Reflecting on Euclid and Mathematical Beauty

When I took a Euclidean or classical geometry course as an undergraduate, we examined Euclid’s axioms in depth. At that time, it struck me that Euclid was truly the forefather of our modern mathematical system- grounding it in truths I had never previously thought to question. What fascinated me most, and what I later came to appreciate even more through reading about The Elements from the St. Andrew’s site, was how carefully Euclid structured his work- from definitions to postulates to logical axioms. This axiomatic method, where each proposition builds upon the previous in a clear deductive sequence, was my first real encounter with a historical mathematical piece that was a systematic presentation of reasoning itself.

I often think about Euclid’s continuing relevance to our current mathematical framework. Euclidean geometry models the geometry of flat space- the geometry that governs the physical world we interact with daily. In fields like architecture and construction, whether building furniture or entire structures, Euclid’s axioms form the invisible foundation for all spatial reasoning. Although non-Euclidean geometries expand beyond these limits, most of my students will only ever work within the Euclidean framework, since it remains central to the curricular content and to how we understand space in practice.

Something that really stayed with me from our class discussions was learning that Western colonizers and settlers viewed The Elements as second only to the Bible in importance. That was new to me, and it made me wonder whether they revered it not only for its content but also for its systematic way of thinking and its pedagogical value. In many ways, Euclid’s work provided a model for what knowledge and reasoning should look like- structured, logical, and ordered- a mindset that profoundly shaped Western education and, through colonial expansion, much of the modern world.

Whenever I think about the term mathematical beauty, I’m reminded of Francis Su’s (2020) Mathematics for Human Flourishing. Su writes about beauty as a human virtue and explains that the first and most accessible kind of mathematical beauty is sensory beauty. In my previous course (EDCP 553) with Susan, we explored embodied mathematics, which I see as a form of sensory beauty itself. Sensory beauty is the kind we experience with our senses- the intricate fractal patterns in cauliflower, the symmetry of Islamic art, or the luminous stained-glass windows of Notre-Dame in Paris (as in the photo attached that I took this summer). These common notions in Euclidean mathematics seem almost simple, so familiar that I’ve often taken them for granted. Yet it wasn’t until I encountered the “counter-universe” of non-Euclidean geometry that my own mathematical universe felt as though it had shattered. And still, it is precisely through the existence- and the limitations- of Euclidean geometry that mathematicians were able to explore and discover the beauty of the non-Euclidean world. Ultimately, Euclid's enduring legacy reminds me that mathematics is not just a study of numbers or shapes, but of how humans seek order, meaning, and beauty in the world around them.