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nomadreid

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In looking at the definition of a Hausdorff dimension of a space S =

inf{d>0: inf{Σ

where i ranges over a countable set, it would appear that it would be acceptable to take the index set to be finite, but I am not sure how you would then get a zero infimum.

The motivation for the question (and thereby a second question) is as follows: whether a finite number of steps in evaluating a fractal algorithm is acceptable. If so, there could be a situation where a process generates a fractal if it stops before the infinite number of steps, but its fractal nature disappears when taken out to infinity if one of the requirements to be a fractal is to have a non-integer fractal dimension. , but remains a fractal if a fractal is only required to be an irregular shape that can be divided into parts in such a manner that the shape of each part

The latter definition would make it seem that a single step would be sufficient, but the zero infimum requirement would seem to nix that. I am not sure if there is an accepted definition of a fractal; the allowance for a finite number of steps certainly allows one to classify finite objects as fractals, but on the other hand it goes against the grain if one wishes the fractal to be a fixed point of a scaling self-similarity. So, the verdict: is a space-filling curve's fractal nature a function of the number of scalings before self-similarity stops?

inf{d>0: inf{Σ

_{i}r_{i}^{d}: there is a cover of S by balls with non-zero radii} =0}where i ranges over a countable set, it would appear that it would be acceptable to take the index set to be finite, but I am not sure how you would then get a zero infimum.

The motivation for the question (and thereby a second question) is as follows: whether a finite number of steps in evaluating a fractal algorithm is acceptable. If so, there could be a situation where a process generates a fractal if it stops before the infinite number of steps, but its fractal nature disappears when taken out to infinity if one of the requirements to be a fractal is to have a non-integer fractal dimension. , but remains a fractal if a fractal is only required to be an irregular shape that can be divided into parts in such a manner that the shape of each part

**resembles**the shape of the whole. (For example: a Penrose tiling decomposition has a non-integer Hausdorff dimension for a construction of a finite number of steps, but at infinity it becomes a space-filling curve with the Hausdorff dimension of 2.)The latter definition would make it seem that a single step would be sufficient, but the zero infimum requirement would seem to nix that. I am not sure if there is an accepted definition of a fractal; the allowance for a finite number of steps certainly allows one to classify finite objects as fractals, but on the other hand it goes against the grain if one wishes the fractal to be a fixed point of a scaling self-similarity. So, the verdict: is a space-filling curve's fractal nature a function of the number of scalings before self-similarity stops?

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