# Fibonacci Numbers

### From Math Images

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|ImageName=Fibonacci Spiral | |ImageName=Fibonacci Spiral | ||

|Image=NAUTILUS.jpg | |Image=NAUTILUS.jpg | ||

- | |ImageIntro=The spiral curve of the Nautilus sea shell follows the pattern of | + | |ImageIntro=The spiral curve of the Nautilus sea shell follows the pattern of a spiral drawn in a Fibonacci rectangle, a collection of squares with sides that have the length of Fibonacci numbers. |

|ImageDescElem=The Fibonacci sequence is the sequence <math>1, 1, 2, 3, 5, 8, 13, 21, 34, 55, \ldots,</math> where the first two numbers are 1s and every later number is the sum of the two previous numbers. So, given two <math>1</math>'s as the first two terms, the next terms of the sequence follows as : <math>1+1=2, 1+2=3, 2+3=5, 3+5=8, \dots</math> | |ImageDescElem=The Fibonacci sequence is the sequence <math>1, 1, 2, 3, 5, 8, 13, 21, 34, 55, \ldots,</math> where the first two numbers are 1s and every later number is the sum of the two previous numbers. So, given two <math>1</math>'s as the first two terms, the next terms of the sequence follows as : <math>1+1=2, 1+2=3, 2+3=5, 3+5=8, \dots</math> | ||

[[image:sunflower.jpg|Image 1|frame]] | [[image:sunflower.jpg|Image 1|frame]] | ||

- | The Fibonacci numbers can be discovered in nature, such as the spiral of the Nautilus sea shell, the petals of the flowers, the seed head of a sunflower, and many other parts | + | The Fibonacci numbers can be discovered in nature, such as the spiral of the Nautilus sea shell, the petals of the flowers, the seed head of a sunflower, and many other parts. The seeds at the head of the sunflower, for instance, are arranged so that one can find a collection of spirals in both clockwise and counterclockwise ways. Different patterns of spirals are formed depending on whether one is looking at a clockwise or counterclockwise way; thus, the number of spirals also differ depending on the counting direction, as shown by <i>Image 1</i>. The two numbers of spirals are always consecutive numbers in the Fibonacci sequence. |

- | Nature prefers this way of arranging | + | Nature prefers this way of arranging seeds because it seems to allow the seeds to be uniformly distributed. For more information about Fibonacci patterns in nature, see [[Fibonacci_Numbers#Fibonacci_Numbers_in_Nature| Fibonacci Numbers in Nature]] |

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[[image:Rabbit.png|thumb|300px|Image 2]] | [[image:Rabbit.png|thumb|300px|Image 2]] | ||

- | On January 1st, there is only 1 pair. On February 1st, | + | On January 1st, there is only 1 pair. On February 1st, this baby rabbits matured to be grown up rabbits, but they have not reproduced, so there will only be the original pair present. |

Now look at any later month. June is a good example. As you can see in <i>Image 2</i>, all 5 pairs of rabbits that were alive in May continue to be alive in June. Furthermore, there are 3 new pairs of rabbits born in June, one for each pair that was alive in April (and are therefore old enough to reproduce in June). | Now look at any later month. June is a good example. As you can see in <i>Image 2</i>, all 5 pairs of rabbits that were alive in May continue to be alive in June. Furthermore, there are 3 new pairs of rabbits born in June, one for each pair that was alive in April (and are therefore old enough to reproduce in June). | ||

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We can build Fibonacci rectangles first by drawing two squares with length 1 next to each other. Then, we draw a new square with length 2 that is touching the sides of the original two squares. We draw another square with length 3 that is touching one unit square and the latest square with length 2. We can build Fibonacci rectangles by continuing to draw new squares that have the same length as the sum of the length of the latest two squares. | We can build Fibonacci rectangles first by drawing two squares with length 1 next to each other. Then, we draw a new square with length 2 that is touching the sides of the original two squares. We draw another square with length 3 that is touching one unit square and the latest square with length 2. We can build Fibonacci rectangles by continuing to draw new squares that have the same length as the sum of the length of the latest two squares. | ||

