THE GOLDEN RATIO
The Divine Proportion

by Florence W Deems
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Golden Ratio, or the Golden Mean, or the Divine Proportion - a special number that appears in many places in nature, art, architecture, and geometry. It is an irrational number (see below) that is approximately 0.618. This number represents the ratio of one side of a rectangle to its adjacent side, for a special rectangle called the Golden Rectangle.

Golden Rectangle - a rectangle where the ratio of the length to the width is the Golden Ratio. To draw a Golden Rectangle, establish the length of the longest side (A) and then multiply that number by 0.618. This smaller number will be the length of the shortest side (B).

Golden Spiral - a special spiral drawn inside of a Golden Rectangle that appears in many places in nature. To calculate the increasingly larger Golden Rectangles needed to draw this Golden Spiral, add the length (A) and the width (B) to get the next measurement for the next side (C) needed.

Irrational Number - a number that cannot be written in the form a/b, where a and b are both integers. An integer means a whole number, rather that a decimal or fraction.

Phi - Φ or φ - a Greek symbol referring to the Golden Ratio of 0.618.

Sequence - a "list" of numbers where there is a special relationship between successive numbers in the list. For example: the Fibonacci sequence (1, 1, 2, 3, 5, 8, 13, 21, ...). This sequence can be continued by adding the last number to the number before it. So 13 plus 21 equals 34. The person who discovered this particular sequence was Leonardo Fibonacci.

Calculating for the Golden Mean sequence: You simply do the same type of addition to establish a Golden Ratio Sequence as we did for extending the Fibonacci sequence. (6, 10, 16, 26, 42, 68, 110, 178, ...). Please note that these numbers have been rounded off.

If you are mathematically inclined and want to learn more about the Golden Ratio, the following website is set up to explain and to teach people how to use the Golden Ratio. http://cuip.uchicago.edu/~dlnarain/golden/

But for those who, like me, don't want to know or can't remember the mathematical formula, here's a great online calculator to figure Golden Ratio proportions: http://www.calculator.com/calcs/calc_golden.html

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The first image above shows how the various Golden Rectangles fit, plus the diagonals discussed earlier.

The next image shows the Golden Spiral drawn in. One of the best examples in nature of this Golden Spiral is the way Nautilus shells grow. Please see This page for an interesting discussion of this spiral in nature.

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