Beauty, Proof and Symmetry
…those things are said to be beautiful which please when seen.
Saint Thomas Aquinas, Summa Theologiae I-II, q. 27, a. 1, ad. 3
Reflect for a moment on a time when you experienced a sense of wonder or beauty. I would suspect that accompanying that experience was a strong, positive emotion drawing you to attend closer to the object of wonder, to repeat the experience, or perhaps to see if further treasures lay beyond the immediate object. This experience of beauty is what is at play in the mathematician’s encounter with mathematics, one which conveys a sense of unveiling the mysteries within the forms and patterns being discerned and deciphered. The type of beauty that is found within a mathematician’s world creates a magnetic attraction that pulls upon the mathematician’s attention, focusing his or her full being and bringing pleasure in all of the ways the object of beauty is viewed, grasped, and sensed.
As a concrete case study in mathematical beauty, we will concentrate on a subject that often provides the easiest gateway to mathematics through the senses, namely geometry. This ancient form of rigorous mathematics helps us encounter beauty in two particular ways, through rigorous proof and symmetry. We will spend more time with the latter, but I should note that rigorous proof is an important source of aesthetic encounters for the mathematician. For example, consider the following observation from Sir Bertrand Russell in his autobiography: “At the age of eleven, I began Euclid, with my brother as my tutor. This was one of the great events of my life, as dazzling as first love. I had not imagined that there was anything so delicious in the world.” Even more poetic and succinct are the words of Edna St. Vincent Millay: “Euclid alone has looked on Beauty bare.”
What is it in the structure of Euclid’s Elements that evokes such responses of wonder in those who take it upon themselves to study this work? Euclid, in putting together the Elements, produced a paradigm for organizing a body of knowledge. Beginning with basic key geometric definitions and self-evident postulates which govern the way these geometric notions relate, the theory of plane geometry is developed from the basic to the more sophisticated through further definitions and constructions, with deeper relations disclosed in propositions. These in turn are established through rigorous proofs that unveil how preexisting relations can be woven together through logic in order to arrive at the desired result. Euclid’s approach to conveying mathematical knowledge in his Elements has been considered the ideal approach to organizing any subject of mathematics and persists today. It thus provides the means to find beauty in all parts of mathematics in ways that resonate with Russell, St. Vincent Millay, and many others.
Now, let’s consider an approach to geometry in which the notion of beauty presents itself in a more common sense fashion, namely through the notion of symmetry. The idea behind symmetry is rather simple. Consider the following square:
Symmetries of this square can be imagined in two different ways. First, imagine rotating the square counterclockwise through a fixed angle. A rotation is a symmetry if the square looks the same after rotation, such that the vertices and edges have moved to the positions of other vertices and edges. If you can imagine this, you should conclude that the rotations through 0, 90, 180, and 270 degrees give all the useful rotational symmetries. Below is a square with some of these rotations rendered:
The second type of symmetry is a reflection, as illustrated in this same diagram. Fixing your attention upon any one of the four lines L1, L2, L3, and L4 crossing through the center point of the square, imagine spinning the square in space around that fixed line until it lies back in the plane. This gives a reflective symmetry. It is a geometric result that all planar symmetries of the square that fix the center point are either one of the four rotations or one of the four reflections, totaling eight possible planar symmetries. Furthermore, a given symmetry can be related to another symmetry through some third symmetry by a method of composition in which performing one symmetry then applying a second symmetry will result in a third symmetry. For example, rotating the square counterclockwise 90 degrees then reflecting through a horizontal line L1 gives a symmetry that is identical to reflecting the square around the diagonal axis L2. Collectively these symmetries, together with this method of composition, form an example of a structure called a group.
Now an analysis of symmetries can be carried out for any geometric shape, not just the square. For example, any polygon in the plane, like the regular pentagon and hexagon below, has symmetries:
The regular pentagon has five rotations and five reflections in its group of symmetries, the regular hexagon has six rotations and six reflections in its group of symmetries, and so on.
If we consider now geometric objects in three spatial dimensions, the analog of regular polygons are the regular polyhedra, also known as the Platonic solids. In contrast to regular polygons, in which their number is infinite, there are exactly five Platonic solids, as shown on the next page. Each of these solids carries a group of spatial symmetries as well. Recall that for planar objects, the rotations and reflections were about lines, but for these solid objects the rotations and reflections are through planes. As a source of inspired beauty, many mathematicians, philosophers, and scientists, such as Euclid, Plato, and Kepler, have found such deep aesthetic pleasure in the Platonic solids that they’ve sought to make them building blocks of the universe. For example, Euclid’s Elements concludes with characterizations and a complete classification of the Platonic solids and Kepler, in his Mysterium Cosmographicum, gave a model rendering the solar At the start of the twentieth century, the revolutions in physics— special and general relativity theory and quantum physics—found in the mathematical theory of symmetry the means to model the quantitative properties and relations in the newly understood nature of space and time of the very large or the very small. For example, in Einstein’s theory of relativity, his principle of invariance asserts that the same experiment conducted at two different points in space and time will have essentially the same outcome once the appropriate space-time symmetry is taken into account. In quantum physics, the most fundamental of particles possess internal symmetries that individually characterize them as the particles they are. Furthermore, the way these particles interact with each other possess a wealth of symmetries that both characterize the relationships between them and provide the means to locate them experimentally in, for example, particle accelerators. In the current regime of theoretical physics research and exploration, the search for a Grand Unified Theory—a model of fundamental particles that accounts for all the forces of nature—has led physicists to extend the theory of symmetries of space-time in order to expand our current accounts of particle physics to also include gravity and relativity theory. For example, superstring theory incorporates a ginned-up version of symmetry known as supersymmetry. Thus, symmetry illustrates all three types of beauty mentioned earlier: unexpected relationships of symmetries into groups, simplicity of the visible geometry of the symmetric relationships, and new possibilities when applied to the physical world.