Most people are familiar with the number Pi, since it is one of the most ubiquitous irrational numbers known to man. But, there is another irrational number that has the same propensity for popping up and is not as well known as Pi. This wonderful number is Phi, and it has a tendency to turn up in a great number of places, a few of which will be discussed in this essay.
One way to find Phi is to consider the solutions to the equation
When solving this equation we find that the roots are
x = 
~ 1.618... or x=
~ -.618...
~ 1.618... or x=~ -.618...We consider the first root to be Phi. We can also express Phi by the following two series.
Phi = 
or Phi = 
or Phi = What is golden ratio
In mathematics and the arts, two quantities are in the golden ratio if 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 is an irrational mathematical constant approximately 1.6180339887.
Two quantities a and b are said to be in the golden ratio φ if:
This equation unambiguously defines φ.
The fraction on the left can be converted to
Multiplying through by φ produces
which can be rearranged to
The only positive solution to this quadratic equation is
a body of literature on the aesthetics of the golden ratio was developed. As a result, architects, artists, book designers, and others have been encouraged to use the golden ratio in the dimensional relationships of their works.
The Parthenon's facade as well as elements of its facade and elsewhere are said by some to be circumscribed by golden rectangles
Leonardo da vinci's illustrations of polyhedra in De Divina Proportione (On the Divine Proportion) and his views that some bodily proportions exhibit the golden ratio have led some scholars to speculate that he incorporated the golden ratio in his paintings.
Fibonacci sequences appear in biological settings, in two consecutive Fibonacci numbers, such as branching in trees, arrangement of leaves on stems, the fruitlets of a pineapple,the flowering of artichoke, an uncurling fern and the arrangement of a pine cone. In addition, numerous poorly substantiated claims of Fibonacci numbers or golden section in nature are found in popular sources, e.g. relating to the breeding of rabbits, the spirals of shells, and the curve of waves.The Fibonacci numbers are also found in the family tree of honeybees.
A Fibonacci spiral created by drawing arcs connecting the opposite corners of squares in the Fibonacci tiling; this one uses squares of sizes 1, 1, 2, 3, 5, 8, 13, 21, and 34; a great example of golden ratio.The use of Golden ratio is great in present era. Try to get more about it.
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