RE: The Unsung Heroes Of Modern Science - Gottfried Wilhelm Leibniz

[clinton19]

There is difficulty in defining in what a fractal is because there are many fractal dimensions (boxcounting, correlation etc.).

Quite interesting.

Is something still a fractal when there exists a fractal dimension which is equal to topological dimension and a fractal dimension which is not equal to topological dimension?

🧐 Why does this sound contradictory. Or are you talking about an object with 2 topological dimensions and above?
Probably in the first dimension of, say a 2D object, there's self-similarity but it's topological dimension is not the same as the fractal dimension and in the second dimension there's also self-similarity but the topological dimension is the same as the fractal dimension. Or is there something I'm missing ?

If it's for the 2D object, then i don't know. Plus i would love to see a visual representation of this kind of pseudo-fractal - if that's the right word to use.

Comments

[mathowl]

So we can just take the real line and select the rationals over that. Then the topological dimension is 0, hausdorf dimension is also 0, but the box-counting dimension is 1.

For adding coordinates you can of course extend this argument to n-dimensions. :3

[clinton19]

So you are saying that supposing we had a set S with points/elements (rational numbers) such as {1.1, 1.2, 1.3, ...} - gotten from a real number line, we could pick any point, say 1.1 and scale/magnify it down by 0.1, and correspondingly we have points which is also a set, given below as

X = { 0.11, 0.22, 0.33,...., 1.1}

X has cardinality of 10

Which means that every point in S when scaled/magnified by a non-zero factor gives a new set of points ( e.g X for 1.1), and every point in this new set also gives it's own corresponding new set when also magnified, and so those the process continues to ad infinitum. The self-similarity here is that points are contained in a point when magnified. However, a single point has a topological dimension of zero which is equivalent to the point's fractal dimension but the fractal dimension (box-counting dimension) of any magnified point, say 1.1 in S above is given by

D = log(N)/log(1/e)

D = fractal/box-counting dimension
N = number of points in the magnified point (1.1)
e = scale/magnification factor.

In our scenario as seen above

N = 10 (cardinality of set x)
e = 0.1
Therefore D = 1

This seems to me like a fractal.

[mathowl]

It works more general for all the rationals. I think it is true that any subset of the rationals gives a box counting dimension \leq 1

The main point is that the Hausdorf dimension is zero. Because it is countable. Thefore, the Hausdorf dimension says it is not a fractal whereas the Boxcounting dimensions says it is a fractal.

[clinton19]

It works more general for all the rationals.

Take it easy comrade, I'm aware

I think it is true that any subset of the rationals gives a box counting dimension \leq 1

It (subset of rationals) actually does.

The main point is that the Hausdorf dimension is zero. Because it is countable. Thefore, the Hausdorf dimension says it is not a fractal whereas the Boxcounting dimensions says it is a fractal.

I think I get where you are coming from.

It's nice learning from you anyways, thanks.

[mathowl]

I am happy to share :3

I think it is true that any subset of the rationals gives a box counting dimension \leq 1

Ah yes, this is obviously true :P

An interesting exercise would be to construct a non-finite subset of the rationals with box counting dimension equal to zero :3

[clinton19]

An interesting exercise would be to construct a non-finite subset of the rationals with box counting dimension equal to zero :3

Ok, good luck with that.