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This is similar to the claim that an orange, say, can be cut into pieces that can then be put together to make two oranges. It turns out some of these pieces wo
by hw-guy 5y ago
This is similar to the claim that an orange, say, can be cut into pieces that can then be put together to make two oranges. It turns out some of these pieces would be infinitesimal, and hence smaller than the atoms making up the orange (or whatever). While such a result may be satisfying to a theoretical mathematician, the engineer in me recoils.
- nmilo 5y agoBut here the pieces aren't infinitesimal. They're fractals, but still measurable.
- mrob 5y agoFractals are still cheating IMO, because fractals have infinitely small features. Whenever you use infinity you can get all kinds of crazy results. It's like the geometric proof that pi=4: Draw a circle of diameter 1 Draw a square touching it on all sides, perimeter 4 Cutting at right angles to the existing edges, cut smaller squares out of all the corners so they touch the circle Perimeter remains 4 Repeat this corner cutting infinity times Perimeter of the cut square (4) matches the circumference of the circle (pi) pi = 4 Unlike traditional geometry, it's just abstract symbol manipulation with no relevance to real shapes.
- mb7733 5y agoThat proof is just plain incorrect, though. It will break down when trying to prove this statement: >Perimeter of the cut square (4) matches the circumference of the circle (pi) Calculus will show that the area of the fractal approaches the area of the circle. But it will not show that the perimeter of the fractal approaches the circumference of the circle. It remains 4 at every step in the iteration, so the limit is still 4.
- contravariant 5y agoThe pieces in this particular example seem to be quite a bit better behaved. In fact they're measurable.
- Someone 5y agoMathematically, it’s quite different. The Banach-Tarski paradox (https://en.wikipedia.org/wiki/Banach–Tarski_paradox https://en.wikipedia.org/wiki/Banach–Tarski_paradox) changes the _volume_ of the objects. That’s requires some of the prices to be immeasurable. It also is about a 3D sphere, and the strong form (cutting a sphere in finitely many parts and reassembling those into two equal-sized spheres) doesn’t work in 2D or 1D (in contrast, in 3D, five pieces suffice. I don’t know whether that is a tight bound)
- thaumasiotes 5y ago> The Banach-Tarski paradox (https://en.wikipedia.org/wiki/Banach–Tarski_paradox https://en.wikipedia.org/wiki/Banach–Tarski_paradox) changes the _volume_ of the objects. That’s requires some of the prices to be immeasurable. It changes the volume by a discretionary amount; you can create two spheres of the same size as the original sphere, or 500 spheres of the same size as the original sphere, or you can create one sphere of double the radius [= four times the size] of the original sphere. I see no reason to believe that you couldn't also make one cube of equal volume to the original sphere?
- OscarCunningham 5y agoIt is a tight bound. http://matwbn.icm.edu.pl/ksiazki/fm/fm34/fm34125.pdf http://matwbn.icm.edu.pl/ksiazki/fm/fm34/fm34125.pdf https://www.irregularwebcomic.net/2339.html https://www.irregularwebcomic.net/2339.html