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FABONACCI SEQUENCES

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The Fibonacci sequence is a set of numbers that starts with a one or a zero, followed by a one, and proceeds based on the rule that each number (called a Fibonacci number) is equal to the sum of the preceding two numbers. 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ... FABONACCI SEQUENCES

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PROCEDURE This computer drawing was created using Fibonacci Numbers. This is called a Fibonacci Spiral

The Fibonacci Numbers : 

The Fibonacci Numbers The Fibonacci Sequence is the series of numbers: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ... 1 1 2 }+

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The Fibonacci Sequence is the series of numbers: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ... 1 1 2 3 The Fibonacci Numbers }+

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The Fibonacci Numbers The Fibonacci Sequence is the series of numbers: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ... 1 1 2 3 5 8 13 21 34 55 }+

GRAPHICAL REPRESENTATION : 

GRAPHICAL REPRESENTATION

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29 little boxes down 15 little boxes across 1 little square Square Spiral

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1 more little square 29 little boxes down 15 little boxes across

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2 x 2 square 29 little boxes down 15 little boxes across

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3 x 3 square

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Now by drawing quarter circles in each square and joining to form the spiral we can draw “Nautilus” shell

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“Nautilus” shell

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APPLICATIONS

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19 13 8 5 3 2 1 Fibonacci’s sequence… in nature

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Fibonacci’s sequence… in nature 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584… Pine Cone

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Fibonacci’s sequence… in nature 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584… Cauliflower

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1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584… Fibonacci’s sequence… in nature 3 petals: lilies 5 petals: buttercups, roses 8 petals: delphinium 13 petals: marigolds 21 petals: black-eyed susans 34 petals: pyrethrum 55/89 petals: daisies

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Fibonacci’s sequence… in nature 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584… Leaves are also found in groups of Fibonacci numbers. Branching plants always branch off into groups of Fibonacci numbers. A plant that grows very much like this is the "sneezewort“.

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Spirals, like the one we saw earlier are common in nature. Fibonacci’s sequence… in nature

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The intervals between keys on a piano are Fibonacci numbers. 2 3 5 8 white13 w & b Fibonacci’s sequence… in Music

Fibonacci .....in human body : 

Fibonacci .....in human body The lengths of bones in a hand are Fibonacci numbers.

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The human arm: The human finger: A Closer Look at the application of Fibonacci Sequences in Human body

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Fibonacci… and his rabbits OK, OK… Let’s talk rabbits… Leonardo Pisano Fibonacci

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Suppose a newly-born pair of rabbits, one male, one female, are put in a field. Rabbits are able to mate at the age of one month. So at the end of its second month a female can produce another pair of rabbits. Suppose that our rabbits never die. And the female always produces one new pair (one male, one female) every month from the second month on. The puzzle that I posed was... How many pairs will there be in ten months?

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Pairs 1 pair At the end of the first month there is still only one pair

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Pairs 1 pair 1 pair 2 pairs End first month… only one pair At the end of the second month the female produces a new pair, so now there are 2 pairs of rabbits

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Pairs 1 pair 1 pair 2 pairs 3 pairs End second month… 2 pairs of rabbits At the end of the third month, the original female produces a second pair, making 3 pairs in all in the field. End first month… only one pair

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Pairs 1 pair 1 pair 2 pairs 3 pairs End third month… 3 pairs 5 pairs End first month… only one pair End second month… 2 pairs of rabbits At the end of the fourth month, the first pair produces yet another new pair, and the female born two months ago produces her first pair of rabbits also, making 5 pairs.

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1 1 2 3 5 8 13 21 34 55 1 2 3 4 8 6 7 5 9 10

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Golden Ratio

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The Golden (or Divine) Ratio has been talked about for thousands of years.People have shown that all things of great beauty have a ratio in their dimensions of a number around 1.618 1 1.618 Golden Ratio

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Here is a surprise. If you take any two successive (one after the other) Fibonacci Numbers, their ratio is very close to the Golden Ratio "φ“ which is approximately 1.618034... In fact, the bigger the pair of Fibonacci Numbers, the closer the approximation. Let us try a few: Golden Ratio

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Golden Ratio

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Leonardo da Vinci showed that in a ‘perfect man’ there were lots of measurements that followed the Golden Ratio. Golden Ratio

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Fibonacci numbers and the Fibonacci sequence are prime examples of "how mathematics is connected to seemingly unrelated things." Even though these numbers were introduced in 1202 in Fibonacci's book Liber abaci, they remain fascinating and mysterious to people today.

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41 “The essence of FIBONACCI SEQUENCE is not to make simple things complicated, but to make complicated things simple.”