- | After building Fibonacci rectangles, we can draw a spiral in the squares, each square containing a quarter of a circle. Such | + | After building Fibonacci rectangles, we can draw a spiral in the squares, each square containing a quarter of a circle. Such spiral is called the Fibonacci spiral, and it can be seen in sea shells, snails, the spirals of the galaxy, and other parts of nature, as shown in <i>Image 6</i> and <i>Image 7</i>. |

[[image:galaxy.jpg|250px|thumb|Image 7|none]] | [[image:galaxy.jpg|250px|thumb|Image 7|none]] | ||

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:{{EquationRef2|Eq. (1)}}<math>F_1+F_2+\dots+F_n=F_{n+2}-1</math> | :{{EquationRef2|Eq. (1)}}<math>F_1+F_2+\dots+F_n=F_{n+2}-1</math> | ||

- | For example, the sum of first <math>5</math> Fibonacci | + | For example, the sum of first <math>5</math> Fibonacci numbers is : |

:<math>F_1+F_2+F_3+F_4+F_5= 1 + 1 + 2 + 3 +5=F_7-1=12</math> | :<math>F_1+F_2+F_3+F_4+F_5= 1 + 1 + 2 + 3 +5=F_7-1=12</math> | ||

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The example is demonstrated below. The total length of red bars that each correspond to <math>F_1, F_2, F_3, F_4, F_5</math> is one unit less than the length of <math>F_7</math>. | The example is demonstrated below. The total length of red bars that each correspond to <math>F_1, F_2, F_3, F_4, F_5</math> is one unit less than the length of <math>F_7</math>. | ||

- | [[image:identity1.gif|thumb|Image | + | [[image:identity1.gif|thumb|Image 9|none|600px]] |

{{HideShowThis|ShowMessage=Click here to show proof.|HideMessage=Click here to hide proof. | {{HideShowThis|ShowMessage=Click here to show proof.|HideMessage=Click here to hide proof. | ||

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This example is shown below. | This example is shown below. | ||

- | [[image:fidentity2.gif|thumb|Image | + | [[image:fidentity2.gif|thumb|Image 10|none|800px]] |

{{HideShowThis|ShowMessage=Click here to show proof.|HideMessage=Click here to hide proof. | {{HideShowThis|ShowMessage=Click here to show proof.|HideMessage=Click here to hide proof. | ||

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This example is shown below. | This example is shown below. | ||

- | [[image:identity3.gif|thumb|Image | + | [[image:identity3.gif|thumb|Image 11|none|600px]] |

To see the proof, click below. | To see the proof, click below. | ||

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{{HideShowThis|ShowMessage=Click here to show proof.|HideMessage=Click here to hide proof. | {{HideShowThis|ShowMessage=Click here to show proof.|HideMessage=Click here to hide proof. | ||

|HiddenText= | |HiddenText= | ||

- | Subtracting {{EquationNote|Eq. (2)}}, the sum of Fibonacci numbers with odd indices, from | + | Subtracting {{EquationNote|Eq. (2)}}, the sum of Fibonacci numbers with odd indices, from the sum of the first <math>2n</math> Fibonacci numbers, we get the identity of the sum of Fibonacci numbers with even indices. |

+ | |||

+ | First, when we find the sum of first <math>2n</math> Fibonacci numbers through {{EquationNote|Eq. (1)}}, we get: | ||

+ | :<math>F_1+F_2+\dots+F_{2n}=F_{2n+2}-1</math> | ||

+ | |||

+ | Now, subtract {{EquationNote|Eq. (2)}} from the above equation, and we get: | ||

