Geometric Planes and Lines

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There is very little on heaven and earth that cannot be somehow represented by artistic expression and likewise, almost everything within our realm of physical awareness-and even in a few realms beyond, can be expressed in mathematical terms. When an artist turns to a mathematician for their help in better understanding art as well as the world as a whole through mathematics, what will they discover? What are the ways in which, for example, the science of math is reflective of the ways of Chinese brush painter might approach their painting? How does a mathematical formula define the very essence of what is considered good design in classical Western arts? Can a design be defined in mathematical terms? Can mathematical concepts help artists in designing and producing their work? What do artists and mathematicians have in common?

Initially, when we consider the connection between math and art, we can start with observing important components of mathematics that are also found in works of art, these being points, lines, curves, angles, planes and geometric shapes (circles, squares, triangles, etc.) Even though most works of art do not simply consist of blatant geometric shapes, we do find that these shapes are often implied in the composition. The apples, oranges, and grapes in a still life are of a circular nature; the door and stairs seen in another composition are based on rectangles. The fine arts student who has used an artists wooden model of the human form has seen how the human body can be translated into geometric shapes. And if we advance in geometry to begin to work with (for example) a cube, we are concerned with the same term that the "life drawing" student strives to achieve in their rendering, and that is volume.

Beyond the basic geometric shapes, the artist is constantly working with points, lines, curves, and various planes. Any shape not easily defined by geometric shapes can be further described with these. After all, how often have we heard people who compliment the design of an object refer with admiration to its (elegant, sleek, classical, modern, or fluid) "line?"

Not only does math play a part in the mechanics of creating a composition, it can be helpful in guiding the artist to find the most pleasing approach, or the most accurate rendering. One of the most frequent subjects of artists through the ages has been of those found in nature, how is it that mathematics and nature are related?

The proportions of what is known as the "golden rectangle," which the Greeks derived using the Pythagorean Theorem, was thought to be the most pleasing and aesthetic proportions for a rectangle. We can see the use of the golden rectangle in the construction of the ancient Parthenon, the Temple of Athena built in 5th century BC in Athens Greece. The use of these special proportions can be seen in art and architecture from the ancient Greeks up to present day. There is another special aspect of proportions in relation to the Fibonacci numbers and the golden rectangle. Take a golden rectangle of say, 34 inches in length and 21 inches in width. Break the rectangle into a square that is 21'' x 21''. Break the remaining area into a square of 13'' x 13'', and the area that remains into 8'' x 8'', continue this process as long as possible; it is a process that theoretically could continue to infinity. "If a quarter circle is added inside each square, the arcs fit together into an elegant spiral. This spiral is a good approximation to the so called logarithmic spiral often found in nature, such as in the shell of a nautilus mollusk. Successive turns of the ...