Reason Why? I Love Maths - Part 1

        I love Maths not as Subject for scoring marks, but for the way it has taken shape. Only lovers of Maths can appreciate things so-called Encryption. Similarly, some interesting facts about Maths.
      
        I believe this is some super power. To show humans that everything is inter-related and nothing is stand-alone. Maths can be seen in nature, as usually i blabber the artists and mathematician of ancient times made it clear " MATHS IS EVERY WHERE".

       Have any one heard of something called golden ratio.If not,Here it is

Expressed algebraically:

The definition is the ratio of the sum of the quantities to the larger quantity is same as the ratio of larger quantity to smaller quantity.

The golden ratio is often denoted by the Greek letter phi (Φ or φ).
 The golden ratio is an irrational mathematical constant, approximately 1.6180339887. The value of phi used is 1.618.

Significance of 1.618:


Hope  everyone knew Fibonacci series, our class guys wont forget( Every lab this will be the first program we do write).

Even then,
Fibonacci numbers are: 0,1,1,2,3,5,8,13,21,34,55,89,....

Just compute the ratio of larger number to smaller one,

1/0 = undermined.
1/1 = 1
2/1 = 2
3/2 = 1.5
5/3 = 1.6666
8/5 = 1.6
13/8 = 1.625
21/13 = 1.61538.

so it seems that the value gets stable approx 1.618...

Golden Rectangle:

When we equate this separately, we will get phi^2 - phi -1 = 0.

Solving the quadratic equation, we get phi = 1.618......

The rectangle that have sides ratio as 1:1.618 is called Golden Rectangle.

The golden rectangle is shown below:

We can see the breadth is 1 units and  length is 1 + sqrt(5) / 2 (which equals 1.618).
So breadth : length = 1 : 1.618.

Golden ratio in Nature:

Click the pic to see animated view...
           

This is also golden rectangle, but u try finding it?

Golden Spiral is a logarithmic spiral whose growth factor b is related to golden ratio.Specifically, a golden spiral gets wider( or further from its origin) by a factor of j for every quarter turn it makes.

                                           A Nautilus shell reveals golden spiral




 Coming back to my Da Vinci, as told he is also a mathematician. So his paintings do portrays this Golden rectangle.

The Vetruvian Man and Monalisa :

In architect too,

The fact that golden ratio exist in human hand,in nature, in animal. Golden ratio is only 0.000.. percent of maths concepts, think of entire maths relation with nature and humans. Unimaginable.