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I was re-reading about the method of exhaustion, and how it can be used to approximate the area of a circle via successive 2n-gons, by basically adding new triangles to each side of the previous polygon.

But then I remembered a weird problem in which a unit circle is approximated by a circumscribed square, from which we remove the corners, and then we remove the corners of the new "step-looking figure", ad infinitum, leading to a perimeter of 4 for the circle, as shown in this figure

https://ibb.co/tQ0NpJ5

This is, of course, ridiculous. But I can't quite put my finger on why. At first I thought it might be some obscure thing between areas and perimeters, but then I ran into a different problem that deals strictly with lengths, and ends by "showing" that '\sqrt{2} = 2', as shown in

https://ibb.co/bPBP27j

so now I'm thinking perhaps something to do with fractals, but I'm not even sure how to start my research on the topic

***I'm not sure the site is loading properly, because of my internet connection, as I don't see the formatting options bar, and the tags won't show the usual suggestions as I type, so I apologize for any weird formatting within my question

Mahoma
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  • You might want to find a reference on rectifiable curves, or just arc length. Rudin definitely has one, although he is annoyingly terse. The same logic behind that $\pi=4$ meme also shows how you can make any curve appear to be equal to any number you want. – pancini Oct 11 '19 at 05:31
  • The rationale behind the exhaustion procedure for the mensuration of the circle is that areas are monotone, that is if $A$ is a subset of $B$ then the area of $A$ is $\le$ the area of $B$. – Angina Seng Oct 11 '19 at 05:35
  • For the conceptual answers, see TCL https://math.stackexchange.com/a/12917 "two functions can be very close, but their derivatives can still be far apart"; Eric Naslund https://math.stackexchange.com/a/22846 "limit of the sum is not the sum of the limit"; and Steve Byrnes https://math.stackexchange.com/a/1856555 "arclength of a curve is a discontinuous function" – Chris Culter Oct 11 '19 at 05:38
  • Same problem, its a nice read! :-) http://coopersquarereview.org/post/how-to-tame-infinity/ – maxmilgram Oct 11 '19 at 05:38

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