The golden ratio, for some, embodies perfection of form. It was and is a strictly mathematical way of working out the proportions of structures. It is still in use today, in art and architecture. Many natural systems seem to demonstrate a basis in the golden ratio as well.

There have always been accepted proportions, whether for the width of a street, or the ratio of an apse to its transept. The creative elite as well as everyday builders have worked and played with these proportions down through history.

Music, most of it, is based on scales, with intervals based in mathematics. Music is repetition and variation through time that is grounded in defined notes. However, different cultures use different systems for defining each note.

Some plastic and reconstructive surgeons use mathematical models, often applied by computer, to envision their goals for patients choosing to change their features.

Computer users themselves are using algorithms, mathematical recipes, every time they create a graphic or decide which will make the most beautiful and useful web page.

The Golden Ratio

The golden ratio is sometimes called Phi, to honor Greek sculptor Phidias. Some art historians believe he used it to structure his works. No existing piece of art or architecture can be definitely attributed to Phidias. Consequently, what we have is a legend of his greatness, built on ruins and the reports of his admirers.

The golden rectangle is a four-sided figure, like all rectangles. It is composed of two parallel sides of length one, and two sides of length phi, such that phi equals 1 divided by the quantity phi minus 1. If a 1×1 square is removed from a golden rectangle, a golden rectangle remains, and the process can be repeated infinitely. Tracing the corners creates the golden spiral, also employed in art. The ratio phi is an irrational number close to 1.6180.

Le Corbusier used it in the Villa Stein, designed for Gertrude Stein, and in other buildings. The ratio is quite common in western architecture. However, it is not universal.

Other Ratios

A system of proportion based on the square root of two is also quite common. It is easily generated; the diagonal of any square is a multiple of the square root of two. Many inspiring examples of Islamic architecture use proportions based on the square root of two.

Many early building systems and village layouts were based on the human form, being made in feet, arm lengths, paces and so on.

Two circles, the circumference of each touching the center of the other, comprise the vescia pisces. It marked and divided sacred spaces before Christianity brought the cross.

Venerable feng shui takes the double tatami mat as one of its fundamental units. This ancient system is not only about esthetics, but its products can be breathtaking.

Nowadays posts and beams are standardized according to engineering discoveries about load, span, and shear. In early Greek culture though, columns and entablatures held to generally accepted proportions, which were established according to esthetic ideals, but replicated by precise measurement.

Music

Notes are the basis of most music. The notes of various scales are well established and based on vibrations that produce varying pitch. The vibrations, in turn are dependent on the length of a plucked string. However, there are many scales.

Ours is composed of seven notes, arranged in a series of steps and half steps that we are so used to that they seem foreordained. However, quartertones are common in the Middle East and do not sound exotic there.

Other scales include a variety used in gamelan music, various pentatonic scales found around the world, and hexatonic scales as well.

Plastic Surgeon’s Esthetics

Many plastic surgeons believe in an esthetic ideal, which is based in mathematics. Charts and graphs exist which purport to illustrate the physical ideal. Probably a symmetrical appearance, youth and health, are components of human beauty. Yet sculptures like She Who was Once the Beautiful Heloise or any Saint-Gaudens Lincoln illustrate the beauty found in weakness and decrepitude. Bonsai is another art with does not worship a simple symmetry, though it clearly does follow esthetic laws.

Ethnicity is another issue. If there is a perfect nose, found in one ethnic group, then most of the people in the world will not come near the standard. Yet utterly beautiful individuals are found in every nation and every tribe.

Computer use

Whether by GIMP or Photoshop, the graphics of the modern world are created by the logic operations of mathematics. Recursion fills areas, or outlines space, not the icon of a paint bucket or pencil. The mathematics is buried deep, but it actualizes every decision a designer makes. People drive these mathematical machines, it seems.

However, our analog brains may someday be understood in a way that traces our elaborate neural nets to their sources in an organic equivalent of ones and zeroes, the neurons that are exclusively off or on, that fire or don’t fire.

Conclusion

Some art can be reduced to mathematics with tools we have now. Energy is a form of matter, matter is a form of energy, and something we know about the universe is that rules underlie transformations. Rules are the soul of mathematics.

However, we do not really know the laws of aesthetics. We have esthetic systems, popular in one place or many, at one time or another. The rules are not fixed, the proportions are not universal, and we do not truly know the mathematics of aesthetics, or for certain, if there is any such math.