Oct 18 Reflection on Euclid's Works

I don’t remember when was the first time I heard of Euclid’s moniker, but there was a math contest from the University of Waterloo named after him. People who were applying there [for math/CS/engineering] in grade 12 were recommended to take that exam. Younger feverish students also took it for fun (or for fame/rewards). The early bits were doable, but the difficulty got harder as the questions progressed. There were some very interesting geometry questions later on that I’ve never seen before. The fact that this contest (which was a supplementary material for evaluating application forms for UWaterloo) was named after him shows how lasting his name is embedded in our academia institutions. I later studied Euclidean geometry in university, a topic I haven’t revisited since grade 9. Euclid’s book (The Elements) has indeed left a great mark on mathematics and the way we think about abstract concepts. His methods are quite systematic. He presented a subject in a logical manner based on definitions, axioms, and theorems. This systematic approach became a model for subsequent mathematical works and the scientific method in general. Starting with small lego blocks, larger things could be built from those blocks which in turn build into even larger things. Geometric proofs are also quite visual, so those who are more visual thinkers would have a better time understanding them than say a proof from real analysis. The geometric proofs are also presented in a clear manner. This clarity has made it a favored choice for educators over time. Overall it lays out the foundations of plane geometry nicely and succinctly. Although beauty is quite subjective, it often appears with elements of simplicity, symmetry, clarity, depth, and harmony. When applied to abstract concepts like mathematics, beauty often emerges from the elegance of a certain topic, the depth of insight it offers, or the surprising connections it makes (how well things go together). Because of this, many find beauty in the Euclidean axioms. Some of the proofs and constructions that use those axioms are thought to be beautiful, like some sort of minimalist perfection where every step feels necessary with no redundancy. Beauty isn’t all about a physical image that is captivating to the eyes, but it’s also about mathematical notions working together and fitting together nicely like jigsaw pieces.