Fibonacci+Sequence

=//__**Fibonacci Sequence and Nature **__//=

=__**What is Fibonacci Sequence?**__=

Fibonacci Sequence is a pattern in which a number equals the two preceding terms added together. So, starting from Fn-2, Fn-1... Fn= (Fibonacci number) F1- 0 F2- 1 (F1+1) F3- 1 (F1+F2) F4- 2 (F2+F3) F5- 3 (F3+F4)... Do you see the Pattern?

Essentially, all you have to do is add the two preceding numbers of the number that you want. If you want the 5th number of the Fibonacci Sequence, you had the 3rd and the 4th number. 3= 2+1.

The Formula for finding Fibonacci Number (Fn)... =Fn= Fn-1+ Fn-2= This Formula only works from the 3rd Fibonacci number, because the 2nd and the 1st Fibonacci number do not have 2 preceding numbers.

=__**Interesting Formulas For Fibonacci Sequence/Numbers**__=

There are lots of different formulas that were developed through the basis of the Theory of Fibonacci Numbers.

These are only a few... 1. **Cassini's Formula**: Fn+1 · Fn-1 - (Fn)**(To the 2nd power) = (-1) (to the nth power)**

To see if this formula actually makes sense, lets try putting in a number. So, the Fibonacci Number I will put in is 3. F(3+1) · F (3-1) - (F3)2= (-1)3. <--- (This isn't 3, it's the the 3rd power, same applies for (F3)2. It's the the 2nd power) (F2 · F4) - (F3)2= (-1)3

The 2nd Fibonacci number of the Fibonacci Sequence=1. The 3rd Fibonacci number of the Fibonacci Sequence= 2 The 4th Fibonacci number of the Fibonacci Sequence= 3.

(1 x 3) - (2)2= (-1)3 3-4= -1 -1=-1. So this Formula Holds True.

2. **Simpson's Relations**: Fn+1 · Fn-1 + (-1)n-1 = (Fn)2 **< (The 2 after Fn, is to the 2nd power, and the n-1 in front of (-1) is to the n-1 power)** To see if this formula actually works, let's try putting in the number 4 of the Fibonacci Number. F (4+1) x F (4-1) + (-1) (4-1 power) = (F4)(To the 2nd power. F5 x F3 - 1= (F4) (to the 2nd power)

The 5th Fibonacci number of the Fibonacci Sequence= 5.
(5 x 2 ) - 1 = 3 (to the 2nd power) 10 - 1= 9 9=9

So this Formula Holds True.

= = =__Golden Rectangle and Leonardo Da Vinci__=

== If you look at the painting of __Mona Lisa__, you may see many rectangles inscribed in one big rectangle, which is almost the size of the painting. This is called the **//__Golden Rectangle.__//

// I will give you a procedure that will help further illustrate the idea behind Golden Ratio.

1. Look at the two smallest squares (adjacent to each other), situated on her nose, and her cheek. 2. Think of each of their side length as 1. Therefore, it will match the first 2 numbers of the Fibonacci Sequence: 1,1. 3. Now look at the square that is on her mouth, to her chin. Since it is double the length of square in number 2, give the side length a two. Therefore this will match the first 3 numbers of the Fibonacci Sequence: 1,1,2. 4. Now, look at the square adjacent to the square on number 2, that is to the left (from the perspective of the viewer). According to the Addition postulate of segments, it states that a+b= c, if the sides a and b add up to be the length of C. Since the side length is square on number 2+Square on number 3= 1+2= 3. Therefore, it will match the first 4 numbers of the Fibonacci Sequence: 1,1,2,3.

This pattern of "Fibonacci square" continues on in the painting of __Mona Lisa!!__

//** =__Golden Rectangle Continued__= Another diagram, (This one is straight forward) is the actual Golden Rectangle.

I will name some of these squares to make it more easier to understand. Square A: Side length of 1. Square B: Side length of 1. Square C: Side length of 2. Square D: Side length of 5. Square E: Side length of 8. Square F: Side length of 13.

A,B,C,D,E,F,= 1,1,2,5,8,13. Do you see the pattern?


 * Something Really Amazing:** If you create a square, with lengths and width of that of a Fibonacci Number, it forms a rectangle: The Golden Rectangle. The rectangle above has dimensions: 13 x 21.

=__Life and Fibonacci Sequence__=

Your question may be... How is our life, in any way shape or form, related to some mathematical sequence? Have you ever seen a flower? Yes/No.

