Pythagoras, a Greek mathematician and cult leader, is credited with discovering the golden ratio. The golden ratio is a ratio of 1 to approximately 1.618. It is sometimes expressed as a "golden rectangle" with a height of 1 and a width of 1.618. The golden ratio has special mathematical properties, but it is perhaps most significant for its application to other fields besides mathematics. In fact, Pythagoras himself was the first to apply the golden ratio to fields other than mathematics. He examined the human body and declared that every body part was related to the others in terms of the golden ratio!
Today the golden ratio is designated by the Greek letter phi, in honor of the Greek sculptor Phidias, who was said to have used the golden ratio extensively in his sculptures, which included one of the Seven Wonders of the Ancient World (the statue of Zeus at Olympia). Unfortunately, none of his sculptures survive to prove or disprove this legend.
While none of Phidias' sculptures have survived, Phidias was said to have directed the construction of the Parthenon of Athens. The Parthenon's facade, and some individual elements of the facade, are laid out according to the golden ratio. A rectangle drawn around the whole of the Parthenon's facade is a golden rectangle, and this is said to be the source of the building's eternal beauty. More recently, the architect Le Corbusier developed an entire system of architectural proportions dubbed "Modulor," based on the golden ratio, that he used in his works.
In the Renaissance, the golden ratio was rediscovered by Luca Pacioli, who published his book "Divine Proportion" in 1509. It has been claimed that Leonardo da Vinci used the golden ratio in composing the "Mona Lisa," as a rectangle drawn around Mona Lisa's face is close to a golden rectangle.
Since the Renaissance, the golden ratio has been used extensively by painters from Edward Burne-Jones to Salvador Dali, whose painting "The Sacrament of the Last Supper" is explicitly laid out in a series of golden rectangles. The painting also contains a dodecahedron, a three-dimensional shape whose sides' relationships to one another are defined by the golden ratio.
Biologists find the golden ratio all over the natural world. For example, when sunflowers grow their seeds, they grow them in spiral formations in which each seed is rotated exactly the golden ratio from its neighbor. The spirals that form in flower petals, plant leaves, and nautilus shells are all related to the golden ratio.
Modern scientists don't give much credence to Pythagoras's claim that the human body follows the golden ratio in its proportions, but some scientists claim that the most attractive human faces are those that display a golden ratio between the height of the whole face and the distance from the chin to the eyebrows. So perhaps Pythagoras was right after all!