+ | :<math>F_2+F_4+F_6+\dots+F_{2n}=F_{2n+2}-F_{2n}-1</math> | ||

+ | |||

+ | By definition of Fibonacci numbers, <math>F_{2n+2}-F_{2n}=F_{2n+1}</math>. Thus, | ||

+ | :<math>F_2+F_4+F_6+\dots+F_{2n}=F_{2n+1}-1 | ||

+ | }} | ||

====Sum of the squares of Fibonacci numbers==== | ====Sum of the squares of Fibonacci numbers==== | ||

The sum of the squares of the first <math>n</math> Fibonacci numbers is the product of the <math>n^{\rm th}</math> and the <math>{(n+1)}^{\rm th}</math> Fibonacci numbers. | The sum of the squares of the first <math>n</math> Fibonacci numbers is the product of the <math>n^{\rm th}</math> and the <math>{(n+1)}^{\rm th}</math> Fibonacci numbers. | ||

- | [[image:goldrectangle.jpg|right|Image | + | [[image:goldrectangle.jpg|right|Image 12|thumb|300px]] |

:<math>\sum_{i=1}^n {F_i}^2=F_n F_{n+1}</math> | :<math>\sum_{i=1}^n {F_i}^2=F_n F_{n+1}</math> | ||

- | This identity can be proved by studying the area of the rectangles in <i>Image | + | This identity can be proved by studying the area of the rectangles in <i>Image 12</i>. |

{{HideShowThis|ShowMessage=Click here to show proof.|HideMessage=Click here to hide proof. | {{HideShowThis|ShowMessage=Click here to show proof.|HideMessage=Click here to hide proof. | ||

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:<math>\gcd(F_n, F_{n+1})=F_{\gcd(n,n+1)}=F_1=1</math>. | :<math>\gcd(F_n, F_{n+1})=F_{\gcd(n,n+1)}=F_1=1</math>. | ||

- | That is, <math>F_n</math> and <math>F_{n+1} </math> | + | That is, <math>F_n</math> and <math>F_{n+1} </math>, or two consecutive Fibonacci numbersare always <balloon title="Two integers are relatively prime if their greatest common divisor is 1"> relatively prime</balloon>. |

To see the proof for this special case, click below. | To see the proof for this special case, click below. | ||

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Because the first sequence of differences of the Fibonacci sequence also includes a Fibonacci sequence, the <balloon title="Second difference is a sequence of differences between two consecutive numbers of the first sequence of differences">second difference </balloon> also includes a Fibonacci sequence. The Fibonacci sequence is thus reproduced in every sequence of differences. | Because the first sequence of differences of the Fibonacci sequence also includes a Fibonacci sequence, the <balloon title="Second difference is a sequence of differences between two consecutive numbers of the first sequence of differences">second difference </balloon> also includes a Fibonacci sequence. The Fibonacci sequence is thus reproduced in every sequence of differences. | ||

- | |||

We can see that the sequence of differences is composed of Fibonacci numbers by looking at the definition of Fibonacci numbers : | We can see that the sequence of differences is composed of Fibonacci numbers by looking at the definition of Fibonacci numbers : | ||

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{{Hide|1= | {{Hide|1= | ||

- | [[image:golden_rectangle_detailed.jpg|150px|thumb|Image | + | [[image:golden_rectangle_detailed.jpg|150px|thumb|Image 13]] |

- | The golden ratio appears in paintings, architecture, and in various forms of nature. Two numbers are said to be in the golden ratio if the ratio of the smaller number to the larger number is equal to the ratio of the larger number to the sum of the two numbers. In <i>Image | + | The golden ratio appears in paintings, architecture, and in various forms of nature. Two numbers are said to be in the golden ratio if the ratio of the smaller number to the larger number is equal to the ratio of the larger number to the sum of the two numbers. In <i>Image 13</i>, the width of A and B are in the golden ratio if<math> a : b = (a+b) : a</math>. |