If you circled no, then I guess the Fibonacci Sequence applies to you in a different way. However, everyone (almost everyone) that has ever seen a flower has encountered the world of Fibonacci Sequence.

=Ex: White Calla Lily- 1 petal Ex: Euphorbia- 2 petals=

 =**Ex: Trillium- 3 petals Ex: Columbine- 5 petals**=

=Ex: Bloodroot- 8 petals Ex: Black-eyed Susan- 13 petals=

=Ex: Shasta Daisy- 21 petals Ex: Field Daises- 34 petals=  =If you look at these flowers, they form a Fibonacci sequence: 1,1,2,3,5,8,13,21,34!!!=

=**__The World and Fibonacci Numbers (Youtube!) This video relates Fibonacci Numbers and the World:__ media type="youtube" key="TAqTfBaqxGM" height="344" width="425" __URL:__ __http://www.youtube.com/watch?v=TAqTfBaqxGM__ **MLA Citation** __**:**__ __"Fibonacci Sequence: Numbers of Life, Matrix of Universe ."__ //__Fibonacci Sequence: Numbers of Life, Matrix of Universe__// __. Web. 16 Dec 2009. .__ **= == == =__Fun Version of Fibonacci Sequence!! Also has information on Golden Sequence, rectangle, and proportion media type="youtube" key="2nAycC7sGVI" height="344" width="425" URL: http://www.youtube.com/watch?v=2nAycC7sGVI __= =MLA CITATION: "Fibonacci sequence / Golden scale ." //Fibonacci sequence / Golden scale//. Web. 16 Dec 2009. . =

=Work Cited (MLA Citation)=  "Art." Art. Web. 17 Nov 2009. .

Britton, Jill. "TITLE ." Fibonacci Numbers in Nature. 7 May 2005. Web. 15 Nov 2009. <__http://britton.disted.camosun.bc.ca/fibslide/jbfibslide.htm__>.

" Fibonacci's Calculator." Fibonacci Number Properties. 18,November 2009. Web. 17 Nov 2009. .

"Fibonacci Numbers ." //Fibonacci Numbers // <span style="color: #000000; font-family: 'Times New Roman',Times,serif; font-size: 120%;">. Web. 14 Jan 2010. <http://www.jimloy.com/algebra/fibo.htm>. <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">"FIBONACCI SEQUENCE." <span style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px; font-family: verdana,helvetica,sans-serif; font-size: 10px; letter-spacing: 1px; line-height: normal;">//<span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">FIBONACCI SEQUENCE // <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">. Web. 15 Jan 2010. <http://www.geom.uiuc.edu/~demo5337/s97b/fibonacci.html>. <span style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px; color: #333333; font-family: verdana,helvetica,sans-serif; font-size: 10px; letter-spacing: 1px; line-height: normal;"> <span style="color: #000000; font-family: 'Times New Roman',Times,serif; font-size: 120%;"> "Golden Ratio ." Golden Ratio. Web. 17 No <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">v 2009. <http://www.fabulousfibonacci.com/portal3/golden-ratio.html>.

Knott, R. "The life and numbers of Fibonacci ." <span style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px; font-family: verdana,helvetica,sans-serif; font-size: 10px; letter-spacing: 1px; line-height: normal;">//<span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">The life and numbers of Fibonacci // <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">. September, 1997. Web. 14 Jan 2010. <The life and numbers of Fibonacci >.

Morris, Stephanie. "Fibonacci Numbers." <span style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px; font-family: verdana,helvetica,sans-serif; font-size: 10px; letter-spacing: 1px; line-height: normal;">//<span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">Fibonacci Numbers in Nature // <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">. Web. 15 Jan 2010. <http://jwilson.coe.uga.edu/emt669/student.folders/morris.stephanie/EMT.669/Essay.3/Fibonacci.Essay.html>.

"The Fibonacci Sequence ." <span style="font-family: 'Times New Roman',Times,serif; font-size: 144%;"> <span style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px; font-family: verdana,helvetica,sans-serif; font-size: 10px; letter-spacing: 1px; line-height: normal;">//<span style="color: #000000; font-family: 'Times New Roman',Times,serif; font-size: 120%;">Fibonacci Sequence (PRIME) // <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">. Web. 15 Jan 2010. <http://www.mathacademy.com/pr/prime/articles/fibonac/index.asp>.

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