A Golden Rectangle

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Let the side of the gray square be called a; then its area is a2.

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Each colored triangle is half of an a - by - 2a rectangle, and so has an area of a 2 .

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The large square has an area of 5a2, and so its side is of length (√5)a.

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If the long side of each colored rectangle is b, then the short side is b-a.

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Each colored area is still equal to a2, so a2 = b(b-a). That is, a/b = (b-a)/a.

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A rectangle with sides a and b is similar to a rectangle with sides b-a and a: in other words, it is a "golden rectangle."

(This paragraph will be easier to follow if the slider is pulled to the right.) The inner square has side a and the outer square has side a√5, so b - a, the short side of the colored rectangles, is a(√5 - 1)/2. The long side, b, is a(√5 + 1)/2. Therefore

τ = b/a = (√5 + 1)/2, and

ρ = a/b = (b-a)/a = (√5 - 1)/2.

Last modified on $Date: 2015-05-02 12:51:16 -0400 (Sat, 02 May 2015) $

Christopher J. Henrich