PHI-Divine Proportion

Phi ( = 1.618033988749895... ), most often pronounced fi like "fly", is simply an irrational number like pi ( p = 3.14159265358979... ), but one with many unusual mathematical properties. Unlike pi, which is a transcendental number, phi is the solution to a quadratic equaton.​

The ratio, or proportion, determined by Phi (1.618 ...) was known to the Greeks as the "dividing a line in the extreme and mean ratio" and to Renaissance artists as the "Divine Proportion". It is also called the Golden Section, Golden Ratio and the Golden Mean#-Link-Snipped-#.​

Just as pi (p) is the ratio of the circumference of a circle to its diameter, phi () is simply the ratio of the line segments that result when a line is divided in one very special and unique way.​

​

the ratio of the length of the entire line (A)
to the length of larger line segment (B)​

is the same as​

the ratio of the length of the larger line segment (B)
to the length of the smaller line segment (C).
​

A is 1.618 ... times B and B is 1.618 ... times C.​

Alternatively, C is 0.618... of B and B is 0.618... of A.​

Phi with an upper case "P" is 1.618 0339 887 ..., while phi with a lower case "p" is 0.6180339887, the reciprocal of Phi and also Phi minus 1.​

What makes phi even more unusual is that it can be derived in many ways and shows up in relationships throughout the universe.​