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.
Specific examples:
- from computer science:
- the existence of undecidable problems, especially simple to state ones, e.g. mortal matrix problem
- Mathematics | 17, 10k, 260
- Ciro Santilli's Homepage | 238, 154k, 2k
- Classification of finite simple groups | 84, 166, 4
- Classification of regular polytopes | 213
- Lists of mathematical problems | 13, 291, 3
- Platonic solid | 38