Great is the LORD and most worthy of praise;
his greatness no one can fathom.
One generation commends your works to another;
they tell of your mighty acts.
They speak of the glorious splendor of your majesty—
and I will meditate on your wonderful works.
In high school, I was a voracious reader, particularly of science fiction and fantasy novels. It was a form of escapism for me. To imagine myself as a knight in medieval times or a space explorer on an unknown planet was my favorite way of passing the time—so much so that if my studies failed to feed my imagination in the same way, they often took second stage.
During my sophomore year, I encountered a genre of science fiction, written in particular by H.P. Lovecraft, that portrayed our everyday experiences as a mere façade. Behind the façade lay deep, ancient mysteries and forces, often nihilistic, that might someday make their appearance to the detriment of humankind. This type of literature fascinated for me for two reasons. First, much of the writing occurred during an era when research in physics was providing startling new insights into the nature of our physical world. Einstein’s theories of special and general relativity were beginning to give us a picture of large-scale space and time as curved rather than flat, as previously thought. Quantum mechanics was illuminating the very small as exhibiting weird physical properties in which particles exist as energy packets, called quanta, which “move” in a discrete fashion.
This view blurred our ability to make observable predictions of both their position and velocity. Ideas like higher dimensions and non-Euclidean geometries provided fertile ground for imaginative stories. Often these stories contained descriptions of strange alien races who achieved bewildering advances in science and technology which we would begin to understand only when our mathematics and science one day reached a suitable maturity, perhaps millennia from now.
The second reason these writings fascinated me was that they often found a way to convey a deep sense of wonder and mystery toward the unknown of our world, while portraying science and mathematics as a way of pulling back the veils that hide these mysteries. While I was reading these stories, my geometry class started to hold more of my attention as I looked for evidence of deeper mysteries in the geometric constructions and deductions discussed in class. During my senior year, as I was considering which colleges and universities to apply to, I looked for opportunities to explore topics like curved space and non-Euclidean geometry in ways that hinted at the possibility of wonder these stories promised. I made it my mission to focus my studies on mathematics and seek out the deeper mysteries that might be hidden with our reality.
When I entered college and embraced Christianity, suddenly mathematics and wonder took on completely new meanings.
From Wonder to Beauty
Where there exists wondrous proportion and primal equality…
Saint Augustine, On the Trinity, vi. 10
As is often the case, a college or university can be a transformative place for young minds. For me, it was the place where my academic studies fed my imagination rather than being a distraction from the fantasies I sought to indulge. In my study of mathematics, I found hints of the wondrous worlds that the stories I read in high school had suggested, a sense of something that transcended our everyday experience.
In my first year, I encountered something else that also pointed to transcendence beyond our world. I made friends whose lives demonstrated connection with a divine Creator who sought a deep relationship with men and women and initiated that relationship through a historical act of incarnation. The faith that they shared with me opened my eyes to a true source of transcendence: a God who divinely created our reality and fills it with wonder and beauty, a God who provided the true way to knowing him through the salvific work of his Son, as incarnated in Jesus Christ. Suddenly, I understood our reality and the potential realities mathematics spoke of as being illuminated by the same light, the light of Christ.
Now, what can be said of these wonders in mathematics that mesmerized me? Are they the same things that attract other people to the study of mathematics? At the root of every subject in mathematics are both a sense of quantity and a sense of relations: geometry explores spatial relations, number theory explores natural number relations, analysis studies relations within continuous quantities, and so on. Within each one of these subjects, mathematicians can unlock the mysteries of deep and elegant patterns. From these patterns, mathematicians develop sophisticated theories that expand the context in which these patterns can be found. These theories in turn provide a broader range of possibilities for applications within mathematics and, possibly, within other sciences. It is this process of pattern exploration, theory building, and application which drives the development of the various subjects in mathematics. Within that process, mathematicians find points that can instill inspiration and wonder. Such points have certain features in common which individually or collectively can be said to portray a sense of beauty. Here are some of those features:
- Unexpected connections: In a study of one or more mathematical subjects, two or more seemingly disparate objects or relations may suddenly be seen as shades of a single web of relations, providing a sense of unity within or across such subjects.
- Simplicity: A mathematical theory aims to explain the logic underlying discovered patterns using basic definitions, intuitive truths, and suitably basic constructions. Within such a framework, deep and unexpected connections are most intensely revealed when relations can be explained with the greatest simplicity, enabling ease both in discerning their hidden truths and in articulating and communicating such patterns to others.
- Openness to new possibilities and deeper connections: Dwelling upon singular patterns and merely giving them a simple explanation is often insufficient and can lead dead ends. What can instill a deeper sense of inspiration to mathematicians is to develop a framework within the theory that not only explains those relations and patterns investigated, but allows for previously unseen connections to unfold.
Let us now move beyond these broad stroke descriptions of what mathematicians find beautiful to look at some concrete examples that demonstrate some of the features associated with beauty in mathematics.