What is the Golden section of a line?

Cindy

the terms golden ratio, golden section, and golden cut date back to Euclid. Let a line AB of length l be divided into two segments by the point C.

A_________________C__________B

If the lengths satisfy AB:AC as AC:CB then C is the golden cut or the golden section of AB. For simplicity if we let AC = x and CB = 1 we find

(x+1)/x =x/1 leading to x^2-x-1 = 0 and, in turn x = (1+SQRT5)/2, about 1.6183.

We usually denote this constant by the greek letter 'phi'. It has sometimes been alluded to as the divine proportion. When you hear of golden rectangles this refers to rectangles whose sides are in the ratio of phi to one. It is said the front of the Parthenon in Greece has such proportions - a proportion that is often adopted by artists for their canvases.

For more information on this and its connection to logarithmic spirals, phyllotaxis etc. I recommend the book, The Divine Proportion - A Study in Mathematical Beauty, by H. E. Huntley, Dover, ISBN 0-486-22254-3.

Best wishes, Denis Hanson