The Golden ratio is obtained when we divide a line into two parts a and b so that:

•        The whole length a+b is divided by the long part a that is also equal to
•        The long part a divided by the short part b

$\dpi{120} \LARGE \frac{a +b}{a}= \frac{a}{b} = \frac{1 + \sqrt5}{2} = 1.618.....=\Phi$

Some artists and architects believe the Golden Ratio makes the most pleasing and beautiful shape.

This rectangle has been made using the Golden Ratio, Looks like a typical frame for a painting, doesn't it?

In mathematics, the Fibonacci sequence is the ordering of numbers in the following integer sequence: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144… and so on forever. Each number is the sum of the two numbers that precede it.

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.

Sunflower seeds grow in Fibonacci spirals.

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.