Ciro Santilli intends to move his beauty list here little by little: github.com/cirosantilli/mathematics/blob/master/beauty.md

The most beautiful things in mathematics are results that are:

- simple to state but hard to prove:
- Fermat's Last Theorem
- transcendental number conjectures, e.g. is $e+π$ transcendental?
- basically any conjecture involving prime numbers:
- many combinatorial game questions, e.g.:

- 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, sometimes via pathological counter-examples. General surprise themes include:Lists:
- classification of potentially infinite sets like: compact manifolds, etc.
- problems that are more complicated in low dimensions than high like:
- generalized Poincaré conjectures. It is also fun to see how in many cases complexity peaks out at 4 dimensions.
- classification of regular polytopes

- 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? Abel-Ruffini theorem

- 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

Whenever Ciro Santilli learns a bit of mathematics, he always wonders to himself:

Am I achieving insight, or am I just memorizing definitions?Unfortunately, due to how man books are written, it is not really possible to reach insight without first doing a bit of memorization. The better the book, the more insight is spread out, and less you have to learn before reaching each insight.