The Golden Ratio

If a line segment is divided in two parts, so that the ratio of the larger part to the smaller equals the ratio of the whole segment to the larger part, then that ratio is called the golden ratio. Here, AC/BC is the golden ratio:

Golden Line

A golden rectangle is one whose sides are in the golden ratio. A golden rectangle can be divided into a square and another golden rectangle:

Golden Rectangle

Here is an interactive page that shows why the golden ratio has the value (√5 + 1)/2.

The golden ratio is often denoted by the letter τ (tau), for the Greek word tomé (section), or by φ (phi), for the Greek sculptor Phidias; but it could be for Fibonacci. The reciprocal ratio, (√5 - 1)/2, is often denoted ρ (rho), apparently for "ratio."

Here is a simpler construction of τ and ρ.

Two triangles, and a regular pentagon, constructed with the golden ratio. From these triangles we can deduce some useful trigonometrical ratios.

Here is another construction of a regular pentagon.

Is it a research tool? Is it educational? Is it a nifty toy? Or all of the above? In any case, Zome is an interesting application of the golden ratio.

Last modified on $Date: 2015-04-23 17:26:33 -0400 (Thu, 23 Apr 2015) $

Christopher J. Henrich