The Art of Mathematics, the Mathematics of Art

THEART OF MATHEMATICS, THEMATHEMATICS OF ART A scientist worthy o f the name, above all a mathematician, experiences in his work the same impressions as an artist; his pleasure is as great and o f the same nature. -H. Poincark Thefinal abstract expression o f a e r y art is number. -W. Kandinsky Mathematicspossesses not on4 truth, but supreme beauty-a beauty cold and austere, like that o f sculpture. -B. Russell What is it about the relationship between art and mathematics that these three modern, seminal thinkers understand and that most of us fail to grasp?What does the Pythagorean theorem share with Michelangelo’sDavid? What could Picasso’s Guernicapossibly have in common with Euclid’sElements? Sadly,one answer is that both disciplines share an obscurity in our modern world. Name a living mathematician. This should be easy;there are more practicing mathematicians alive today than the total sum of mathematicians who have already perished. Name a living artist. This should be easy; museums and galleries proliferate, public art decorates our cities, the National Endowment for the Arts hands out millions in grants. Even worse, the general populace has neither an understanding of basic mathematical principles nor an appreciation of works of art. Somewhere between ancient Greece and twentiethcentury America, these disciplines, long thought to be the hallmarks of a civilized person, became marginalized. Rather than analyzingwhat went wrong, I want to present an argument for reintegrating these disciplines and, in the process, for making a larger place for both in our lives. The argument is really quite simple. At their essential cores, mathematics and art are engaged in the same vital, important, intellectual activity-interpreting the fundamental nature of both the universe and our place within it. As they proceed with this task, the mathematician and artist progress through a five-stage relationship. SHARED TOOLS Every form of human activity uses mathematics as a tool-for counting, measuring, modelling . Art is no different. Ask the painter who needs to stretch a canvas, or the glass blower concerned with annealing, or the photographer absorbed by focal distance. But mathematicians also use artistic tools: Prior to Descartes’sinvention of analytic geometry (i.e. the notion that points and lines could be described by coordinates in a plane), all geometry was numberless and proceeded by constructions with ruler and pencil. Even today, many mathematicians rely on visual techniques (e.g. diagrams of the four-color map problem, models of nonEuclidean spaces, wire sculptures to demonstrate how to invert a sphere) to help explain their weird, mind-boggling problems. 0 1994ISAST LEONARDO, Vol. 27. No. I , pp. 87-89,1994 87 MATHEMATICAL FOUNDATIONS Like the hard sciences, the foundation of many of the fundamental concepts of art (e.g. perspective ,proportion, symmetry) is mathematically based. According to Leonardo da Vinci (quite a mathematician and artist himself), “Perspectiveis the rein and rudder of painting.” But perspective is a relatively recent concept to the world of art, only having been discovered in the fifteenth century by two Florentines: Alberti and Brunelleschi. By imagining a picture’s surface as a plane cutting through a pyramid of visual rays, they were able to produce a gridwork that allowed the illusion of three-dimensionality to be produced on a flat canvas. Proportion has been a mathematical staple for so long that Euclid stole Eudoxus’swork on the subject and included it in his Ekments. The Pythagoreans (a mystic Greek brotherhood that believed the order and harmony of Nature was to be found in the science of numbers) discovered the divine proportion: Divide any line AB by a point C; if AC/CB = CB/AB, then the entire line is divided by C into the extreme mean ratio, whichJohannes Kepler called “the divine proportion.” This ratio, closely modelled by our modern 3x5-inch index card, is the driving aesthetic behind the Parthenon and thousands of well-known works of art. Symmetry-a transformation in which the original figure and its image are congruent-is a staple of all art, from aboriginal through high-Renaissanceto postmodern. It is also a basic mathematical principle. Mathematicians recognize four basic types of symmetry (bilateral reflection, rotation, translation, glide reflection) and have...