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Fibonacci Numbers andFibonacci Numbers and

Say What?Say What?

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What is theWhat is theGolden RatioGolden Ratio??

'',,

examine an interesting sequence (or list)examine an interesting sequence (or list)..

0,1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144,0,1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144,

233, 377, 610233, 377, 610

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The series is derived by the following formula:The series is derived by the following formula:

== -- --

oror

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Now, I know what you might be thinking:Now, I know what you might be thinking:

"What does this have to do with the"What does this have to do with the oldenolden

Ratio?Ratio?

s sequence o num ers was rsts sequence o num ers was rstdiscovered by a man named Leonardodiscovered by a man named Leonardo

Fibonacci, and hence is known asFibonacci, and hence is known as

Fibonacci'sFibonacci's sequence.sequence.

Math GEEK

Leonardo FibonacciLeonardo Fibonacci

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The relationship of this sequence to theThe relationship of this sequence to the

GoldenGolden Ratio lies not in the actual numbersRatio lies not in the actual numbers

o t e sequence, ut n t e rat o o t eo t e sequence, ut n t e rat o o t econsecutive numbers. Let's look at some ofconsecutive numbers. Let's look at some of

the ratios of these numbers:the ratios of these numbers:

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 6101, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610

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== ..3/2 = 1.53/2 = 1.5

nce a a o s as ca y a rac on or

a division problem) we will find the

ratios of these numbers by dividing

the larger number by the smaller

= .= .8/5 = 1.68/5 = 1.6

number that fall consecutively in the

series.

So, what is the ratio of the 2nd and

13/8 = 1.62513/8 = 1.625

==

3rd numbers?

Well, 2 is the 3rd number divided by the ..

34/21 = 1.61934/21 = 1.6192nd number which is 1

2 divided by 1 = 2

..89/55 = 1.61889/55 = 1.618

And the ratios continue like this.

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= .= . (bigger)(bigger)

3/2 = 1.53/2 = 1.5 (smaller)(smaller)

we continue downwe continue down 5/3 = 1.675/3 = 1.67(bigger)(bigger)8/5 = 1.68/5 = 1.6(smaller)(smaller)

,,

ratios seem to beratios seem to be

13/8 = 1.62513/8 = 1.625 (bigger)(bigger)

==converging upon oneconverging upon one

number (from bothnumber (from both

..

34/21 = 1.61934/21 = 1.619 (bigger)(bigger)

sides of the number)!sides of the number)! ..(smaller)(smaller)

89/55 = 1.61889/55 = 1.618

Fibonacci Number calculator

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Depiction of Golden Ratio

2

2.1

1.8

1.9

1.5

1.6

.

1.3

1.4

1.1

1.2

1

0 5 10 15 20 25 30

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irrational number: it has an infiniteirrational number: it has an infinite

number of decimal places and it nevernumber of decimal places and it never

the Golden Ratio to 1.618.the Golden Ratio to 1.618.

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We work with another importantWe work with another important

,,which is approximately 3.14. Sincewhich is approximately 3.14. Since

we don't want to make the Goldenwe don't want to make the Golden

own Greek letter: phi.own Greek letter: phi.

Phi

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which is equal to:

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One more interesting thing about Phi is itsOne more interesting thing about Phi is its

reciprocal. If you take the ratio of any number inreciprocal. If you take the ratio of any number in

the reverse of what we did before), the ratio willthe reverse of what we did before), the ratio will

reciprocal of Phi: 1 / 1.618 = 0.618. It is highlyreciprocal of Phi: 1 / 1.618 = 0.618. It is highly

unusual for the decimal integers of a number andunusual for the decimal integers of a number andits reciprocal to be exactly the same. In fact, Iits reciprocal to be exactly the same. In fact, I

cannot name another number that has thiscannot name another number that has this

proper y s on y a s o e mys que o eproper y s on y a s o e mys que o eGolden Ratio and leads us to ask: What makes itGolden Ratio and leads us to ask: What makes it

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The Golden Ratio is not just some number that mathThe Golden Ratio is not just some number that math

eac ers n s coo . e n eres ng ng s a eepseac ers n s coo . e n eres ng ng s a eepspopping up in strange placespopping up in strange places -- places that we may notplaces that we may not

ordinarily have thought to look for it. It is important to noteordinarily have thought to look for it. It is important to note

that Fibonacci did not "invent" the Golden Ratio; he justthat Fibonacci did not "invent" the Golden Ratio; he just

discovered one instance of where it appeared naturally. Indiscovered one instance of where it appeared naturally. In

Egyptians, the Mayans, as well as the Greeks discoveredEgyptians, the Mayans, as well as the Greeks discovered

the Golden Ratio and incorporated it into their own art,the Golden Ratio and incorporated it into their own art,arc ec ure, an es gns. ey scovere a e o enarc ec ure, an es gns. ey scovere a e o en

Ratio seems to be Nature's perfect number. For someRatio seems to be Nature's perfect number. For some

reason, it just seems to appeal to our natural instincts. Thereason, it just seems to appeal to our natural instincts. The

most basic example is in rectangular objects.most basic example is in rectangular objects.

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Look at the following rectangles:Look at the following rectangles:

ow as yourse , w c o em seems o e e mosow as yourse , w c o em seems o e e mos

naturally attractive rectangle? If you said the first one, thennaturally attractive rectangle? If you said the first one, then

you are probably the type of person who likes everything toyou are probably the type of person who likes everything to

be symmetrical. Most people tend to think that the thirdbe symmetrical. Most people tend to think that the thirdrectangle is the most appealing.rectangle is the most appealing.

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''

and width, and compare the ratio of length to widthand width, and compare the ratio of length to width

for each rectangle you would see the following:for each rectangle you would see the following:Rectangle one: Ratio 1:1Rectangle one: Ratio 1:1

Rectangle two: Ratio 2:1Rectangle two: Ratio 2:1

ec ang e ree: a o . :ec ang e ree: a o . :

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Have you figured out why the third rectangle is theHave you figured out why the third rectangle is the

most appealing? That's rightmost appealing? That's right -- because the ratio ofbecause the ratio ofits length to its width is the Golden Ratio! Forits length to its width is the Golden Ratio! For

centuries, designers of art and architecture havecenturies, designers of art and architecture have

their work.their work.

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Fibonacci made the following assumptionsFibonacci made the following assumptions

based on the breeding habits of rabbitsbased on the breeding habits of rabbits

He imagined a pair of rabbits in a field on theirHe imagined a pair of rabbits in a field on their

own.own.

ne ma e one ema e.ne ma e one ema e.

Rabbits never die.Rabbits never die. Female rabbits always produces one new pairFemale rabbits always produces one new pair

(one male, one female) every month from the(one male, one female) every month from the

.. The gestation period for rabbits is one month.The gestation period for rabbits is one month.

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..

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8

there be after 5 months?

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male and one female.male and one female.

month.month.

n ree ng.n ree ng.

Rabbits die and get killed.Rabbits die and get killed.

EtcEtc

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Practise questionPractise question

How many she calves after 10years?

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In this activity, we study the ancestors of aIn this activity, we study the ancestors of a

oney ee.oney ee.

In a family of honeybees not al