The Pythagorean paradox

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While area is easily approximated by squares, length resists being defined out of simpler parts! A diagonal isn't just right-angle steps, no matter how tiny. An ever finer staircase converges to the area of the triangle, but NOT to its perimeter —a foundational fractal paradox.

It's unexpected because the method of exhaustion does calculate the perimeter of a circle out of simpler shapes: infinitesimal diagonals approximate the length of a curve, a keystone of calculus from Greek antiquity. But this method breaks down for diagonals themselves!

In a geometric space, the "easier" concept of 1D length turns out to be "harder" than 2D area. It is the paradoxical genius of the Pythagorean theorem that it uses area to define length, building down the simple from the complex.

This is all very related to Weyl's tile argument that in a discrete universe the Pythagorean theorem doesn't apply. Though here you're not counting steps as square sides, the squares are distance atoms, so the diagonal of our triangle has leg length (vs × 2 or √2).

Weyl's 1949 argument is simple yet obscure. It seems to be of serious interest to philosophers & physicists more than mathematicians. en.wikipedia.org/wiki/Weyl%27s_tile_argument Of the precious few to ever tweet about it, there's of course @prathyvsh, referencing a fun thread on this meme: twitter.com/prathyvsh/status/1594768105309237249

I gather from this comment twitter.com/PeterSaveliev/status/1662109644431433732 by @PeterSaveliev that this recent thread seems to be a key to this whole discussion, but I'm still trying to figure out the math. twitter.com/gabrielpeyre/status/1661960353809580034?s=20

See also years of discussion in the Math StackExchange where the most upvoted question calls it the staircase paradox math.stackexchange.com/questions/12906/the-staircase-paradox-or-why-pi-ne4 > The problem is it doesn't approach in a "smooth" way. > By splitting the path you essentially create lots of little triangles.