# PHI RATIO GEOMETRY

The golden ratio of phi has been claimed to have held a special fascination for at least 2,400 years. Ancient Greek mathematicians first studied what we now call the golden ratio of phi because of its frequent appearance in geometry; the division of a line into "extreme and mean ratio" (the golden section) is important in the geometry of regular pentagrams and pentagons.^{ }

In mathematics, two quantities are in the golden ratio of phi, if their ratio is the same as the ratio of their sum to the larger of the two quantities. The golden ratio of phi is also called the golden mean or golden section (Latin: *sectio aurea*).^{ }Other names include phi ratio, divine proportion, divine section (Latin: *sectio divina*), golden proportion, and golden number.

The golden ratio of phi has also been used to analyze the proportions of natural objects as well as man-made systems such as financial markets, in some cases based on dubious fits to data.^{ }The golden ratio of phi appears in some patterns in nature, including the spiral arrangement of leaves and other plant parts.

Some twentieth-century artist have proportioned their works to approximate the golden ratio of phi—especially in the form of the golden rectangle, in which the ratio of the longer side to the shorter is the golden ratio—believing this proportion to be aseptically pleasing.

Leonardo da Vinci's illustrations of polyhedron *Divina proportione*^{ }have led some to speculate that he incorporated the golden ratio of phi in his paintings. But the suggestion that his Mona Lisa, for example, employs golden ratio of phi proportions, is not supported by Leonardo's own writings.^{ }Similarly, although the Vitruvian Man is often shown in connection with the golden ratio of phi, the proportions of the figure do not actually match it, and the text only mentions whole number ratios.^{ }

Salvador Dali, influenced by the works of Matila Ghyka, explicitly used the golden ratio of phi in his masterpiece, The Sacrament of the Last Supper. The dimensions of the canvas are a golden rectangle. A huge dodecahedron, in perspective so that edges appear in golden ratio of phi to one another, is suspended above and behind Jesus and dominates the composition.