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:
Good lists of such problems Lists of mathematical problems.
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.
One of the most beautiful things in mathematics are theorems of conjectures that are very simple to state and understand (e.g. for K-12, lower undergrad levels), but extremely hard to prove.
This is in contrast to conjectures in certain areas where you'd have to study for a few months just to precisely understand all the definitions and the interest of the problem statement.

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A beautiful quote fom Edward Titchmarsh:[ref]
It can be of no practical use to know that pi is irrational, but if we can know, it surely would be intolerable not to know.
Ciro Santilli believes that it is perhaps for this reason that simple to state but hard to prove theorems are so attractive.
In mathematics, a "classification" means making a list of all possible objects of a given type.
Classification results are some of Ciro Santilli's favorite: Section "The beauty of mathematics".

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Oh, and the dude who created the en.wikipedia.org/wiki/Exceptional_object Wikipedia page won an Oscar: www.youtube.com/watch?v=oF_FLN-TmCY, Dan Piponi, aka @sigfpe. Cool dude.
Cool examples:

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He's a bit overly obsessed with polynomials for the taste of modern maths, but it's still fun.

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Ciro Santilli would like to fully understand the statements and motivations of each the problems!
Easy to understand the motivation:
Hard to understand the motivation!

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