Third definition of fractal - a relaxed definition

[binjiang]

Thanks for your advice!

My definition of fractal or the so called third definition of fractal is recursive in nature: A set or pattern is fractal if there are far more small things than large ones, or to be more precise the scaling of far more small things than large ones recurs at least twice with ht-index being at least three. This definition was inspired by head/tail breaks which is recursive:  
Recursive function Head/tail Breaks:
    Break the input data (around mean or average) into the head and the tail;  
    // the head for data values greater the mean
    // the tail for data values less the mean
    while (head <= 40%):
        Head/tail Breaks(head);
End Function

https://en.wikipedia.org/wiki/Head/tail_Breaks

[Chillheimer]

I still don't see the difference to just saying fractals always follow a power law.
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.
Is your definition "even more relaxed" and includes other things than power laws?

the scaling of far more small things than large ones recurs at least twice with ht-index being at least three.
https://en.wikipedia.org/wiki/Head/tail_Breaks

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..

[binjiang]

> 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

[binjiang]

@0Encrypted0 Many thanks for the tip!

I tried to add one picture by modifying this original question, but failed. Let me try to see if I am able to add one picture in this test post.

This picture illustrates the third definition of fractal using a cartographic curve consisting of 8 segments or 9 vertices. If one sees this curve as a collection of line segments or vertices, then this is an Euclidean geometric perspective. Instead, if one sees the curve as a collection of 7 bends with far more small bends than large ones, then this is a fractal geometric perspective.
 


Sorry it appears not working. However, in a browser, I can see this picture via the URL http://www.fractalforums.com/index.php?action=gallery;sa=view;id=20485. Any idea what is wrong I made? Thanks!

Let me know try again with another method here.

[binjiang]

@Chillheimer
I have a picture uploaded to this site you suggested: https://ibb.co/maGZUa

Now let me try to insert it into this post. I failed previously, so I would like to try it again.

[0Encrypted0]

Sorry it appears not working. However, in a browser, I can see this picture via the URL http://www.fractalforums.com/index.php?action=gallery;sa=view;id=20485. Any idea what is wrong I made? Thanks!

Above URL is for the web page.
Copy image address from context menu in Chrome or Copy Image Location in Firefox.


[url=https://ibb.co/maGZUa][img]https://preview.ibb.co/itEM9a/Figure0Y.jpg[/img][/url]



[img]http://nocache-nocookies.digitalgott.com/gallery/20/15173_12_07_17_12_53_25.jpeg[/img]