how will you define PHI = 1.618... ??? what the relationship?

Answer 1

It's an achetype embedded in biology and consequently used in art & architecture.
The value is approximately 1 : 1.61803398875 (half of the sum of the square root of five and one).

The 'number' is also a solution to the equation x^2 - x - 1 = 0 (the other solution is actually the reciprocal of the 'number,' also equal to the 'number' minus one!).

In a well-drawn five-sided star, the ratio is everywhere.

The ratio can be approximated by successive terms in the Fibonacci sequence (1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ...), the higher the terms you use, the better the approximation:
55/34 = 1.61765...
89/55 = 1.61818...
144/89 = 1.61798... notice the approximations 'oscillate' around the value.

You can find the ratio in plants (most famously sunflowers and pine cones), insects (ratio of a bee's thorax to abdomen), and even the human body (ratio of height to navel to height of head, lenght of forearm to length of hand, even the bones in your fingers are loosely tied to this ratio)!
Even entire galaxies fit the form of a 'golden spiral!'

Answer 2

Although the number may be defined in terms of Fibonacci numbers, there is a closed formula for it. It is:

phi = (sqrt(5) + 1)/2

Also, you don't HAVE to start with 1 in the series to calculate phi. For example, you could start with 7.

7, 7, 14, 21, 35, 56, 91, 147, 238 . . .

14/7 = 2
21/14 = 1.5
35/21 = 1.66+
56/35 = 1.6
91/56 = 1.625
147/91 = 1.615+
238/147 = 1.619+
etc.

I don't remember why, but it works. Good night!

Answer 3

phi = [sqrt(5) + 1] / 2;

derivation:
Taking a length of (1+x) which consists of 2 sub-lengths of 1 and x respectively. The definition of phi is the ratio in such a way that, the ratio of the full length to the larger sub-length = the ratio of larger sub-length to the smaller sub-length, ie.
(1+x) / x = x / 1
x^2 - x - 1 = 0
which give the ans x = phi = [sqrt(5)+1]/2