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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That at some point you'll run out of corners when you run up against the circle?
Or am I just not learned enough in math/science to follow along with the point?
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the way I see it is like, the folds you get by infinitely cutting in corners will make the perimeter always 4, but the shape will never be a true circle. ~Kinda similar to how mendelbrot set has an infinite perimeter by definition but the area is finite. Or kinda like intenstines
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