Summary: Although traditionally used in art and architecture, the golden ratio can be referenced to design aesthetically pleasing interfaces.
Proportional systems are based on ratios and have been used for centuries in architecture and art. The golden ratio was first mentioned as early as about 500 BC by Phidias, Plato, and then Euclid. It is fair to assume that this ratio has been discovered several times throughout history — hence its many names, including golden mean, golden ratio, golden section, divine proportion (coined by Leonardo Da Vinci) and the Greek symbol φ.
The exact mathematical definition is as follows (according to Wikipedia):
Golden ratio: Two quantities a and b (a>b) are in the golden ratio φ if their ratio is the same as the ratio of their sum to the larger of the two quantities: