= The beauty of mathematics
<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.