Sun, 16 Mar 2003 23:33:00 EST (-0500)

I recently worked out the value of "The Golden Ratio", since I had heard the definition on the radio while driving.

The author of a new book by that name explained that if you take a line (segment) and divide it into a large part and a small part such that the ratio of the lengths of the large to the small part is equal to the ratio of the length of the whole segment to that of the large part, that ratio is the Golden Ratio, "Phi".

This means that Phi must be equal to one over Phi (ie the proportion of the large part, taking the whole as 1 -- Phi is the whole to the large part, meaning the large part must be the whole over Phi) divided by the difference of 1 less one over Phi, which difference must be the size of the small part.

Solving, you first get Phi equals 1 divided by the difference Phi minus one, then you get Phi squared minus Phi equals 1, and then you get the quadriatic Phi squared minus Phi minus one equals zero.

Using the formula for the roots of a quadriatic, ie the roots equal 2a divided into the sum (or difference) of negative b plus or minus the square root of the difference b squared minus 4ac, or "completing the square" if you don't remember the formula, you get Phi equals half of the sum of one plus the square root of 5, discarding the result given by the negative square root of five as inadmissible since it would give a negative ratio. This gives approx. 1.618034 (etc.).

The value is listed to a bunch of decimal places the Handbook of Chemistry and Physics.

Have you heard of any alternate definitions of Phi? I wonder how the Greeks worked it out. Did they have some marvelous method using only compass and straighedge (and paper and pencil) (and maybe a flat surface on which to work)?

Last updated April 12, April 2, 2003