Ciro Santilli intends to move his beauty list here little by little: https://github.com/cirosantilli/mathematics/blob/master/beauty.md
The most beautiful things in mathematics are results that are:
- simple to state and understand (pre-high school, high-school, lower undergrad), but extremely hard to prove, e.g. Fermat's
- surprising results: we had intuitive reasons to believe something as possible or not, but a theorem shatters that conviction and brings us on our knees.General surprise themes include:
- classification of potentially infinite sets like: compact manifolds, etc.
- problems that are more complicated in low dimensions than high like: classification of complex manifolds. It is also fun to see how in many cases complexity peaks out at 4 dimensions
- unpredictable magic constants. Why is the lowest dimension for an exotic sphere 7? Why is 4 the largest degree of an equation with explicit solution?
- applications: make life easier and/or modeling some phenomena well, e.g. in physics. See also: explain how to make money with the lesson
Good lists of such problems Lists of mathematical problems.