The mathematical constant known as "phi" is abbreviated by the lower-case Greek letter "φ".

This constant is known by many names such as the golden ratio, the golden section, the golden mean, the divine proportion or the extreme and mean ratio.

The concept of the phi ratio has existed for thousands of years, and has been utilized by artists, designers and a great many other professions.

Basically it is the belief that the most pleasing artistic proportion (for horizontally-oriented art) is 1.62 (x axis) by 1 (y axis) or 1 by 1.62 for vertically-oriented art.

Perhaps the best way to describe the phi ratio is to state its two properties:
1) The reciprocal of the number φ is phi minus 1 or stated mathematically:

              1 ÷ φ = φ minus 1

2) The number phi squared equals phi plus 1 or stated mathematically:

            φ² = φ + 1

Here is another way to conceptualize the number "phi".
A line of length (L) is bisected into 2 line sections (M and N) such that

the length of the original line (L) divided by the longer line segment (M) equals the length of line segment (M) divided by the length of line segment (N).
Putting that into an equation:

There are some other interesting mathematical properties of the number "phi".

You are no doubt familiar with the "Fibonacci Number Sequence" in which a number sequence is built where each number is the sum of the previous two numbers.
For example these numbers are in a Fibonacci sequence:

1 + 1 = 2     3 + 5 = 8     13 + 21 = 34

If we take Fibonacci #10 and divide it by Fibonnacci #9, we get 55 / 34 = 1.61764705882353...
Fibonacci #25 divided by Fibonnacci #24 produces 75,025 / 46,368 = 1.6180339889579...
Further down the sequence we see Fibonacci #41 divided by Fibonacci #40 yields 1.618033988749894...
Basically, the further we go down that sequence, we get a more accurate value of "phi".

Geometrically speaking, the phi ratio appears in many geometric diagrams.
For example, the phi ratio appears many times in a pentagram (a five-pointed star).

We can see that:

line segment a ÷ line segment b = φ

line segment b ÷ line segment c = φ

line segment c ÷ line segment d = φ

By using this quadratic equation φ² -φ -1 = 0 we can solve for phi using the quadratic formula where a = 1   b = -1   c = -1 :

φ = [--1 +- Sqr Root (-1)² - 4 * 1 * -1] / 2*1

φ = [(1 + Sqr Root (1 + 4))] / 2

***********************************************************