Where Mathematics, Science, and Art Converge

The fine arts are tied into every facet of the human experience. Among these interconnected facets mathematics and science are prime, even when their links to art are not immediately apparent. Discovering and enjoying these connections can empower teachers to place the arts deeper into the core of teaching and learning.

No matter the historical era or geographical region, visual art is dependent on the same principles that underlie arithmetic, geometry, trigonometry, chemistry, physics, and even meteorology (yes, humidity and temperature matter). The sister arts of music and dance have similar essential connections. Modern education has blithely lost sight of the relationship of the arts to mathematics and science. That is all the more reason for teachers engaged in classical education to move with urgency to restore this understanding.

One of the easiest ways to begin this restoration involves learning the basic principles of two-dimensional art (painting, drawing). An artist must make two initial, critical decisions in establishing a horizon and a vanishing point, both based on geometry. Horizon involves sketching a horizontal line (hence its name). The line could indicate an actual horizon, as in a landscape where the earth meets the sky, or it may be a line drawn across what will become an interior scene, street scene, portrait, still life, or any other genre of two-dimensional art. The line can be placed high, low, or in the middle of the canvas. Ultimately this horizon will be erased or obscured by mountains, trees, or other objects like buildings or furniture, yet the line will guide everything about the placement of images in the work.

The second geometric decision is called a vanishing point. Here the artist picks a point from which the background—and hence the entire painting—will emanate. The easiest example to consider would be paintings where the vanishing point is that spot where a railroad track or path disappears into the distance. The point could be a doorway in the back wall of a banquet hall. It could be a dark corridor running along the far side of a parlor or a distant pasture that seems to lie “behind” a pastoral frolic. In short, any painting that employs perspective (a critical concept of Western art perfected in the Renaissance) will have a vanishing point. No matter where it is, that vanishing point—and the geometric facts that rule the intersection of two lines—will organize everything else in the painting.

The critical visual trick called perspective creates a three-dimensional effect in two-dimensional art. Perspective is based on addition and subtraction in the form of adjusting the sizes and profiles of objects so as to make them look close or far away. Young children can create a successful example of perspective in art by drawing a set of similar trees scattered in a loose line. The trees “in the back” (probably around the top of the page) can be made small (foreshortening), while trees in the foreground can be made the tallest and broadest. Once crayons are brought out, outlines of a few hills are added (tiny arcs in the distance for faraway mountains; bigger shapes for the hills that lie nearer), and the sky is colored, the child will have created a reasonable example of perspective in art.

Three-dimensional art, like sculpture, employs similar principles of geometry. Perhaps a useful comparison would be the need for a builder to establish the plumb line in construction, where an absolutely straight vertical line, laid at perfect right angles, guides what the builder will do. A sculptor must work out all proportions before beginning to chisel. He needs to understand his materials, his tools, and the mathematical dimensions that will define the very substance of his work, not to mention the issues of weight and gravitational stresses that a sculpture must endure.

Color in art provides an example of art’s scientific basis. Everything from the horticultural and mineralogical components of pigment and ink in medieval times, to the advanced chemistry of today’s acrylics, from measured techniques that determine the thickness of the color (called texture), to the optical science that explains our perceptions of color—all of this falls within the umbrella of mathematics or science.

For students who wish to delve into more extreme expressions of mathematics in art, it might be rewarding to explore what artists have done with hyperbolic (hyperdimensional) geometry. This movement began largely  with Dutch artist M. C. Escher (1898-1972), whose fantastic engravings were spurred by his friendship with the eminent British and Canadian mathematician and geometer Harold Scott MacDonald Coxeter (1907-2003). Coxeter started life intending to become a composer, and his artistic flair never waned. Escher’s work adopted Coxeter’s enthusiasm for hyperbolic geometry and Escher’s resulting fantastic engravings continue to delight, intrigue, and puzzle us today.

A similar geometric procedure underlies what became one of the most influential new musical styles ever to develop: minimalism. Often based on a simple musical idea, minimalism stretches that idea into longer forms with fascinating mathematical patterns of repetition and subtle variation. From a theoretical style of music to a broad rhythmic current, minimalism has affected our musical palate in everything from classical compositions to film scores and pop hits.

A great deal of modern art is derided upon first impression, and some of it (particularly in terms of theme and message) deserves that derision. But some art that seems abstract actually does form around geometric models, such as fractals, whose geometric structures repeat on different scales. Thus, whether it is obvious or not, some artists who produce seemingly abstract, undisciplined works are working in terms of complex equations whose properties are repeated in carefully measured patterns.

We can look back to classical Greece for arguably our greatest form of art: architecture. Architecture really is the supreme reflection of mathematics in art, wherein every aspect of a building depends on a precise  understanding and application of mass and void, tensile strength, geometry, algorithms, and a dazzling array of algebraic and scientific equations that determine whether a building stands or not. Computers do a great deal of this work today, but let us not forget that the greatest buildings in our Western tradition—those that still stand and those etched solely in the heritage of our memories— began with a pencil that drew geometrical lines and scratched out mathematical equations.

Undeniably, the nineteenth-century Romantic image of the frenzied or dreamy artist, fueled passionately by emotion, still prevails in the popular admiration. That image is a far cry from the realities an artist handles daily when creating a work of visual art. The romanticized image may be glamorous, but recognizing the artist as handmaiden to mathematics and science comes far closer to the truth.