061 Fibonacci Slides

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PERSPECTIVE INMATHEMATICS VI

The Decline and Revival of Learning:Fibonacci Sequence


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Who is Fibonacci?

The "greatest European mathematician of the middleages", his full name was Leonardo of Pisa, orLeonardo Pisano in Italian since he was born in Pisa(Italy), the city with the famous Leaning Tower, about

1175 AD.

In Fibonacci's Liber Abacibook (1202), chapter 12, heintroduces the following problem

Another Mathematical Contribution:

Introducing the Decimal Number system into Europe
http://www.emmeti.it/Welcome/Toscana/Pisano/Pisa/index.uk.htmlhttp://www.emmeti.it/Welcome/Toscana/Pisano/Pisa/index.uk.html


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Leonardo of Pisa (c. 1170 c. 1250)aka Fibonacci

Brought Hindu-Arabicnumeral system to Europethrough the publication of hisBook of Calculation, the

Liber Abaci.

Fibonacci numbers,constructed as an example inthe Liber Abaci.

3


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The Fibonacci Numbers

The number pattern that you have been using is known asthe Fibonacci sequence.

1 1

}+

2


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1 1 2

}+

3


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1 1 2 3

}+

5 8 13 21 34 55

These numbers can be seen in many natural situations


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Fibonacci Sequence

Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233,377, 610, 987, etc.

Look at ratios:

1/1 = 1.0

2/1 = 2.0 3/2 = 1.5

5/3 = 1.666

8/5 = 1.6

13/8 = 1.625

21/13 = 1.615385 34/21 = 1.619048

55/34 = 1.617647

89/55 = 1.61812

7


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The Fibonacci Problem

How Many Pairs of Rabbits Are Created byOne Pair in One Year

If you begin with one pair of rabbits on the first

day of the year, how many pairs of rabbits willyou have on the first day of the next year?

It is assumed that each pair of rabbits

produces a new pair every month and eachnew pair begins to produce two months afterbirth.


10/40Fig. 2-69, p. 113


11/40Fig. 2-70, p. 114


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The Fibonacci Problem

The solution to this question is shown in the table below.

The sequence that appears three times in the table, 1, 1, 2,

3, 5, 8, 13, 21, is called the Fibonacci Sequence.


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Fibonacci Numbers In Nature

The Fibonacci numbers are found manyplaces in the natural world, including: The number of flower petals.

The branching behavior of plants. The growth patterns of sunflowers and

pinecones,

It is believed that the spiral nature ofplant growth accounts for thisphenomenon.


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The Fibonacci Numbers in Plants

Lilies have 3 petalsButtercups have 5Many delphiniums have 8

Marigolds have 13Asters have 21Daisies commonly have 13, 21,34, 55 or 89


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The number of petals on a flowerare often Fibonacci numbers


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On many plants, the number of petals is a Fibonaccinumber:

Buttercups have 5 petals; lilies and iris have 3 petals;some delphiniums have 8; corn marigolds have 13petals; some asters have 21 whereas daisies can befound with 34, 55 or even 89 petals.

13 petals: ragwort, corn marigold, cineraria, somedaisies21 petals: aster, black-eyed susan, chicory34 petals: plantain, pyrethrum55, 89 petals: michaelmas daisies, the asteraceaefamily.

Some species are very precise about the number ofpetals they have - e.g. buttercups, but others havepetals that are very near those above, with theaverage being a Fibonacci number.
http://affiliates.allposters.com/link/redirect.asp?aid=910266&item=376240


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Another Problems

Suppose a tree starts from one shoot thatgrows for two months and then sprouts asecond branch. If each established branch

begins to sprout a new branch after onemonths growth, and if every new branch

begins to sprout its own first new branch aftertwo months growth, how many branches does

the tree have at the end of the year?


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Plants grow in a spiral pattern. The ratio of the number of spirals

to the number of branches is called the phyllotactic ratio.

The numbers in the phyllotactic ratio are usually Fibonacci

numbers.


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Example: Thebranch at right has aphyllotactic ratio of

3/8. Both 3 and 8 are

Fibonacci numbers.


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One plant in particular shows theFibonacci numbers in the numberof "growing points" that it has.

Suppose that when a plant puts outa new shoot, that shoot has to growtwo months before it is strongenough to support branching.

If it branches every month after thatat the growing point, we get thepicture shown here.

A plant that grows very much likethis is the "sneezewort.


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Fibonacci in Nature

The lengths of bones ina hand are Fibonaccinumbers.


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Fibonacci in Nature


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Fibonacci in Music

23

5

8 white13 w & b

The intervals between keys ona piano are Fibonaccinumbers.


