That was a nice read. I didn't know the contributions of Leibniz outside math :3
mathematical development of fractal geometry - a geometric figure that is self-similar at all scales
But a straight lines is also self-similar at all scales ;)
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I see what you're trying to do comrade 😜.
Yes it is but it's hardly a fractal, technically speaking it's fractal dimension is the same as it's topological dimension - they have to be different for it to be a fractal. The self-similar definition is mainly used for qualitative understanding/description.
Self-similar is more of a property. But then again not all fractals are self-similar. Closest to being correct and easy to understand is saying that fractals are typically self-similar.
Ok boss. I'm not well grounded in this fractal concept, so i may not be familiar with some of the things you just stated. However, are you saying conceptually speaking, there's some difficulty generalizing what a fractal actually is ?
There is difficulty in defining in what a fractal is because there are many fractal dimensions (boxcounting, correlation etc.). 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?
Quite interesting.
🧐 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.
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
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.