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18 Oct 2010

The Phi and The Fibonacci Number

I got so attached to this number Phi after reading the novel "The Da Vinci Code" by Dan Brown.The novel is full of twists and turns and surprises you with its mathematical involvement.As a lover of maths,I started to research about Phi and Fibonacci and here I am posting them the way it amused me.The links between Phi and Fibonacci is so beautiful that one cannot ignore it.
What is Phi ?
"Phi" is the ratio of the sum of the quantities to the larger quantity is equal to the ratio of the larger quantity to the smaller one.
The golden ratio an irrational mathematical constant, approximately 1.6180339887.It is also called Golden section or Golden Mean, divine proportion, divine section, golden proportion, golden cut,golden number, and mean of Phidias.You can come across these terms in Dan Brown's Book.

It is denoted by





The geometrical figure that is commonly associated with Phi is the Golden Rectangle. This rectangle has sides A and B that are in proportion to the Golden Ratio. It is believed that some architects and buildings have been built with this golden ratio. If we take a golden rectangle and bisect one of the base angles ,we should get another golden rectangle.If we continue in this fashion we should get a set of Whirling Rectangles.Out of these Whirling Triangles, we are able to draw a logarithmic spiral that will converge at the intersection of the the two lines like shown below

If we calculate the ratios of their length of the whirling rectangle as shown ,from the figure,we will get the value of "Phi"
2/1 = 2
3/2=1.5
5/3=1.66
8/5=1.6
13/8=1.625 and so on where the results give you the value of " Phi "

As we increase the number of squares we get a figure to look more and more like the Golden Rectangle.And if you notice the sides of the squares,the left side in bold,we can see a familiar sequence called the Fibonacci sequence.
The Fibonacci numbers are 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584,......
1
0+1=1
1+1=2
1+2=3 ..and so on adding the last subsequent numbers

Here when you divide one number in the sequence by the number before it, you obtain numbers very close to one another. In fact, this number is fixed after the 13th in the series. This number is known as the "golden ratio."or "Phi" as explained above in the golden rectangle.
Thus ,the Fibonacci sequence and the golden ratio are intertwined with each other.

Some Interesting Facts:

  • Leonardo da Vinci's drawings of the Human Body is said to be Golden Ratio measuring from foot to naval and from naval to head .Also the human face is said to fall under the Golden Ratio.
  • It is believed that architects and artists throughout history used this Golden ratio to produce objects of great beauty like Michelangelo's "David" and the Greek temples .
  • It is also believed that the Golden Ratio is spread in nature from the leaves to the flowers ,pine cones,spirals ,etc...

1 comments:

  1. I liked this post.Thanks for sharing!

    ReplyDelete

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