The golden ratio was called the extreme and mean ratio by Euclid,[2] and the divine proportion by Luca Pacioli,[3] and also goes by several other names.[b] Mathematicians have studied the golden ratio's properties since antiquity.

It is the ratio of a regular pentagon's diagonal to its side and thus appears in the construction of the dodecahedron and icosahedron.[7] A golden rectangle—that is , a rectangle with an aspect ratio of may be cut into a square and a smaller rectangle on. Some 20th-century artists and architects, including Le Corbusier and Salvador Dalí, have proportioned their works to approximate the golden ratio, believing it to be aesthetically pleasing. These uses often appear in the form of a golden rectangle.