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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Hmmm. I wonder if i took calclulus and have since forgot why limits make sense. Or if i just learned to use them without ever understanding them
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Huh. I guess I would have considered it to approach pi asymptotically as n approached inf, but that's not the case, is it? That's... yeah, that's interesting.
Now I kinda wish I had a kid to teach this to, with jars and boxes and strings, just point 'em at it see what happens.
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is the answer that the resulting circle is infinitely wrinkly with the extra length
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i would say it more simply - there just is no “resulting” shape at all, circle or otherwise
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the problem is that the perimeter of the area is wrong, though the area itself converges properly. riemann sums are about area not perimeter!
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well, I'm not concerned with the length of the edges of the rectangles in the riemann sum...
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