# Fibonacci

# Fibonacci

The Fibonacci Sequence

Leonardo Pisano (1170-1250) was born in Italy, probably in Pisa. He is better known by his nickname Fibonacci. In 1202 he wrote his famous book “Liber abaci” in which he formulates the following problem:

A certain man put a pair of rabbits in a place surrounded on all sides by a wall. How many pairs of rabbits can be produced from that pair in a year if it is supposed that every month each pair begets a new pair, which from the second month on becomes productive?

The solution to this problem leads to the following sequence

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144 … (1)

This sequence, in which each number is the sum of the two preceding numbers, has proved extremely fruitful and appears in many different areas of mathematics and science. It is called The Fibonacci Sequence.

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The Golden Section

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The Golden Section F (phi)

The Fibonacci Sequence is intimately related to the age old Golden Section. To show this, let us first explain what the Golden Section is. The Golden Section is also known as the Golden Mean, Golden Ratio and Divine Proportion. It is a ratio or proportion defined by the number F

F = 1.618033988749895… (2)

F can be obtained geometrically by dividing a line segment as follows:

The Ratio AB/AC of the whole line AB to the large segment AC is the same as the Ratio AC/CB of the large segment AC to the small segment CB

That is

AB/AC = AC/CB = F (3)

It can be shown (see proof) that such a division occurs only when AB = 1.618 AC and AC = 1.618 CB.

The Golden Ratio F as given by equation (2) has been used throughout history.

The Egyptians may have used it in the pyramids:

The Greeks used it to balance their architecture as in the Parthenon below

Da Vince and many more Renaissance artists called it the Divine Proportion

You can also find it in the design of La Notre Dame de Paris

Now that we know what the Golden Section is we can relate it to The Fibonacci Sequence. Simply stated, the ratios of the successive numbers in the Fibonacci Sequence quickly converge to F.

That is,

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144 … (1)

1/1, 2/1, 3/2, 5/3, 8/5, 13/8, 21/13, 34/21, 55/34, … -> 1.6180

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The Golden String

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There also exists an interesting sequence of 0s and 1s which is closely related to the Fibonacci Sequence and to F:

1 0 1 1 0 1 0 1 1 0 1 1 0 1 0 1 1 0 1 0 1 1 0 1 1 0 1 0 1 1 0 1 … (4)

To see where this series comes from let us look again at Fibonacci’s original Rabbit problem:

A certain man put a pair of rabbits in a place surrounded on all sides by a wall. How many pairs of rabbits can be produced from that pair in a year if it is supposed that every month each pair begets a new pair, which from the second month on becomes productive?

Let us define A as an Adult pair, i.e., a pair that has lived for more than 1 month. B is defined as a Baby pair, i.e., a pair that has lived for a month or less.

At the end of Month 1 (M 1) the original pair of rabbits has lived for one month or less, so we indicate that pair by B.

End of M 1

B

At the end of Month 2 (M 2) the above pair B has lived for more than one month so we now represent it by A. Note that pair B has not yet lived two full months so it has not reproduced yet. Therefore,

End of M 2

A

At the end of Month 3 (M 3) the above pair A has reproduced and is replace by A + B where A is the pair itself and B is the new offspring pair that obviously has not lived more than one month. So now we have,

End of M 3

A + B

At the end of Month 4 (M 4) A will become A+B and B will become A (without an offspring pair). Or,

End of M 4

A + B + A

By now the pattern is clear. To go from one month to the next, we must replace A by A + B and B by A. For example for Month 5 (M 5), Month 6 (M 6) and Month 7 (M 7) we have,

End of M 5

A + B + A + A + B

End of M 6

A + B + A + A + B + A + B + A

End of M 7

A + B + A + A + B + A + B + A + A + B + A + A + B

Therefore, if we write the number of rabbit pairs for each month sequentially we have

1, 1, 2, 3, 5, 8, 13, 21, 34, … (1)

Which is The Fibonacci Sequence. We can also construct another sequence by taking A equal to 1 and B equal to 0, i.e.,

1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1 … (2)

This sequence is called The Golden String and, as we showed, is closely related to The Fibonacci Sequence. Note that the first part of the Golden String for a particular month is a copy of the Golden String of the previous month:

End of Golden String

M 2 1

M 3 1, 0

M 4 1, 0, 1

M 5 1, 0, 1, 1, 0

M 6 1, 0, 1, 1, 0, 1, 0, 1

M 7 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0

This is the reason why Fibonacci numbers also show up in the familiar Mandelbrot set shown here in black and white:

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Examples of the Golden Section

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Applications of the Golden Section can be found in many areas of life. Many proportions in the human body are based on the Golden Section. Certain heart and brain rhythms show the golden proportions. It has penetrated even our DNA. Of course nature is an expert in the golden section. Many designs of animals, plants, mountains and heavenly bodies exhibit the magical number 1.618.

The arts (music, painting and sculpture) and architecture areas are where the Golden Section is quite often applied. Then there are also some odd places where you will run into the Golden Section such as the stock market, population growth and theology.

Here are some useful links:

Golden Museum http://www.goldenmuseum.com/

Unis http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibInArt.html

Math World http://mathworld.wolfram.com/GoldenRatio.html

Images http://images.google.com/images?q=golden+section&hl=en&lr=&safe=off&sa=N&tab=ii&oi=imagest

Or, just check out Google (http://images.google.com/) with keywords Golden Section or Fibonacci.

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Golden Spiral

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The Golden Spiral is also based on the Golden Section. Consider the construction of rectangles below:

As you can see, we start the division of the large rectangle such that the ratio AB/AC = 1.618. Then we divide CD in such a way that the ratio CD/CE = 1.618. And we keep on going with similar (smaller and smaller) division while keeping the same Golden Ratio 1.618. Then we can pencil in the Golden Spiral as follows:

The Golden Spiral is routinely manifested in nature in the spiraling bracts of a pinecone, the development of a nautilus, the path a fly follows as it approaches an object, or most anything.

In manâ€™s attempt to replicate this beauty of nature, we see the Golden Spiral in the fact that the Sphinx and Pyramids of Giza (The Great Pyramids) all lie on a Golden Spiral. Below, see if you can see the spiral in this flower:

There are also computer programs that use the principal of the Golden Spiral to create beautiful pictures. Here are some examples: