About Phi and its unusual properties

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.

The ratio, or proportion, determined by Phi (1.618...) was known to the Greeks as the "Golden Section" and to Renaissance artists as the "Divine Proportion" It is also called the Golden Ratio and the Golden Mean.

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.

 

Divide a line so that:

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). More simply, A is to B as B is to C.

This happens only at the point where 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 1.6180339887..., while phi with a lower case "p" is 0.6180339887, which is the reciprocal of Phi and also Phi minus 1.  Phi is the only number whose reciprocal has a difference in value of 1.  Phi can be calculated by adding 1 to the square root of 5 and then dividing the sum by 2.

What makes phi unusual is that it can be derived in many ways mathematically and geometrically and shows up in relationships throughout the universe.  Phi appears in the proportions of the human body and face, the proportions of many other animals, plants, DNA, the solar system, population growth the stock market and in many aspects of life and the universe.  Our perceptions of beauty in the humans are based on how close facial proportions conform to phi.

Some believe phi is the most efficient outcome, the result of natural forces. Some believe it is a universal constant of design, the signature of God.  Whatever you believe, the pervasive appearance of phi in all we see and experience creates a sense of balance, harmony and beauty in the design of all we find in nature.  It should be no surprise then that mankind would use this same proportion found in nature to achieve balance, harmony and beauty in its own creations of art, architecture, colors, design, composition, space and even music.

The Golden Section, the phi relationship, has long been used by craftsmen and professionals of many fields to understand and achieve beauty, balance and harmony in their work. Artists, architects, designers, engineers, musicians, photographers, sculptors, surgeons and stock analysts alike use this relationship to achieve excellence in their results.

The study and application of the golden section is now much easier with PhiMatrix software.  Put your ruler and calculator away and let the phi relationships just jump out at you as you work or play with the software to bring your work to a new level in any field.

Learn more about phi at the http://goldennumber.net, also developed by Gary B. Meisner, the developer of PhiMatrix software.