Is fractal nature dependent on the number of steps?

[nomadreid]

In looking at the definition of a Hausdorff dimension of a space S =
inf{d>0: inf{Σ_ir_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?

 

[nomadreid]

Or, to rephrase simply: is it correct to say that, strictly, in nature there are no true fractals? Or has the word "fractal" come to be more loosely defined?

 

[jim mcnamara]

Do you know about L-systems? They model fern fronds, for example. So I am not getting what you are saying. Seems to me you are saying something akin to: 'there are no differential equations in physical matter'. Do I understand correctly?
L is for Lindenmayer IIRC.
See: https://en.wikipedia.org/wiki/L-system lots of nice graphics, too. L-system Hausdorff dimensions are between 3 and 4 - I think.

Check out the Barnsley fern.

 

[nomadreid]

Thanks for the reply, jim mcnamara. As I understand it from your reference, L-systems are computer systems which can carry through the recursion up to a finite limit, and so, in the words of your link, "fractal-like", so the products are not actually fractals. The Barnsley fern does not exist in nature; there are natural ferns that approximate them. Likewise, when one says that a cauliflower or broccoli or the coast of England has a non-integer Hausdorff dimension, one means that they are good approximations to a fractal with this dimension. Similarly, physical objects can be "fractal-like" solutions to differential equations , but to be a fractal (as I understand it, but I am posting this to get any false impression corrected), one would need to get the self-similarity down at all scales; otherwise the figure is only approximately self-similar. Complete self-similarity does not exist in our measurements of physical objects. That is, a fractal (by the definition in my original post) is an abstraction, just as "pi" is. The value of the ratio of circumference to diameter in physical measurements can at best be an approximation to pi; similarly to fractals. Or is my definition wrong?

 

[jim mcnamara]

The point I am trying to make, badly, is: just because we cannot measure something down to some arbitrary $\epsilon$ does not invalidate mathematical models. Modern math first came to be in order to explain the physical world, to model some physical entity or process. Works amazingly well.

IMO, the the model of Asplenium M. Barnsley that produced models an Asplenium adiatum-nigrum frond well.
https://en.wikipedia.org/wiki/Asplenium_adiantum-nigrum
https://en.wikipedia.org/wiki/Barnsley_fern

 

[WWGD]

An informal definition of a fractal of dimension , say, 1.5 is that it is not quite like a line but not a surface either. Ditto for the case between 0 and 1. I don't see how the method would fail in the limit to give e.g the Cantor set a fractional dimension.

 

[jim mcnamara]

Hmm. I think we are on the fringe of Philosophy of Science. My perception. So let's move the thread to Discussion.