Golden Section

Golden Section, in art and mathematics, a geometric proportion based on a specific ratio in which the greater part is to the lesser what the whole is to the greater. It is most clearly expressed as a line intersected in such a way (see diagram below) that the ratio of AC to CB is the same as that of AB to AC.

This ratio has the numerical value 0.618..., which can be derived as follows: If AB = 1, and the length of AC = x, then AC/CB = AB/AC becomes x/(1 - x) = 1/x. Multiplying both sides of this equation by x(1 - x) gives x² = 1 - x; therefore, x² + x - 1 = 0. This equation can be solved by using the quadratic formula (see Quadratic Equation), which yields the equation x = (- 1 + Ä)/2 = 0.6180339.... Recognized as an aesthetically pleasing ratio, the Golden Section has been used as the basis on which the elements in a painting, or in an architectural scheme, are arranged.

Plato is generally credited with establishing the study of the Golden Section, and the Greek mathematician Euclid, writing in the 4th century bc, defined this proportion in his chief work, Elements. The Golden Section was of great interest to artists and mathematicians of the Renaissance, when it was known as the Divine Proportion and was regarded as the almost mystical key to harmony in art and science. Divina Proportione, a treatise on the subject written by the great Renaissance mathematician Luca Pacioli and illustrated with 60 drawings by Leonardo da Vinci, was published in 1509, and influenced artists and architects of the age, among them Leonardo, Piero Della Francesca, and Leon Battista Alberti.

The Golden Section has also been used by later artists. Experiments suggest that human perception exhibits an innate preference for proportions that accord with the Golden Section. This in turn implies that artists may almost subconsciously arrange the elements of a picture according to those ratios.