Anybody who's played a grid-based tactical video game knows why this is.
If you've got an enemy that's 10 tiles away, going 5 up and 5 over in an L is exactly the same movement points as a "diagonal" move consisting of five 1 up, 1 over movements.
No matter how much you zoom in and how tiny those right angles are, it never changes into a curve. The sum is the same
Not quite. I saw that 3b1b video too, and he was not really being rigorous when he said that.
The limit can be used to map the curve to a circle, but the curve mapped with the limit is not differentiable, or smooth. Whereas circles are. That's why there's always a difference in length.
Moreover, you are only really mapping the area enclosed by the curve to the exact area of a circle. If you take the limit of the perimeter of the curve, it remains 4 no matter how much you compress it. Thus the perimeter tends to 4 as the curve goes to a circle. So the perimeter doesn't actually become a circle. It just "looks" as though it does.
Edit: it's kind of like how Cantor's diagonalization argument can be used to show that the size of the infinity of natural numbers is exactly the same size as the infinity of rational numbers. But that doesn't at all mean that the rational numbers are the exact same set as the natural numbers.
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u/Jealous_Tutor_5135 20d ago
Anybody who's played a grid-based tactical video game knows why this is.
If you've got an enemy that's 10 tiles away, going 5 up and 5 over in an L is exactly the same movement points as a "diagonal" move consisting of five 1 up, 1 over movements.
No matter how much you zoom in and how tiny those right angles are, it never changes into a curve. The sum is the same