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Patterns in Fibonacci Numbers

1+1 = 2

1+2 = 3

2+3 = 5

3+5 = 8

5+8 = 13

8+13 = 21

13+21 = 34

Complete the Fibonacci Numbers sheet using column addition

Colour all the even numbers in blue and all the even numbers in red.What do you notice about the patterns in the colouring?


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Patterns in Fibonacci Numbers

55+89 = 144

34+55 = 89

21+34 = 55

13+21 = 34

8+13 = 21

5+8 = 13

3+5 = 82+3 = 5

1+2 = 3

1+1 = 2

ratio1.618

1.618

1.618

1.619

1.615

1.625

1.61.666

1.5


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The Golden Ratio


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The Golden Ratio

The ratios of pairs of consecutive Fibonaccinumbers are also represented in the graphbelow.

The ratios approach the dashed line whichrepresents a number around 1.618.


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Fibonacci Sequence

The Fibonacci sequence is thesequence of numbers 1, 1, 2, 3, 5, 8, 13,21,

Any number in the sequence is called aFibonacci number.

The sequence is usually written

F1, F2, F3, , Fn,


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Recursion

Recursion, in a sequence, indicates thateach number in the sequence is foundusing previous numbers in the sequence.

Some sequences, such as the Fibonaccisequence, are generated by a recursionrule along with starting values for the first

two, or more, numbers in the sequence.


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Fibonacci Sequence (cont.)

For the Fibonacci sequence, the startingvalues are F1= 1 and F2= 1.

The recursion rule for the Fibonacci sequenceis:

Example: Find the third number in thesequence using the formula. Let n= 3.

1 2n n nF F F

3 3 1 3 2 2 11 1 2F F F F F

Th ti f ti f th
http://mathworld.wolfram.com/GeneratingFunction.htmlhttp://mathworld.wolfram.com/GeneratingFunction.html


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The generating function for theFibonacci numbers is

Fib i N b i P l
http://mathworld.wolfram.com/GeneratingFunction.htmlhttp://mathworld.wolfram.com/GeneratingFunction.html


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Fibonacci Number in PascalTriangle


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Pascals triangle

0

1

2

3

4

5

6

7

8

n= 1

2

4

8

16

32

64

128

256

sum = = 2n


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Some Famous Numbers

37

091798057638117720345868343654989484820871.61803398

9572470936995266249775360287471384590452352.71828182e

1071693993757950288419462643383235897932383.14159265


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Fibonacci died in 1240

He died in 1240 andthere is now a statuecommemorating him

located at the LeaningTower end of thecemetery next to theCathedral in Pisa.


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Sources Utilized

Burton, David M., The History of Mathematics, An Introduction, p203-226, McGraw-Hill, New York, NY, 2003.Eves, Howard, An Introduction to the History of Mathematics, p

179-182, Saunders College Publishing, Orlando, FL, 1990.Eves, Howard, Great Moments in Mathematics, p 117-120, The

Mathematical Association of America, 1980.Eves, Howard, In Mathematical Circles, p 59-62, Prindle, Weber &

Schmidt, Inc., Boston, 1969.Grattan-Guinness, Ivor, The Norton History of the Mathematicl

Sciences; The Rainbow of Mathematics, p. 80-82, W.W. Norton &Company, New York, 1997.

http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Diophantus.html

http://www-gap.dcs.st-and.ac.uk/~history/Quotations/Diophantus.html

http://www.lib.virginia.edu/science/parshall/diophant.html
http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Diophantus.htmlhttp://www-gap.dcs.st-and.ac.uk/~history/Quotations/Diophantus.htmlhttp://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Diophantus.htmlhttp://www-gap.dcs.st-and.ac.uk/~history/Quotations/Diophantus.htmlhttp://www.lib.virginia.edu/science/parshall/diophant.htmlhttp://www.lib.virginia.edu/science/parshall/diophant.htmlhttp://www.lib.virginia.edu/science/parshall/diophant.htmlhttp://www.lib.virginia.edu/science/parshall/diophant.htmlhttp://www-gap.dcs.st-and.ac.uk/~history/Quotations/Diophantus.htmlhttp://www-gap.dcs.st-and.ac.uk/~history/Quotations/Diophantus.htmlhttp://www-gap.dcs.st-and.ac.uk/~history/Quotations/Diophantus.htmlhttp://www-gap.dcs.st-and.ac.uk/~history/Quotations/Diophantus.htmlhttp://www-gap.dcs.st-and.ac.uk/~history/Quotations/Diophantus.htmlhttp://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Diophantus.htmlhttp://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Diophantus.htmlhttp://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Diophantus.htmlhttp://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Diophantus.htmlhttp://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Diophantus.html


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