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Golden Rectangle History

Date: 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 

-Doctor Jerry,  The Math Forum
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Associated Topics:
High School Fibonacci Sequence/Golden Ratio
High School History/Biography
Middle School History/Biography

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