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Golden Rectangle ( Trivia # 23 )

on Monday, June 22, 2009

A golden rectangle is one whose side lengths are in the golden ratio, 1: φ (one-to-phi) or 1 : ( 1 + √5 )/2 or approximately 1 : 1.618 .
A distinctive feature of the golden rectangle is that when a square section is removed, the remainder is another golden rectangle; that is, with the same proportions as the first. Square removal can be repeated infinitely, in which case we will get golden rectangle again and again with decreasing size .


Ratio of new sides =

1 / (φ-1) = (φ+1) / (φ+1)(φ-1) = (φ+1) / (φ^2 -1)

Now we all know φ^2 = φ+1

Also 1/φ = φ - 1 ...... property of phi

(φ+1) / (φ^2 -1) = (φ+1) /φ = 1 + 1/φ = 1 + φ - 1 = φ : 1

This is equal to the ratio of sides of the original rectangle . Hence we proved the fact that when a square section is removed, the remainder is another golden rectangle. This property makes it unique .

1 comments:

archana said...

moreover der is a relationship between golden ratio and fibonacci series.....its very interesting.find out
http://www.friesian.com/golden.htm

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