**Historical Perspective**

The intertwining of mathematics and art has a long history, dating back past the Egyptians and their pyramids. From the goal of drawing (mathematically) the ideal male figure to M. C. Escher’s optical illusionary-type math to fractals, different variations which have pattern that can mathematically be described have been created.

The pyramids, for example, are found to describe the golden ratio, the most pleasing appearance of a triangle, where the ratio of the opposite over adjacent ratio is 1.619 if interpreted correctly as described. The 3-4-5 right-triangle, based upon the Pythagorean theorem, has also been used. This is close to “Kepler’s triangle” as well. Similarly, Phidias, main Greek sculptor for Parthenon (honoring Athena), is thought to have the golden ratio designation as “phi” named after him; similarly, the “golden rectangle” has been adopted from the golden triangle.

Elsewhere, the first African mosque in Tunisia utilizes nearly the golden ratio as well, in division of the courtyard and patio regions.

Further, Polykleitos, a Greek sculptor, utilized the following scheme: distal phalanges of little finger measured as “x”, then all other areas of body based upon that (multiply by square root of 2 to get distance of second phalanges, then again to get distance of third… with some less precision in noting the general body height as “8 heads high and 2 heads wide.”

Renaissance artists were fascinated by the need for lighting to be described by math (in terms of perspective; Piero della Francesca’s work looked at the eye as an apex of the pyramid. Prior to this time, it was noted that Notre Dame (of Laon and Paris) were designed according to the golden ratio. Leonardo da Vinci’s work of “de devina proportione” relied upon the golden triangle, with works such as Mona Lisa exemplifying this ratio (as did Vitruvian man). In many ways, da Vinci’s adoption of the golden ratio to the human body furthered work on dimensions by Polykleitos.

**Artistic Mathematics of the Present**

Artists of the 20^{th} and 21^{st} centuries relied upon mathematical shapes to show figures – for example, M.C. Escher used polyhedrons and other styles to depict nature despite mathematical objects. Also, the juxtaposition of different spaces, as evidenced by his “Three Intersecting Planes” and “Waterfall” depicted these practical optical illusions. Escher worked with mathematician H.S.M. Coxeter using hyperbolic geometry; (works of these as well as composer Johann Sebastian Bach resulted in a Pulitzer Prize-winning book, *Gödel, Escher, Bach*).

In addition, Palazuelo focused geometric shapes with patterns and coloring to depict “purity” with a cubist style. Sculptors such as John Robinson (1935-2007) had work with topological interest (with toruses) depicting, as some say, humanity. Roger Penrose, after whom Penrose tiles are named (non-periodic tiles generated from a simple base tile), 36- and 72-degree rhombuses have been generated – with golden ratio (thick to thin rhombuses in 1.618 ratio); distances between repeated patterns in tiling grow as Fibonacci numbers.

**Future Math + Art**

It becomes difficult to define the true transition from present to future styles, as there remains a continued evolution. Perhaps one such transition may be in the form of fractal art, which utilizes computerized algorithms to depict shapes (e.g. the Julia set and Mandelbrot sets).

The ability of mathematic modeling into art, and vice-versa, has allowed insight into creation of new art forms as well as analysis (e.g. finding the rhombicuboctahedron in Pacioli’s portrait painted in 1495 by Jacopo de Barbari). It remains intriguing that objects found hundreds, even thousands of years ago, have respected the golden triangle, and other patterns.

Perhaps adding a fourth dimension, such as time, to art has its role in the future.

Today, organizations such as the Bridges Organization (bridgesmathart.org) focus upon development of greater relationships between art and mathematics, as cited in Scientific American as well (scientificamerican.com, at http://www.scientificamerican.com/article/bridging-the-gap/); in addition, educational efforts to introduce mathematics into the liberal arts are being undertaken (www.artofmathematics.org).

*(Portions of this text are from sources of the internet, which may include scientificamerican.com, Wikipedia.org, bridgesmathart.org, artofmathematics.org)*