actually a very good exercise to understand why this argument doesn’t work. you’ll learn something about the nature of continuity and what it really means to repeat a process infinitely many times
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you actually don’t need a background in real analysis or topology even, although it would help you go deeper; the basic conceptual mistake can be explained more simply
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although having understood the basic conceptual mistake, if you did take a calculus class you may subsequently wonder why the same objection doesn’t apply to the definition of an integral in terms of riemann sums. this is a good question and surprisingly subtle imo!
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hm, sorry, looks like i was too vague about what i meant by "the same objection" - it's not exactly the same objection since lengths are not areas. the question is why this limiting procedure is a sensible way to define areas but not lengths
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hm interesting, yea
we can see from the first few egs that "removing corners" doesn't reduce the perimeter, meaning it's basically still the "same 4 lines". and if you repeat that infinitely you still never actually get "to" the circle
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It's interesting how the area does equal the circle's at the end, but the perimeter does not.
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okay i'm gonna be real
i have a math degree but don't really understand why this is the case. perhaps a poor reflection on me/my school
i get that while the distances between the curves goes to 0, that doesn't mean the curve lengths do, but just saying that feels handwavy
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i see there would be some weirdness with the derivatives - slopes are well-defined on a circle but not so on the outer shape, so the derivatives of the curves do not match
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I remember coming up with this independently for the diagonal of a square as infinite steps
I assume it has something to do with “infinity is not a theoretical largest number, it is all numbers”
Idk tho it seems like limits are just made up to me
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my mental model for this, after bouncing around a few links, is like an asymptote: the points on the asymptote approach a curve as n->\infty but never meet it
this process approaches a circle's perimeter but never meets it
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the sequence does converge pointwise to the circle. what fails is the derivative. in the limit, the perimeter is an integral over the derivative of the parameterized curve, but that derivative isn't defined
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