by Eva Brann, St. John's College

The title comes from a sonnet by Edna St. Vincent Millay: "Euclid alone has looked on Beauty bare." It's not the greatest line of poetry, and if you visualize its image with a hint of malice, you have to smirk. Still, it's suggestive of some really good questions.

"Beauty bare." That surely does not mean beauty nude, but rather beauty denuded, stripped of something that veils it. The poet is suggesting that when you look at Euclidean objects, like circles, triangles, and rectangles, you see something revealed that incarnate shapes don't show, and that can move you as physical beauty might. What about a simple shape—devoid of body, of color, of the delights of irregularity—might get to you? With what organ do you see what Euclid saw? Through the eyes in your head, or with the eye of the mind, or on the immaterial tablets of your imagination?

Is Euclidean bareness rightly called abstract? Once, when I was going through Euclid's Elements with a class of freshmen at my college, I brought in a Mondrian painting, the one called Tableau I, a rectangular canvas subdivided by straight, thick black lines that are anything but "breadthless lengths" (Euclid's definition of a geometric line), and pleasing rectangular partitions filled in with primary colors. The class knew that such painting was called "abstract." What, we asked ourselves, were we to think of this esthetically vivid abstraction, almost riotously sensuous compared to the Euclidean diagrams we were studying, representing his "geometric algebra" of rectangles divided and composed to embody such equations or identities as ax - x² = b² or (a — b)² = a² + b² - 2ab? What is the difference between a pleasing picture of rectangles and an ingenious proof that uses them?

The pleasure and profit of a roundtable on "Mathematics and Beauty" is that the most naive and the most sophisticated questions can be entertained together by experienced practitioners and engaged amateurs. Amateurs experience perplexities that professionals may have leaped over too quickly; professionals offer approaches, insights, and illustrations that a non-mathematician could never summon. In such a conversation, problems may not be solved definitively, but possibilities are stirred up endlessly.

Here are a slew of questions that we might bring to the table. What—and try to be as precise as possible—did your moment of seeing "Beauty bare" feel like? Is there a difference between the esthetics of beautiful sensory things and that of sense-pure mathematical objects? Are the criteria for beauty in mathematics articulable, and do mathematicians tend to agree on the beauty of a piece of mathematics? Should we call the object beautiful and the proof elegant? (Does beauty attach more to stabilities of insight, and elegance to motions of thinking?) Are mathematical objects made, like artifacts, so that their beauty must meet criteria of construction invented by knowledgeable critics, or are they given by a sort of intellectual nature, so that the finder might be as surprised by a novel, unexpected beauty, as is an explorer who comes on a hidden enchanted valley? Is there beauty in all branches of mathematics—can symbolic abstractions like algebraic equations or tables be beautiful? What esthetic significance do you see in the fact that so much of mathematics appears as equations? Is there a sort of esthetic satisfaction in the balance that an equation maintains about its fulcrum, i.e., the equal sign? Are all mathematical structures visualizable to a practitioner who has lived long enough with them? Are there plain or even ugly mathematical objects, or is there not one that someone doesn't love? Is truth beauty and beauty truth in mathematics, or are there things in the mathematical world that can be shown to exist and even to be useful, but that are repulsive? What is the meaning of the word "powerful" when mathematicians use it, and can a procedure be powerful but unbeautiful?

To return to Euclid, are those objects he deals with, the simplest and most elementary structures, more or less beautiful because we can easily imagine them and often see them embodied in the world around us, especially the man-made world of four-square structures and wheeled traffic?

Finally, are there mathematical figures that take the crown of beauty? It used to be thought that the circle was the perfection of beauty. That is why Ptolemy put up with the complexities of epicycles that produced absurd-looking real orbits, and why Kepler was reluctant to accept his own greatest discovery, the elliptical orbits of the planets. What makes a circle beautiful? Circles of all sizes look exactly the same, yet they have very different curvatures, in that they are like the tones that compose the rising scales on a keyboard. Each tone has just the same quality as the one an octave above, but the pitch is totally different. Yet no one thinks that the consonance of the octave is particularly beautiful; it's too much of a unity for that.

Furthermore, the circle's circumference defines a center that can be found in various ways. If the circle is set spinning, it stays stock-still, and if it is set rolling it describes a straight line, which is why you can get a chariot body on the axle between two wheels and step in. It's useful, but who thinks it's particularly beautiful? The fact that any one circle is self-congruent—any segment of the circumference can be slid onto another—could be regarded as boring rather than beautiful.

But the circle does show an unequivocally pleasing aspect when embodied in a roundtable; it brings about what might be called the Arthurian effect. King Arthur seated his chivalrous knights about a roundtable because he wanted to function as a first among equals, rather than as the head of the table. Roundtables give everyone an equal chance to talk about a topic—in this case, the elementary question of the beautiful.

This paper was commissioned by the Philoctetes Center as part of a grant from The John Templeton Foundation. This and three other commissioned pieces are featured in a special issue of Dialog, the newsletter of the Philoctetes Center.