What are your favorite (potentially long) mathematician quotes? Bonus points if it's not just a pithy restatement of "math is beautiful", but something which changes how you think.
E.g. here's a nice one, h/t @Looking_glass_u:

“If one proves the equality of two numbers a and b by showing first that a ≤ b and then that a ≥ b, it is unfair; one should instead show that they are really equal by disclosing the inner ground for their equality.” Emmy Noether
Another good one: “All mathematicians are familiar with the concept of an open research problem. I propose the less familiar concept of an open exposition problem. Solving one means explaining a mathematical subject in a way that renders it totally perspicuous” - Timothy Chow

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More from @3blue1brown

19 Jul
Okay, follow through with me on this one and I guarantee you're in for a fun surprise.

We'll start forming rows of numbers, kind of like Pascal's triangle, but with a different rule.

The first row is simple "1, 1"
For row 2, copy the first row, but insert a 2 between any elements that add to 2.

In this case that just means sticking a 2 between the two 1's.
For row 3, copy row 2, but insert a 3 between every pair of elements adding to 3.
Read 19 tweets
7 Jul
In light of the very sad news about Ron Graham's passing, I thought I'd share an interesting tidbit about his famous constant which I only learned recently (thread).
This number has captured the imaginations of many people with its unfathomable size, myself included. I distinctly remember when I was a kid first learning about it spending hours trying to wrap my mind around it and using it as inspiration for writing ever-larger numbers.
The existence of huge numbers is not in and of itself interesting, but what captivated me was how abstraction and recursion even let us _describe_ such numbers. Honestly, it was one of the first times I appreciated how powerful good definitions could be.
Read 11 tweets
30 Jun
Of all the videos I've made, one of my favorites topics to have covered was a proof of the inscribed rectangle problem by H. Vaughan using a Mobius strip.

But now there's a new result!
@QuantaMagazine recently did a great article about recent work by Greene and Lobb using a beefed-up version of the same idea, letting a Mobius strip encode geometric properties of the curve to solve a more general result, check it out:

From recent comments on the video above, it looks there's a little confusion where some people thought this means the inscribed square problem (i.e. the Toeplitz conjecture) has been solved, but that's not quite the case.
Read 5 tweets
7 Nov 19
The birthday paradox is very famous in probability. If you take 23 people, there's about a 50/50 chance that two of them share a birthday. With 50 people, it's a 97% chance.

We could make many other fun examples to illustrate the same counterintuitive phenomenon (thread).
Choose a random card from a deck of 52 cards. Put it back, shuffle well, and choose another. Do this for only 9 draws, and more likely than not, you've pulled the same card twice.

Do it 16 times, and your chances are over 90%. Try it!
Next time you're in an event with more than 118 people, think to yourself that there's a >50% chance that two people there have phone numbers with the same last four digits (assuming those are uniformly distributed).

With more than 250 people, its >95%.

Ditto for ATM pin codes.
Read 12 tweets

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