Mathematicians, on the other hand, have no qualms in defining their calculations and proofs as elegant. It’s an aesthetic sense that’s proven to be surprisingly universal, spanning cultures and eras: A Babylonian mathematician and a modern student might both enjoy learning a permutation.
The first is an ethereal elegance expressed by formal constructs and patterns. It’s a sense of awe about how the mathematical universe arranges itself in an inexorable order. Consider how the natural numbers line up exactly in an endless row. Consider the series of increasing-dimensional Euclidean spaces: a line, a plane, a vacuum, and so on. Alternatively, formal logic’s rigour and accuracy. These systems are extremely effective and useful, and they can also be breathtaking from a certain viewpoint.
However, for those on the other side of the divide — which seems to include most individuals and definitely most non-mathematicians — the notion of a vector space in n dimensions or a continuous function on the real line is difficult to get genuinely excited about. To grasp these ideas, one must appreciate a type of abstraction, and this aesthetic sense can be cold and formal. It’s the ice queen’s grace, better enjoyed from a discreet distance rather than up close.
The second kind of mathematical elegance is easier to understand. It’s for the exceptions to the rules, the things that don’t fall into any of the other categories. There are the oddities, one-of-a-kinds, and mathematical manifestations of the enthralling fossils and strange minerals that adorned natural history cabinets in the 17th and 18th centuries. This beauty has a distinct vibe: it’s exotic, quaint, personal, and, of course, highly subjective.
Trying to appreciate mathematics without knowing how it works is akin to reading a review of Beethoven’s Fifth Symphony rather than hearing it.
Consider the dodecahedron, which is a common item in many mathematical cabinets of curiosities. It’s a regular solid made up of 12 pentagons that’s just one of the five perfectly symmetric solids. Its allure has been characterised to me as “complicated but not too complicated.” The form has a long tradition as an esoteric symbol, dating back to Plato’s suggestion of a connection between the five objects, now known as the Platonic solids, and the physical universe. The dodecahedron represented all of the heavenly bodies, including the stars and planets, which were all perfectly formed and moving. Since then, this mathematical shape has come to represent the extraterrestrial, and