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Ask HN: Examples of beautiful mathematics?

6 points by tjr ↗ HN
I was amazed to learn the fundamental theorem of calculus, both for its intrinsic abstract elegance, and the fact that it applies to the motion of objects in the real world.

What other concepts in mathematics are similarly beautiful and remarkable? Perhaps more advanced mathematical concepts, or perhaps intriguing ways of looking at the basics?

7 comments

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jgrahamc
Cantor's Diagonal Argument: http://en.wikipedia.org/wiki/Cantors_diagonal_argument It's simple and it reveals the surprising result that, to put it crudely, 'some infinities are bigger than others'.
Jacquass12321
I don't know about beauty, but I realized I was done with math during my functions of a real variable class when we did a proof to the extent of, "There exists a rational number between any two irrational numbers. There exists an irrational number between any two rational numbers. Prove that their are more irrational numbers then rational numbers." Re countably infinite versus uncountably infinite.

Thankfully my crypto course came along a few years later and restored my faith in math.

zander1229
Worrydream.com has a few great examples of displaying complex information intuitively.
quadlock
while at university I witnessed a fellow student loose his faith from a lecture on fractals.
drewrv
I think there's a lot of beauty in the fact that e(i*pi) + 1 =0