I'm not even sure when I recognized these properties I enjoyed as beauty--probably early on before even labelling it as such. High school when exposed to different kinds of math was probably the most eye-opening. Always did have an intrinsic feel for it, such as coming up Pi is dumb (use Tau) before ever hearing of it.
log(a*b) = log(a) + log(b)
i^2 = -1
This was an eye opener for me. Math is designed to simplify things. To represent concepts in an easy way to read and manipulate and tinker with.
I tried at some point to invent an extension of the complex numbers to represent a 3D space with i and j, only to rediscover that this is not possible, you need to go deeper, with i, j and w to represent 3D rotations along an axis. and that's how i began reading about quaternions.
The beauty of maths is also the fact that it is kind of universal, even if it was invented.
3Blue1Brown
The connections and visuals are stunning.
I am a simple soul like that.
Currently doing advanced high school math in order to fix my fundamentals.
Fundamental weaknesses that I discovered:
- I hated trig and was bad at it
- I didn’t understand certain algebraic manipulations
- Integrals, I still need to learn how to do them
That’s what I am fixing with my high school advanced math course. And it’s working!
But the explanations kinda sucked, especially trig explanations.
They did stuff like:
sin(2a) = 2sin(a)cos(a)
With no explanation why!
Just suck it up and memorize.
In high school I refused this treatment and got told off which is how I started to dislike math.
Now I am older.
I am wiser.
I still refuse :P
I still get told off (in the course I am doing now).
But I know it’s not their fault, I’ve simply learned that most math teachers can’t help me as they are not in the business of teaching the why. They teach the how (which I need as well, I need both the why and the how).
Enter 3Blue1Brown
He taught me:
To feel why 1/x is the derivative of ln(x). Yea, I can derive it, but now I feel it.
To feel why the midpoint plus/minus a deviation is a way of expressing the quadratic formula.
To feel why double angle identities are the way they are (he shows it visually with complex numbers, it’s so obvious! How can I forget something that I can visualize in my mind as easy as visualizing my dreams?).
To feel why sin(pi/4) = sqrt(2)/2
Khan Academy did it for a few other sin(...) values.
Suddenly, this high school math course transformed from not understandable to super duper obvious!
3Blue1Brown shows the why visually
The course shows the how.
I need both.
But there are a lot of courses of similar quality, but there is only one 3Blue1Brown.
(There are others like Khan Academy, Numberphile, etc. but 3B1B has shown me the beauty of math, not just the context. Animations with piano music really help!)
If you study enough, you might end up finding some results or proofs elegant and beautiful and so on, and that's great. But in a lot of online materials (e.g. youtube, quanta magazine) beauty is a trick used by people who think that math is important to push math on people who don't care about it and there's something very offputting about the whole thing.
I think people don't need to worry about the beauty in mathematics that much - study math as much as you need/want and if something ends up looking beautiful to you, that's just a nice bonus.