Golden Ratio

The golden ratio is a number, commonly known as phi ($\varphi$), defined as $\varphi ^{2}=\varphi +1$. The value of phi is

${\frac {1+{\sqrt {5}}}{2}}$

Phi is an irrational number. In decimal form it is often written

1.618...

The golden ratio was first defined by Euclid of Alexandria around 300BC. His definition reads:

A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the lesser.

A dodecahedron, a solid in which each side is a pentagon, is related to the Golden Ratio in that both the surface area and the volume of a dodecahedron of unit edge length are functions of the Golden Ratio. Plato described the dodecahedron as the solid which the gods used for embroidering the constellation on the whole heaven.

The golden ratio in art

It has been observed that the golden ratio is evident in the arrangement of branches along the stems of plants, and of veins in leaves. The golden ratio also has a place in art when used to determine what arrangements of a composition are aesthetically pleasing to the eye. Some would suggest that the "perfect" human face adheres to the golden ratio .

Leonardo da Vinci was a close friend of Luca Pacioli, who published a treatise (Divina Proportione) on the Golden Ratio in 1509. It has been claimed that he used the Golden Ratio in his composition of the Mona Lisa, but there is no documentation to prove this.

Salvador Dali deliberately used the golden ratio in his painting Sacrament of the Last Supper, incorporating also a huge dodecahedron around the supper table.

Le Corbusier designed an entire design system called the Modulor, based on the Golden Ratio, which aimed to give harmonious proportions to everything about a building.