.

40. A golden rectangle is a rectangle where the ratio of its length to width is the golden ratio. That is whose sides are in the ratio 1:1.618

41. The golden rectangle has the property that it can be further subdivided in to two portions a square and a golden rectangle This smaller rectangle can similarly be subdivided in to another set of smaller golden rectangle and smaller square. And this process can be done repeatedly to produce smaller versions of squares and golden rectangles

42. Golden Rectangle

45. Relation between Fibonacci Sequence and Golden ratio

46. Aha! Notice that as we continue down the sequence, the ratios seem to be converging upon one number (from both sides of the number)! 2/1 = 2.0 (bigger) 3/2 = 1.5 (smaller) 5/3 = 1.67 (bigger) 8/5 = 1.6 (smaller) 13/8 = 1.625 (bigger) 21/13 = 1.615 (smaller) 34/21 = 1.619 (bigger) 55/34 = 1.618 (smaller) 89/55 = 1.618 The Fibonacci sequence is 1,1,2,3,5,8,13,21,34,55,….

47. If we continue to look at the ratios as the numbers in the sequence get larger and larger the ratio will eventually become the same number, and that number is the Golden Ratio !

48. 1 1 2 3 1.5000000000000000 5 1.6666666666666700 8 1.6000000000000000 13 1.6250000000000000 21 1.6153846153846200 34 1.6190476190476200 55 1.6176470588235300 89 1.6181818181818200 144 1.6179775280898900 233 1.6180555555555600 377 1.6180257510729600 610 1.6180371352785100 987 1.6180327868852500 1,597 1.6180344478216800 2,584 1.6180338134001300 4,181 1.6180340557275500 6,765 1.6180339631667100 10,946 1.6180339985218000 17,711 1.6180339850173600 28,657 1.6180339901756000 46,368 1.6180339882053200 75,025 1.6180339889579000

49. Golden ratio in Nature

50. Nautilus Shell

51. Golden ratio in Art Many artists who lived after Phidias have used this proportion. Leonardo Da Vinci called it the "divine proportion" and featured it in many of his paintings

53. Golden Ratio in the Human Body

54. Golden Ratio in Fingers

55. Golden Ratio in Hands

59. Golden Mean Gauge

61. More Examples of Golden Sections

71. Thank you