• The sequence begins with one. Each The sequence begins with one. Each subsequent number is the sum of the subsequent number is the sum of the two preceding numbers.two preceding numbers.

• Fib(n) = Fib(n-1) + Fib(n-2)Fib(n) = Fib(n-1) + Fib(n-2)

• Thus the sequence begins as follows:Thus the sequence begins as follows:

• 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144….144….

• Fibonacci applied his sequence to a problem Fibonacci applied his sequence to a problem involving the breeding of rabbits.involving the breeding of rabbits.

• Given certain starting conditions, he Given certain starting conditions, he mapped out the family tree of a group of mapped out the family tree of a group of rabbits that initially started with only two rabbits that initially started with only two members.members.

• The number of rabbits at any given time was The number of rabbits at any given time was always a Fibonacci number.always a Fibonacci number.

• Unfortunately, his application had little Unfortunately, his application had little practical bearing to nature, since incest and practical bearing to nature, since incest and immortality was required among the rabbits immortality was required among the rabbits to complete his problem.to complete his problem.

• The Fibonacci sequence has far more The Fibonacci sequence has far more applications than immortal rabbits.applications than immortal rabbits.

• Fibonacci numbers have numerous Fibonacci numbers have numerous naturally-occurring applications, naturally-occurring applications, ranging from the very basic to the ranging from the very basic to the complex geometric.complex geometric.

• Many aspects of Many aspects of nature are grouped nature are grouped in bunches equaling in bunches equaling Fibonacci numbers.Fibonacci numbers.

• For example, the For example, the number of petals on number of petals on a flower tend to be a flower tend to be a Fibonacci number.a Fibonacci number.

• 3 petals: lilies3 petals: lilies

• 5 petals: buttercups, roses5 petals: buttercups, roses

• 8 petals: delphinium8 petals: delphinium

• 13 petals: marigolds13 petals: marigolds

• 21 petals: black-eyed susans21 petals: black-eyed susans

• 34 petals: pyrethrum34 petals: pyrethrum

• 55/89 petals: daisies55/89 petals: daisies

• Leaves are also Leaves are also found in groups of found in groups of Fibonacci numbers.Fibonacci numbers.

• Branching plants Branching plants always branch off always branch off into groups of into groups of Fibonacci numbers.Fibonacci numbers.

• Think about Think about yourself. You yourself. You shouldshould have:have:

• 5 fingers on each 5 fingers on each handhand

• 5 toes on each foot5 toes on each foot• 2 arms2 arms• 2 legs2 legs• 2 eyes2 eyes• 2 ears2 ears

• 2 sections per leg2 sections per leg• 2 sections per arm2 sections per arm

I could go on, but I I could go on, but I think you get the think you get the point.point.

• Fibonacci numbers Fibonacci numbers have geometric have geometric applications in applications in nature as well.nature as well.

• The most The most prominent of these prominent of these is the Fibonacci is the Fibonacci spiral.spiral.

• The Fibonacci The Fibonacci spiral is spiral is constructed by constructed by placing together placing together rectangles of rectangles of relative side relative side lengths equaling lengths equaling Fibonacci numbers.Fibonacci numbers.

• A spiral can then A spiral can then be drawn starting be drawn starting from the corner of from the corner of the first rectangle the first rectangle of side length 1, all of side length 1, all the way to the the way to the corner of the corner of the rectangle of side rectangle of side length 13.length 13.

CauliflowerCauliflower Pine ConePine Cone

• Music involves Music involves several applications several applications of Fibonacci of Fibonacci numbers.numbers.

• A full octave is A full octave is composed of 13 composed of 13 total musical tones, total musical tones, 8 of which make up 8 of which make up the actual musical the actual musical octave.octave.

Fibonacci Ratio Calculated Frequency

Tempered Frequency

Note in Scale Musical Relationship

1/1 440 440.00 A Root

2/1 880 880.00 A Octave

2/3 293.33 293.66 D Fourth

2/5 176 174.62 F Aug Fifth

3/2 660 659.26 E Fifth

3/5 264 261.63 C Minor Third

3/8 165 164.82 E Fifth

5/2 1,100.00 1108.72 C# Third

5/3 733.33 740 F# Sixth

5/8 275 277.18 C# Third

8/3 1173.33 1174.64 D Fourth

8/5 704 698.46 F Aug. Fifth

End Part 1……End Part 1……

Beginning Part 2…..Beginning Part 2…..

• One of the most significant One of the most significant applications of the Fibonacci applications of the Fibonacci sequence is a number that sequence is a number that mathematicians refer to as Phi (mathematicians refer to as Phi ().).

• No, you Physics duals, NOT flux. In No, you Physics duals, NOT flux. In this case, this case, refers to a very refers to a very important number that is known as important number that is known as the the golden ratiogolden ratio..

• Phi is defined as the Phi is defined as the limit of the ratio of a limit of the ratio of a Fibonacci number i and Fibonacci number i and its predecessor, Fib(i-1).its predecessor, Fib(i-1).

• Mathematically, this Mathematically, this number is equal to:number is equal to:

or approximately or approximately 1.618034. 1.618034.

• Phi can be derived by the equation:Phi can be derived by the equation:

• With some fancy factoring and division, With some fancy factoring and division, you get:you get:

• This implies that Phi’s reciprocal is This implies that Phi’s reciprocal is smaller by 1. It is .618034, also known smaller by 1. It is .618034, also known as phi (as phi ().).

0012 xxx

xx 11

• Is there anything Is there anything mathematically mathematically definitive about definitive about when when used in geometry? You used in geometry? You bet there is.bet there is.

• A rectangle whose sides A rectangle whose sides are in the are in the golden ratiogolden ratio is referred to as a is referred to as a golden rectanglegolden rectangle..

• When a When a golden golden rectanglerectangle is squared, is squared, the remaining area the remaining area forms another forms another golden golden rectanglerectangle!!

• Without Without in order to find any in order to find any Fibonacci number, you would need to Fibonacci number, you would need to know its two preceding Fibonacci know its two preceding Fibonacci numbers.numbers.

• But with But with at your service, you can at your service, you can find any Fibonacci number knowing find any Fibonacci number knowing only its place in the sequence!only its place in the sequence!

5

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)(

nn

nFib

5

)()(

)(

nn

nFib

5

1

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n

nn

nFib

• Remember how flowers have leaves Remember how flowers have leaves and petals arranged in sets of and petals arranged in sets of Fibonacci numbers?Fibonacci numbers?

• This ensures that there are This ensures that there are leaves leaves and petals per turn of the stem, and petals per turn of the stem, which allows for maximum exposure which allows for maximum exposure to sunlight, rain, and insects.to sunlight, rain, and insects.

• How about your How about your body?body?

• You have NO IDEA You have NO IDEA how many how many segments of the segments of the human body are human body are related in size to related in size to each other by each other by !!

• The human arm:The human arm:

• The human finger:The human finger:

• When used in dimensioning objects, When used in dimensioning objects, it has always been thought that it has always been thought that produces the most visually appealing produces the most visually appealing results.results.

• Many marketers have used Many marketers have used in their in their products over the years to make products over the years to make them more attractive to you.them more attractive to you.

• An extremely basic example: 3 x 5 An extremely basic example: 3 x 5 greeting cards.greeting cards.

There are numerous other applications of the Fibonacci sequence, Fibonacci numbers, and that were not covered in this presentation—simply because there are far too many to list. Feel free to research on your own if you found any of this interesting.

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

http://library.thinkquest.org/27890/golhttp://library.thinkquest.org/27890/goldenRatio2p.htmldenRatio2p.html

Thank you for your kind attention!Thank you for your kind attention!