One thing that definitely feels like "magic" is Monstrous Moonshine (https://en.wikipedia.org/wiki/Monstrous_moonshine) and stuff related to the j-invariant e.g. the fact that exp(pi*sqrt(163)) is so close to an integer (https://en.wikipedia.org/wiki/Heegner_number#Almost_integers_and_Ramanujan.27s_constant). I hardly understand it at all but it seems mind-blowing to me, almost in a suspicious way.
this post was submitted on 03 Aug 2023
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Integrals. I can have an area function, integrate it, and then have a volume.
And if you look at it from the Rieman sum angle, you are pretty much adding up an infinite amount of tiny volumes (the area * width of slice) to get the full volume.
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An arithmetic miracle:
Let's define a sequence. We will start with 1 and 1.
To get the next number, square the last, add 1, and divide by the second to last. a(n+1) = ( a(n)^2 +1 )/ a(n-1) So the fourth number is (2*2+1)/1 =5, while the next is (25+1)/2 = 13. The sequence is thus:
1, 1, 2, 5, 13, 34, ...
If you keep computing (the numbers get large) you'll see that every time we get an integer. But every step involves a division! Usually dividing things gives fractions.
This last is called the somos sequence, and it shows up in fairly deep algebra.
The infinite sum of all the natural numbers 1+2+3+... is a divergent series. But it can also be shown to be equivalent to -1/12. This result is actually used in quantum field theory.
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Szemeredis regularity lemma is really cool. Basically if you desire a certain structure in your graph, you just have to make it really really (really) big and then you're sure to find it. Or in other words you can find a really regular graph up to any positive error percentage as long as you make it really really (really really) big.
Small nit: you don't compute sigma, you prove a value for a given input. Sigma here is uncomputable.
was not aware of the machines halting only iff conjectUres are true, tho. Thats a flat out amazing construction.
This one isn't terribly mind-blowing, though it does have some really cool uses. I always remember it because of its name: Witch of Agnesi
The birthday paradox