(Question) 3D(+) space-filling

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Offline FractalDave

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« on: November 11, 2018, 10:09:41 PM »
Can't seem to find a proof against this, but then again I can't find an example either.....

Are there any 3D IFS that are genuinely 3D space-filling like the cesaro sweep in 2D (or multiple copies will tile space like the various Dragons) using just two affine transforms ?
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Offline pauldelbrot

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« Reply #1 on: November 12, 2018, 03:38:55 AM »
An obvious 8-transform one would subdivide a cube into eight half-size copies.

This suggests a 2-transform one where each transform is rotated in a way that permutes the faces in two cycles of three (like spinning around a body diagonal 1/3 turn), then shrinks the cube to half its size along just one dimension; the two transforms stack these next to each other to fill the original cube's volume. Because of the 1/3 turn part, a second iteration should hopefully then split these into four long narrow bars, and a third should hopefully break those in two, and after all three iterations you'd have the same result as from one iteration of the 8-transform version.


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