Just in to say, I know exactly what you mean, and I love the subject, to the point I did a PhD in pure maths. The whole "golden ratio" in nature, and a lot of other adjacent stuff, leaves me indifferent at best, and really irritates me at worst. It's often cheapo mathy wank to feel clever when you talk to your friends. There is nothing wrong in being interested in it, but I'd hope someone into maths would eventually go beyond that.

I also am not a fan of several very useful branches of maths, like calculus, but it's a tool you need to have. Some people love it though, and I scratch my head at it as much as you do, if not more probably, because I have had to use it so much.

There are probably as many reasons to find maths beautiful (or ugly) as there are people, but for me it boiled down to the fact that:

• with relatively few assumptions, we can logically and iteratively build an abstract machinery that is consistent (well, with caveats...);
• a lot of these abstract theories provide good enough frameworks for other sciences to base theories on and be successful at explaining the world with these (I could talk about the fifth axiom of Euclid, non-euclidian geometries, and how we eventually arrive to something that is a formal setting for the theory of relativity for hours, it's fascinating);
• it provided me with abstract objects that I could reason about, explore in different ways, and with different points of view, until something clicked and I got to understand the objects better;
• some proofs, even of quite complex theorems, have such a simple and elegant initial idea (when others can feel quite forced), that it is hard not to marvel at how things fall neatly into place (sometimes...).

So to me maths provided a setting in which things worked and made sense, and you could essentially just enjoy an endless supply of puzzles in that setting, whose solutions you could formally prove.

Unlike a lot of maths nerds, I don't necessarily think that that's totally limited to maths though. I think most people do their abstract thinking and puzzle solving on whatever it is that they find beautiful. Or I hope they do, it's a wonderful feeling. The formal aspect of proofs though (and I don't necessarily mean computations), that's the unique thing that can set it apart.