Is everything that has a fractal dimension automatically a fractal?

[Chillheimer]

just the short question in the title.
opinions? explanations?
thx!

[Khashishi]

Yes. If the fractal dimension is greater than the topological dimension, then it's a fractal. Since the topological dimension is always an integer and less than or equal to the fractal dimension, a non-integer fractal dimension is always fractal. But it is possible to have integer dimension fractals, such as space filling curves.

[FractalStefan]

May I add another question?

Do all fractals have self-similarity?

[PieMan597]

It depends on the definition of self-similar.

The Mandelbrot set, for instance, is a fractal, but it is not perfectly self-similar. Also, fractals in nature can't be perfectly self-similar because there is a resolution they can't go higher than.

Even though not all fractals are perfectly self-similar, some are, with an example of a completely self-similar fractal being the Sierpinski Gasket.

So to answer your question, not all fractals are perfectly self-similar, but they all do exhibit some degree of self-similarity.

[rsidwell]

There is an interesting presentation on this subject on youtube: "Fractals are typically not self-similar"

<a href="https://www.youtube.com/v/gB9n2gHsHN4&rel=1&fs=1&hd=1" target="_blank">https://www.youtube.com/v/gB9n2gHsHN4&rel=1&fs=1&hd=1</a>

[hgjf2]

With "UK coastline measuring" method , was been calculated the length of Mammoth Cave and of Odessa's catacombs.
The hidden fractal dimension is proof that catacombs of Odessa was reached at the performance of 2500kilometers length rendered

[Chillheimer]

was about to post that, rsidwell - thank you!

great to see a lot of newbie-activity recently!

hgjf, could you elaborate on how the catacombs of Odessa have a fractal dimension or how it was used to estimate the total lenght? any sources? I couldn't find any..

[FractalStefan]

Thanks PieMan597 and rsidwell, this clarified a lot.