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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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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that's my understanding - it's similar to a fractal coastline, where you can get (handwaving) arbitrarily close to the area of the shape you're describing with as much length of perimeter as you want
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I suck at calculus but I feel like I ought to be able to say something helpful about this with Green's theorem, specifically the smoothness constraints (although that's where I'm getting tripped up in wikipedia rn)
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oh this is interesting, i hadn’t thought about this as a possible angle. not sure the details work out but it would be cool if they did
I'm just grasping at memories half a lifetime ago and hoping they pay off lol
my more recent appropriate math is, idk, differential geometry or something but that doesn't play nicely with the tangents. I think I'm good on an intuition of the problem but miles away from a proof
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I got this far and realized I didn't have the tools to figure out the actual fractal dimension of the perimeter
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Replying to @ThanksThoth
the extra 0.85841... of the perimeter is hidden in the fractal dimension of the "circle"
a circle's perimeter is smooth, fractal dimension of 1
the pseudo-circle's perimeter is 'crinkly' and has a fractal dimension > 1
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