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Hi, greetings from me - a new comer!
I re-defined fractal as a set or pattern in which the scaling of far more small things than large ones recurs at least twice or with ht-index being at least three.

The following attached 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. The notion of far more small bends than large ones recurs twice: x1+x2+x3>x4+x5+x6+x7, and x1>x2+x3.

https://www.researchgate.net/publication/309428627_The_third_definition_of_fractal
The curve shown in slide 8 (of the above presentation) contains 7 recursively defined bends. It is not fractal, using traditional definition, since the number of bends is too small (=7) to meet a power law. Instead, there are far more small bends than large ones, or the scaling of far more small bends than large ones recurs twice.
I welcome your comments and criticisms on the new definition.

Many thanks for your reply!

As for things, they are coherent geometric entities or simply sets. For example, bends are defined by three vertices as shown in slide 8 of the following presentation
https://www.researchgate.net/publication/309428627_The_third_definition_of_fractal

Let's take another example. Given 100 numbers, the first is 1, the second is 1/2, and the third is 1/3,... and the last is 1/100. This set of 100 numbers is fractal, since there are far more small numbers than large ones, or alternatively the notion of far more small things than large ones recurs three times, with ht-index being 4.

I have previously demonstrated that all geographic features are fractal under the new definition, given a right scope and perspective:
https://www.researchgate.net/publication/270634544_Headtail_Breaks_for_Visualization_of_City_Structure_and_Dynamics
https://www.researchgate.net/publication/236627484_Ht-Index_for_Quantifying_the_Fractal_or_Scaling_Structure_of_Geographic_Features

Back to your concern on Mandelbrot's set, there are far more small things (coherent entities) than large ones. Importantly, small things are embedded recursively in large things; see Figure 2 of this paper https://www.researchgate.net/publication/270634544_Headtail_Breaks_for_Visualization_of_City_Structure_and_Dynamics

Any comments and criticisms in particular are more than welcome.

I'm on mobile and it's past midnight here, but I wanted to chime in to say that I find this very interesting, will check and comment as soon as I find appropriate time.

In what way does a line NOT contain far more small things than large ones, no matter what perspective one takes?

I say that it contains many more line segments than whole lines!

I also say that each line segment contains many more points than line segments.

So a straight line does, indeed, meet your definition of a fractal.

In what way are line segments and points not things?

Even if I give you the point that a line does not meet your definition  (which I don't believe)  (consider Cantor Dust - then just keep the line continuous and only "count" the parts that the function breaks the line into)  - then what about a triangle?

A Sierpinksi Triangle?

Or a circle, sphere. or, octahedron?

A circle breaks down into many smaller arcs, each of which has a different orientation in space, and that surely meets your definition even more than a line does.

I don't think we'll ever agree on this one, but I rarely agree on these sorts of things with anyone.  The best I can do is hope to see what you are saying, and in this case I really don't see any reasons why this definition is not too broad to be practical.

Hello & welcome to the forum Bin Jiang!
What a nice coincidence - I am working on an episode for a new youtubechannel about fractals. The topic is power laws and we are investigating the strong connection with fractals.
I'm not sure if your definition is actually "defining enough" but I share your viewpoint.


@Sockratease: Nothing stops a single line to be fractal. It depends what you do with it. A simple line is just that. No iteration, nothing is happening.
You need to put points on it to give it "more meaning". You DO something with it. Be it a boring Add point A and point B. Or adding scale invariance through a powerlaw. Then the distribution of points undeniably has fractal characteristics - no matter if they are on a straight line or not.

now you will come again with "but it's not infinite".
that depends on iteration. if you iterate infinite times, it will be infinite. if you iterate just x times, as Bin Jiang suggests in slide 8, of course it will not be infinite, because your RESOLUTION is limited by ITERATION.
Take the second iteration of the koch curve here:

you say: this single image is not a fractal.
but the fractalness it is not about that still frame. the fractalness is emergent from the recursive process.
I ask you: when does this koch-curve become fractal?
at what iteration? the 3rd? the last one? infinite?

we go out in nature and have a look at still frames like a tree, growing very slowly.
you say it is not a fractal. but thats just the same thing as watching the 2nd iteration of the rkoch curve and not recognizing the fractal principle that created it.
you have to watch the RECURSIVE PROCESS that is responsible for fractals.

In natural fractals, TIME is the eqiuvalent to mathematiocal Recursion and Iteration.
That's one of the core principles of my personal fractal theory.

What your big mistake is Sockratease , you leave that core principle out of the whole picture. Time/Iteration!
Without it, NOTHING is fractal. Just as you say.


so to bring this together:
I am getting more and more confident that we should include powerlaws into the definition.
I think that this is what Bin Jiangs attempt for a relaxed definition is basically saying.

The question that arises is:
1. Do all fractals follow power laws?
I'd say yes
2. Are actually all power-laws fractals?
I'd say yes again.
3. Are these two concepts basically the same "thing"? what is the difference?
I'm not sure yet.

Opinions?

In this thread I came up with a similar definition of fractals in nature (self similar over at least 3 orders of magnitude:
http://www.fractalforums.com/fractals-in-nature/are-powerlaws-fractals/



Nice discussion Bin Jiang!

@Chillheimer, I am curious how you could insert figure into a post.

Is it possible to insert a figure from my computer? Thanks!

Maybe a Stash/Scraps folder in each member's User Gallery to upload images for linking to threads?

I have just been leaving attached images hanging at the bottom of posts.
I will use your method to insert the image into the body of the post.

Edit: Created an Attachments category in my User Galleries for future use.