# A Golden Rectangle

 Highlight this Let the side of the gray square be called a; then its area is a2. Highlight this Each colored triangle is half of an a - by - 2a rectangle, and so has an area of a 2 . Highlight this The large square has an area of 5a2, and so its side is of length (√5)a. Move the slider to rotate the square. Highlight this If the long side of each colored rectangle is b, then the short side is b-a. Highlight this Each colored area is still equal to a2, so a2 = b(b-a). That is, a/b = (b-a)/a. Highlight this 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