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WELCOME Seena.V Assistant  Professor Department of Mathematics
Fibonacci Numbers And   Golden Ratio
Who Was Fibonacci?
[object Object],[object Object],[object Object]
About the  Origin of  Fibonacci Sequence
Fibonacci Sequence was discovered after an investigation on the  reproduction of rabbits.
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?
Pairs 1 pair At the end of the first month there is still only one pair
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
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
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.
Thus  We  get the following sequence of numbers :  1, 1, 2, 3, 5, 8, 13, 21, 34  ,55,89,144. ...   ,[object Object],[object Object],[object Object],[object Object],[object Object]
So  144  Pairs will be there  at the end of One Year….
Fibonacci sequence in   Nature
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Spirals seen in the arrangement of seeds in the head of this sunflower number  34  in a counterclockwise direction
and  55  in a clockwise direction
Note that 34 and 55 are the ninth and tenth Fibonacci numbers respectively .
Also note that The flower itself has 34 petals.
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Note that 8 and 13 are Consecutive Fibonacci numbers
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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
 
The  Golden Ratio
The  golden ratio  is an irrational mathematical constant, approximately  equals to 1.6180339887
The  golden ratio  is often denoted by the Greek letter  φ  (Phi) So  φ =  1.6180339887
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Two quantities are in the   golden ratio  if the ratio between  the sum of those quantities and  the larger one is the same as the ratio between the larger one and the smaller ,[object Object]
a b a+b a+b a =  a  b =  φ
φ = 1+ √5 2 =  1.618
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.
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
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
Golden Rectangle
Golden Spiral ,[object Object]
Golden Triangle ,[object Object]
Relation between Fibonacci Sequence and  Golden ratio
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,….
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 !
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
Golden ratio  in   Nature
Nautilus Shell
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
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Golden Ratio   in the  Human Body
Golden Ratio  in Fingers
Golden Ratio   in Hands
Golden ratio   in the Face ,[object Object],[object Object],[object Object],[object Object]
[object Object],[object Object]
Golden Ratio   in Human body ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Golden Mean Gauge
 
More Examples of  Golden Sections
 
 
 
 
 
 
 
 
 
Thank you

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Fibonacci Sequence and Golden Ratio

  • 1. WELCOME Seena.V Assistant Professor Department of Mathematics
  • 2. Fibonacci Numbers And Golden Ratio
  • 4.
  • 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.
  • 12.
  • 13. So 144 Pairs will be there at the end of One Year….
  • 15.
  • 16.  
  • 17.  
  • 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.
  • 22.
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  • 26. Note that 8 and 13 are Consecutive Fibonacci numbers
  • 27.
  • 28.  
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  • 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
  • 31.  
  • 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
  • 35.
  • 36.
  • 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
  • 43.
  • 44.
  • 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
  • 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
  • 52.
  • 53. Golden Ratio in the Human Body
  • 54. Golden Ratio in Fingers
  • 55. Golden Ratio in Hands
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  • 60.  
  • 61. More Examples of Golden Sections
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  • 70.