The golden ratio is represented by the Greek lowercase phi ,<math>\varphi</math>, and the exact value is | The golden ratio is represented by the Greek lowercase phi ,<math>\varphi</math>, and the exact value is | ||

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This value can be found from the definition of the golden ratio. To see an algebraic derivation of the exact value of the golden ratio, go to [[Golden_Ratio#An_Algebraic_Representation| Golden Ratio : An Algebraic Representation]]. | This value can be found from the definition of the golden ratio. To see an algebraic derivation of the exact value of the golden ratio, go to [[Golden_Ratio#An_Algebraic_Representation| Golden Ratio : An Algebraic Representation]]. | ||

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An interesting fact about golden ratio is that the ratio of two consecutive Fibonacci numbers approaches the golden ratio as the numbers get larger, as shown by the table below. | An interesting fact about golden ratio is that the ratio of two consecutive Fibonacci numbers approaches the golden ratio as the numbers get larger, as shown by the table below. | ||

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Taking the limit of <math>r_n=1+\frac{1}{r_{n-1}}</math> we get : | Taking the limit of <math>r_n=1+\frac{1}{r_{n-1}}</math> we get : | ||

- | |||

:<math>r=1+\frac{1}{r}</math> | :<math>r=1+\frac{1}{r}</math> | ||

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<span id="continuedf" style="display:none">A continued fraction is a fraction in which the denominator is composed of a whole number and a fraction. An infinite continued fraction of the golden ratio has the form : <math>\varphi = 1 + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1} {1 + \ddots\,}}}} </math></span>go to [[Golden_Ratio#Continued_Fraction_Representation_and_Fibonacci_Sequences|Continued Fraction Representation and Fibonacci Sequences]] | <span id="continuedf" style="display:none">A continued fraction is a fraction in which the denominator is composed of a whole number and a fraction. An infinite continued fraction of the golden ratio has the form : <math>\varphi = 1 + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1} {1 + \ddots\,}}}} </math></span>go to [[Golden_Ratio#Continued_Fraction_Representation_and_Fibonacci_Sequences|Continued Fraction Representation and Fibonacci Sequences]] | ||

- | [[image:vitruvianman.jpg|250px|thumb|Image | + | [[image:vitruvianman.jpg|250px|thumb|Image 14: Vitruvian man]] |

- | Many people find the golden ratio in various parts of nature, art, architecture, and even music. However, there are some people who criticize this viewpoint. They claim that many mathematicians are wishfully trying to make | + | Many people find the golden ratio in various parts of nature, art, architecture, and even music. However, there are some people who criticize this viewpoint. They claim that many mathematicians are wishfully trying to make a connection between the golden ratio and other parts of the world even though there is no real connection. |

One example of the golden ratio that mathematicians found in nature is the human body. According to many, an ideal human body have proportions that show the golden ratio, such as: | One example of the golden ratio that mathematicians found in nature is the human body. According to many, an ideal human body have proportions that show the golden ratio, such as: | ||

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*distance between the shoulder line and top of the head : length of the head. | *distance between the shoulder line and top of the head : length of the head. | ||

- | Leonardo da Vinci's drawing <i>Vitruvian man</i> shown in <i>Image | + | Leonardo da Vinci's drawing <i>Vitruvian man</i> shown in <i>Image 14</i> emphasizes the proportion of human body. This drawing shows the proportions of an ideal human body that was studied by a Roman architect Vitruvius in his book De Architectura. In the drawing, a man is simultaneously inscribed in a circle and a square. The ratio of the square side to the radius of the circle in the drawing reflects the golden ratio, although the drawing deviates from the real value of the golden ratio by 1.7 percent. The proportions of the body of the man is also known to show the golden ratio. |

- | Although people later found the golden ratio in the painting, there is no evidence whether Leonardo da Vinci was trying to | + | Although people later found the golden ratio in the painting, there is no evidence whether Leonardo da Vinci was trying to show the golden ratio in his painting or not. For more information about the golden ratio, go to [[Golden_Ratio|Golden Ratio]] |

