Golden Ratio

In mathematics and the arts, two quantities are in the golden ratio if the ratio between the sum of those quantities and the larger one is the same as the ratio between the larger one and the smaller.

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

$\varphi^2 - \varphi - 1 = 0$

$\varphi = \frac{1 + \sqrt{5}}{2}\approx 1.61803\,39887\dots\$

$\varphi = 1 + \frac{1}{1 + \frac{1}{1 + \frac{1}{1 + \frac{1}{\ddots}}}}$

$\varphi = \sqrt{1+\sqrt{1+\sqrt{1+\cdots}}\infty}$

Some of the greatest mathematical minds of all ages, from Pythagoras and Euclid in ancient Greece, through the medieval Italian mathematician Leonardo of Pisa and the Renaissance astronomer Johannes Kepler, to present-day scientific figures such as Oxford physicist Roger Penrose, have spent endless hours over this simple ratio and its properties. But the fascination with the Golden Ratio is not confined just to mathematicians. Biologists, artists, musicians, historians, architects, psychologists, and even mystics have pondered and debated the basis of its ubiquity and appeal. In fact, it is probably fair to say that the Golden Ratio has inspired thinkers of all disciplines like no other number in the history of mathematics.

— Mario Livio, The Golden Ratio: The Story of Phi, The World’s Most Astonishing Number

(Almost all the above information is from wikipedia… I just wanted to populate this article at the moment; the golden ratio has been quite a fascinating constant! I have this Livio book, but haven’t had a chance to read it yet!)