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Fibonacci Number

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Nachorng York

on 9 January 2014

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Transcript of Fibonacci Number

What are Fibonacci numbers?
Fibonacci numbers are 0,1,1,2,3,5,8,13,21,34,55...And so on.
How do they work?
Fibonacci numbers are like a pattern they work like so:
0+1=1+1=2+1=3+2=5+3=8+5=13+8=21+13=34+21=55 and so on.

Who was Fibonacci?
Fibonacci Rabbit Problem
The original problem that Fibonacci investigated (in the year 1202) was about how fast rabbits could breed in ideal circumstances.
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 that at the end of its second month a female can produce another pair of rabbits. Suppose that our rabbits never die and that the female always produces one new pair (one male, one female) every month from the second month on. The puzzle that Fibonacci posed was...


How many pairs will there be in one year?


Fibonacci in Nature
Fibonacci in Famous
Art
In addition to in nature, you can find many examples of
of the Fibonacci in famous works of art. This is because
the Fibonacci sequence is considered a naturally
elegant number sequence and using it is said to create
very aesthetically pleasing images, so many artists have
used it to increase the visual appeal of their artwork.

Let’s take a look of some of the notable examples
of the Fibonacci sequence and golden ratio in artwork.
Fibonacci in Famous
Architecture
The ancient Greeks, Romans, Egyptians, and many other civilizations knew
about the golden ratio. They called it by different names, but they all saw
this particular ratio as very pleasing and designed architecture that
utilized it. The Parthenon, which the Greeks constructed, has the golden
ratio present in many different places. The ratio of the width to the
height of the building and the ratio of the height of the building to the
height of the roof are both golden. Plus, the pillars in the front are placed
so that the width of the building is split into a golden segment. The
Egyptians used the golden ratio to construct the Pyramids of Giza. The
ratio of the side length of the pyramid to half the length of the base is,
you guessed it, the golden ratio.
FIBONACCI NUMBER
Fibonacci was one of the ”greatest” European mathematicians of the middle ages.
His full name is Leonardo Pisano, and some people called him Leonardo of Pisa.
The very strange things about him are:
- He grew up in north africa
- He has about 50 names
One other thing about Fibonacci was that nobody knows his birth and death date.
The Rule is: x(n) = x(n-1) + x(n-2)
where:
x(n) is term number "n"
x(n-1) is the previous term (n-1)
x(n-2) is the term before that (n-2)

Example: term 9 would be calculated like this:
x(9) = x(9-1) + x(9-2) = x(8) + x(7) = 21 + 13 = 34


The Rule
The Fibonacci Sequence can be written as a "Rule"
First, the terms are numbered from 0 onwards like this:
By: Reasey
Sokun
Risemoon
Nachorng

So term number 8 is called x8 (which equals 21).
Example: the 8th term is the 7th term plus the 6th term:
x8 = x7 + x6
So we can write the rule:
At the end of the first month, they mate, but there is still only 1 pair.
At the end of the second month the female produces a new pair, so now there are 2 pairs of rabbits.
At the end of the third month, the original female produces a second pair, making 3 pairs in all.
At the end of the fourth month, the original female has produced yet another new pair, the female born two months ago produced her first pair also, making 5 pairs.


The number of pairs of rabbits in the field at the start of each month is 1, 1, 2, 3, 5, 8, 13, 21, 34, ...
So in 12 months (1year) there will be 233 pairs of rabbits
.
The Fibonacci numbers are Nature's numbering system.
They appear everywhere in Nature.
The Fibonacci numbers are therefore applicable to the growth of every living thing, including a single cell, a grain of wheat, a hive of bees, and even all of mankind.
Plants do not know about this sequence - they just grow in the most efficient ways.
White calla lily
1 petal
Euphorbia
2 petals
Shasta daisy
21 petals
A banana has
3 sections
An apple has
5 sections
Conch Shells:
An obvious example of the Fibonacci sequence
in nature is the conch shell. The spiral pattern
of these shells traces out a perfect Fibonacci
spiral shape.

Florets:

Many of those little seed like parts of
flowers that you see, known as florets,
also behave according to the golden
ratio and are arranged in a Fibonacci
spiral pattern.

The Mona Lisa:
Leonardo Da Vinci’s famous portrait of a mysterious woman incorporates the Fibonacci sequence into its design in a very subtle way. As you can see highlighted in the image to the left, the woman’s figure traces out a Fibonacci spiral

Vitruvian Man:
Another piece of artwork by Leonardo Da
Vinci, The Vetruvian Man, uses the Fibonacci
sequence in a different way. The ratio of the
distance from the man’s feet to his stomach
to the distance from the man’s stomach to
his head is approximately the golden ratio.

The Parthenon
Stained Glass
Fibonacci Patterns
A few companies have come up with a new way to incorporate the Fibonacci sequence and the golden spiral in their artwork. They are stained glass manufacturers and the golden spiral is the main focus of a few of the pieces they have made.
A few examples of these glass masterpieces are shown here.
CONTENTs
Who was Fibonacci?
What are Fibonacci numbers?
Fibonacci Rabbit Problem
Fibonacci in Nature
Fibonacci in Famous Art
Fibonacci in Famous Architecture
Stained Glass Fibonacci Patterns
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