Story Teller: Weird Mathematics Fact: THE GOLDEN RATIO

The golden ratio is a special number approximately equal to 1.618

It appears many times in geometry, art, architecture and other areas.

We find the golden ratio when we divide a line into two parts so that:the longer part divided by the smaller part is also equal to the whole length divided by the longer part.

Beauty
This rectangle has been made using the Golden Ratio, Looks like a typical frame for a painting, doesn't it?
Some artists and architects believe the Golden Ratio makes the most pleasing and beautiful shape.

Many buildings and artworks have the Golden Ratio in them, such as the Parthenon in Greece, but it is not really known if it was designed that way.

The Actual Value
The Golden Ratio is equal to:
1.61803398874989484820... (etc.)
The digits just keep on going, with no pattern. In fact the Golden Ratio is known to be an Irrational Number.

Calculating It
You can calculate it yourself by starting with any number and following these steps:

· A) divide 1 by your number (=1/number)

· B) add 1

· C) that is your new number, start again at A

With a calculator, just keep pressing "1/x", "+", "1", "=", around and around. I started with 2 and got this:
1.6154...
It is getting closer and closer!

Drawing It

Here is one way to draw a rectangle with the Golden Ratio:

· Draw a square (of size "1")

· Place a dot half way along one side

· Draw a line from that point to an opposite corner (it will be √5/2 in length)

· Turn that line so that it runs along the square's side

Then you can extend the square to be a rectangle with the Golden Ratio.

The Formula
That rectangle above shows us a simple formula for the Golden Ratio.
When one side is 1, the other side is:

The square root of 5 is approximately 2.236068, so the Golden Ratio is approximately (1+2.236068)/2 = 3.236068/2 = 1.618034. This is an easy way to calculate it when you need it.
Interesting fact: the Golden Ratio is also equal to 2 × sin(54°), get your calculator and check!

Fibonacci Sequence
There is a special relationship between the Golden Ratio and the Fibonacci Sequence:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ...
(The next number is found by adding up the two numbers before it.)
And here is a surprise: when we take any two successive (one after the other)Fibonacci Numbers,their ratio is very close to the Golden Ratio.
In fact, the bigger the pair of Fibonacci Numbers, the closer the approximation. Let us try a few:

A B B/A
2 3 1.5
3 5 1.6666..
5 8 1.6
8 13 1.625
.. .. ...
144 233 1.618055556
233 377 1.618025751

The Most Irrational ...
I believe the Golden Ratio is the most irrational number. Here is why ...
One of the special properties of the Golden Ratio is that
it can be defined in terms of itself, like this:

(In numbers: 1.61803... = 1 + 1/1.61803...) That can be expanded into this fraction that
goes on for ever (called a "continued fraction"):

So, it neatly slips in between simple fractions.
But many other irrational numbers are reasonably close to rational numbers (for example Pi = 3.141592654... is pretty close to 22/7 = 3.1428571...)

Pentagram

No, not witchcraft! The pentagram is more famous as a magical or holy symbol. And it has the Golden Ratio in it:

· a/b = 1.618...

· b/c = 1.618...

· c/d = 1.618...

Other Names

The Golden Ratio is also sometimes called the golden section, golden mean,golden number,divine proportion, divine section and golden proportion.

In 1509, Luca Pacioli wrote a book that refers to the number as the "Divine Proportion," which was illustrated by Leonardo da Vinci. Da Vinci later called this sectio aurea or the Golden section. The Golden ratio was used to achieve balance and beauty in many Renaissance paintings and sculptures. Da Vinci himself used the Golden ratio to define all of the proportions in his Last Supper, including the dimensions of the table and the proportions of the walls and backgrounds. The Golden ratio also appears in da Vinci's Vitruvian Man and the Mona Lisa. Other artists who employed the Golden ratio include Michelangelo, Raphael, Rembrandt, Seurat, and Salvador Dali.

HUMAN BODY

( NOTE:- After this pic, This is a request to all men here, please scroll down there are a lot of things lined up. I can understand your concern, after all i’m a man as well! and this girl is so hot :P )

The Golden ratio also appears in all forms of nature and science. Some unexpected places include:

Flower petals: The number of petals on some flowers follows the Fibonacci sequence. It is believed that in the Darwinian processes, each petal is placed to allow for the best possible exposure to sunlight and other factors.

Seed heads: The seeds of a flower are often produced at the center and migrate outward to fill the space. For example, sunflowers follow this pattern.

Pinecones: The spiral pattern of the seed pods spiral upward in opposite directions. The number of steps the spirals take tend to match Fibonacci numbers.

Tree branches: The way tree branches form or split is an example of the Fibonacci sequence. Root systems and algae exhibit this formation pattern.

Shells: Many shells, including snail shells and nautilus shells, are perfect examples of the Golden spiral.

Spiral galaxies: The Milky Way has a number of spiral arms, each of which has a logarithmic spiral of roughly 12 degrees. The shape of the spiral is identical to the Golden spiral, and the Golden rectangle can be drawn over any spiral galaxy.

Hurricanes: Much like shells, hurricanes often display the Golden spiral.

Fingers: The length of our fingers, each section from the tip of the base to the wrist is larger than the preceding one by roughly the ratio of phi.

Animal bodies: The measurement of the human navel to the floor and the top of the head to the navel is the Golden ratio. But we are not the only examples of the Golden ratio in the animal kingdom; dolphins, starfish, sand dollars, sea urchins, ants and honeybees also exhibit the proportion.

DNA molecules: A DNA molecule measures 34 angstroms by 21 angstroms at each full cycle of the double helix spiral. In the Fibonacci series, 34 and 21 are successive numbers.

THE TAJ MAHAL