}} | }} | ||

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Thus, the sequence defined by <math>c=0</math> is bounded and <math>0</math> is included in the Mandelbrot set. | Thus, the sequence defined by <math>c=0</math> is bounded and <math>0</math> is included in the Mandelbrot set. | ||

- | |||

On the other hand, when we test<math>c=1</math>, | On the other hand, when we test<math>c=1</math>, |

## Revision as of 10:38, 8 July 2010

{{Image Description |ImageName=Fibonacci Spiral |Image=NAUTILUS.jpg |ImageIntro=The spiral curve of the Nautilus sea shell follows the pattern of a spiral drawn in a Fibonacci rectangle, a collection of squares with sides that have the length of Fibonacci numbers. |ImageDescElem=The Fibonacci sequence is the sequence where the first two numbers are 1s and every later number is the sum of the two previous numbers. So, given two 's as the first two terms, the next terms of the sequence follows as :

The Fibonacci numbers can be discovered in nature, such as the spiral of the Nautilus sea shell, the petals of the flowers, the seed head of a sunflower, and many other parts. The seeds at the head of the sunflower, for instance, are arranged so that one can find a collection of spirals in both clockwise and counterclockwise ways. Different patterns of spirals are formed depending on whether one is looking at a clockwise or counterclockwise way; thus, the number of spirals also differ depending on the counting direction, as shown by *Image 1*. The two numbers of spirals are always consecutive numbers in the Fibonacci sequence.

Nature prefers this way of arranging seeds because it seems to allow the seeds to be uniformly distributed. For more information about Fibonacci patterns in nature, see Fibonacci Numbers in Nature

## Contents |

## Origin

The Fibonacci sequence was studied by Leonardo of Pisa, or Fibonacci (1770-1240). In his work Liber Abacci, he introduced a problem involving the growth of the rabbit population. The assumptions were

- there is one pair of baby rabbits placed in an enclosed place on the first day of January
- this pair will grow for one month before reproducing and produce a new pair of baby rabbits on the first day of March
- each new pair will mature for one month and produce a new pair of rabbits on the first day of their third month
- the rabbits never die, so after they mature, the rabbits produce a new pair of baby rabbits every month.

The problem was to find out how many pairs of rabbits there will be after one year.

On January 1st, there is only 1 pair. On February 1st, this baby rabbits matured to be grown up rabbits, but they have not reproduced, so there will only be the original pair present.

Now look at any later month. June is a good example. As you can see in *Image 2*, all 5 pairs of rabbits that were alive in May continue to be alive in June. Furthermore, there are 3 new pairs of rabbits born in June, one for each pair that was alive in April (and are therefore old enough to reproduce in June).

This means that on June 1st, there are 5 + 3 = 8 pairs of rabbits. This same reasoning can be applied to any month, March or later, so the number of rabbits pairs in any month is the same as the sum of the number of rabbit pairs in the two previous months.

This is exactly the rule that defines the Fibonacci sequence. As you can see in the image, the population by month begins: 1, 1, 2, 3, 5, 8, ..., which is the same as the beginning of the Fibonacci sequence. The population continues to match the Fibonacci sequence no matter how many months out you go.

An interesting fact is that this problem of rabbit population was not intended to explain the Fibonacci numbers. This problem was originally intended to introduce the Hindu-Arabic numerals to Western Europe, where people were still using Roman numerals, and to help people practice addition. It was coincidence that the number of rabbits followed a certain pattern which people later named as the Fibonacci sequence.

# Fibonacci Numbers in Nature

|ImageDesc===Symbolic Definition of Fibonacci Sequence==

The Fibonacci sequence is the sequence where

- ,

and

- .

The Fibonacci sequence is recursively defined because each term is defined in terms of its two immediately preceding terms.

## Identities and Properties