<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 but hard to prove>:
  * Fermat's Last Theorem
  * <transcendental number conjectures>, e.g. is $e + \pi$ transcendental?
  * basically any conjecture involving <prime numbers>:
    * <twin prime conjecture>
  * many combinatorial game questions, e.g.:
    * <first-move advantage in chess>
* 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 (mathematics)> counter-examples. General surprise themes include:
  * <classification (mathematics)> of potentially <infinite> sets like: compact <manifolds>, etc.
    * <classification of finite simple groups>
    * <classification of regular polytopes>
    * <classification of closed surfaces>, and more <generalized Poincaré conjectures>
    * classification of <wallpaper groups> and <space groups>
  * 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>
  Lists:
  * https://math.stackexchange.com/questions/139699/what-are-some-examples-of-a-mathematical-result-being-counterintuitive
  * https://math.stackexchange.com/questions/2040811/what-are-some-counter-intuitive-results-in-mathematics-that-involve-only-finite/2055458\#2055458
* applications: make life easier and/or modeling some phenomena well, e.g. in <physics>. See also: <how to teach/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:
\Q[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.


Discussion on "The beauty of mathematics" by Ciro Santilli 32