Golden Rectangle HistoryDate: 9/7/96 at 2:18:49 From: Peter Jackson Subject: Golden Rectangle History Can you tell me anything about the history of the golden rectangle? When was it discovered and by whom? Date: 9/12/96 at 10:31:26 From: Doctor Jerry Subject: Re: Golden Rectangle History There are entire books written about the golden rectangle or the golden section. Some books concentrate on the applications of the golden section or ratio to architecture. Try any good book on the history of mathematics. Another source is Sir Thomas L. Heath's edition of Euclid's Elements. He gives many historical comments in footnotes and other discussion. For example, in vol. 1, page 137 of the Dover paperback edition, Heath mentions that Plato refers to the golden section (closely related to the golden rectangle) and Euclid includes the construction in proposition 11 of Book II of his Elements. Heath also says that the idea is most probably Pythagorean, which means that it was known ca. 500 BC. I doubt if it is known or will ever be known exactly who discovered the golden section. As Heath says, "As the solution of this problem [see below] is necessary to that of inscribing a regular pentagon in a circle, we must necessarily conclude that it was solved by the Pythagoreans, or, in other words, that they discovered the geometrical solution to the quadratic equation a(a-x)=x^2." Proposition 11 of Book II is: To cut a given straight line so that the rectangle contained by the whole and one of the segments is equal to the square on the remaining segment. If AB is the given line, we want to cut it at H so that AB*BH = AH*AH. If AB has length 1 and AH = x, then Prop. 11 is, in modern terms, x^2+x-1 = 0. The positive root is (-1+sqrt(5))/2, which is approx 0.62. This is the golden ratio, on which the golden rectangle is based. -Doctor Jerry, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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