> I still don't see the difference to just saying fractals always follow a power law.

The first two definitions of fractal require power law, either strictly (the first) or statistically (the second). The third definition has relaxed this requirement, so there is no need for a fractal to follow a power law. It requires only the scaling of far more small things than large ones recurs at least twice, or equivalently with ht-index being 3 (see below for my explanation on ht-index).

> Isn't that saying exactly the same? Basically what you describe is a power law. And the linked wikipedia article you linked kind of says the same thing.

>Correct me if I'm wrong-my head is filled with a lot of stuff today.

The third definition is inclusive, so what are considered to be fractal under the first two definitions are still fractal under the third definition. However, it does not hold true reversely; for example, a highway was not fractal under the second definition, but it is under the third definition.

> Is your definition "even more relaxed" and includes other things than power laws?

Yes, exactly! my definition extends to other heavy tailed distributions such as lognormal, exponential, and even right-skewed normal distribution, as long as ht-index = or > 3.

> to keep me in this discussion, would you mind explaining the ht-index? I'm very busy this week and so little time for researching myself..

Ht-index indicates the number of times the scaling of far more small things than large ones occurs plus one. For example, given the 100 numbers of 1, 1/2, 1/3, ... and 1/100, the first mean of these 100 numbers is about 0.052, which puts the 100 numbers into two parts: those above the first mean called the head - the first 19, and those below the first mean called the tail - the remaining 81; far more small numbers than large ones. For the head or the first 19, the average or mean is 0.19, which put the first 19 numbers into two parts: those above the second mean called the head - the first 5, and those below the second mean called the tail - the remaining 14; again far more small numbers than large ones. For the first 5, the mean is 0.46, which put the first 5 into two parts: those above the third mean called the head - the first 2, and those below the third mean called the tail; again far more small numbers than large ones.

Seen from the above recursive partition process, the notion of far more small numbers than large ones recurs three times, so the ht-index = 3+1 = 4, meaning 4 hierarchical levels for the 100 numbers.

To put the discussion into a context, herewith the first paper that develops the ht-index idea:

https://www.researchgate.net/publication/236627484_Ht-Index_for_Quantifying_the_Fractal_or_Scaling_Structure_of_Geographic_Features