The Golden Proportion & Phi


The Golden Proportion is that point at which a line divides such that the long segment is to the short segment as the whole line is to the long segment:
a : b : : a + b : a

 
The Golden Ratio is represented by the Greek symbol  Phi 

One definition of Phi is that to square it you just add 1 or, in mathematics:   Phi2 = Phi + 1 

In fact, there are two numbers with this property, one is Phi and another is closely related to it when we write out some of its decimal places. 

Here is a mathematical derivation of the two values:-

Multiplying both sides by Phi gives a quadratic equation:
                  Phi2 = Phi + 1 or 
                  Phi2 - Phi - 1 = 0

By solving this quadratic equation, you find that the 2 possible values of Phi are:

1·6180339887...(Phi) and   –0·6180339887... (phi)

Phi is a mathematical constant used in the calculation of Fibonacci numbers. The nth Fibonacci number, Fn, is calculated as follows :

F0 = 0
F1 = 1
Fn = Fn-1 + Fn-2

& 

The Golden Ratio to the first 50 decimal points is :

1.618033988749894848204586834365
63811772030917980576......

Other designs of the golden proportion :


The Pentagram is the favourite of the Phythagoreans (followers of Phythagoras and his work). The number five was peculiarly associated by the Pythagoreans with the art of healing, and the pentagram, or five-pointed star, was recognised as a symbol of health. Note that this pentagram forms the basic structure of the Sarva Dharma Symbol which has 5 petals and carries the shape of the lotus. Thus, the Sarva Dharma symbol is a symbol of Life and Unity.

 Ancient Greeks adopted the Divine Ratio as sacred measure and aesthetic proportion in art and architecture. Greeks sought divine guidence at the Oracle of Del-PHI, whose symbol was a serpent coiled around an egg. The Greeks thought it was the most pleasing proportion to the human eye. Parthenon was designed using this golden proportion. 

18 is again another unique number, because 
1 + 8 = 9, where 9 represents Brahman


Temple of Neptune, Paestum

"The Golden Ratio manifests in the whole of creation. Take the ratio of the length of a man and the height of his navel. The ratio of the sides of the Great Temple. The ratio between the long and short sides of a pentagram. Why is this? Because the ratio of the Whole to the Greater is the ratio of the Greater to the Lesser." 
- Pythagoras (Mathematician & Philosopher)

References:

Golden Ratio by Alex Bonjour, Alex.Bonjour@Rose-Hulman.Edu (source: internet)

PHI: The Divine Ratio - Life's Archetypal Architecture by David Yarrow, October 1996

The Golden Section Ratio: Phi by Dr Ron Knott,    R.Knott@surrey.ac.uk (source: internet)