Tuesday, 5 June 2012

Golden rectangle

Not all rectangles are created equal.  Some are more lovable than others.  Regarded historically by many as the most aesthetically pleasing among all rectangles, the golden rectangle is a rectangle which has its length and width in the ratio φ : 1, where

\varphi = \frac{1+\sqrt{5}}{2} = 1.61803\,39887\ldots
is called the golden ratio.  A remarkable fact about the golden rectangle, ABCD in the following diagram, is that when the largest possible square AEFD is removed from it, the remaining rectangle CFEB is also golden.  If the same process is repeated with rectangle CFEB, a third golden rectangle EBHG will be formed.  This process can be continued indefinitely, with all infinitely many golden rectangles thus formed spiralling around point O:

φ : 1 = AB : AD = CB : CF = EB : EG = ...

The value of φ can be easily calculated by solving the equation

 1 + \frac{1}{\varphi} = \varphi.

It is equivalent to the quadratic equation

{\varphi}^2 - \varphi - 1 = 0

which yields φ as its positive root.

The golden rectangle/ratio has been the obsession of numerous artists and architects through the ages, from ancient till present times, including Le Corbusier.

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