Welcome to Fractal Forums

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 (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.

Hello and welcome to the forums   O0

I'm afraid I have to reject your definition.  It is too vague.

How does one define a "Thing" in this context?

Example : The Mandelbrot Set is a continuous function and therefore a single thing.

It does not have any other "things" in it, so it does not meet your definition - at least according to the commonly accepted definition of the word "Thing"

It is a start, but it need a lot more rigor to be accepted.

Have fun trying though!

And just so you don't get discouraged, I'm our resident skeptic.  I flatly refuse to admit to the existence of fractals in nature at all!!  I maintain they are purely Mathematical Constructs with no corollary in reality whatsoever.

Many things use fractal geometry to form their shapes, but that does not make them Fractals in my view...

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 (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/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 (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 (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 not big on following outside links to reinforce an argument.

Just summarize what you have to say here, or post the image you reference here.

I'm in and out of many sites, half of which I help run, so while I'll stop and chat, I wont bother with outside links.

That said, I think now your definition is too broad  (a consequence of being too vague?).

If that number sequence is a fractal, then what stops a simple straight line from being fractal?

It contains many more line segments than lines, right?  And they come in many lengths, thus it is self similar, right?

And if a straight line is a fractal, then what is there that is *not* a fractal?

I like this definition least of any definition I have seen so far.

It's a good start, but it needs a whole lot more work to be useful!

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.

Thanks for your tips!

>That said, I think now your definition is too broad  (a consequence of being too vague?).
Yes, my definition is a bit broad, but not too broad, since it is governed by head/tail breaks classification scheme. Given a data with a heavy tailed distribution, those values above the mean or average are called the head, and those values below the mean are called the tail. Interestingly, this head/tail breaks process can continue recursively for the head. It means that the head is sub-whole, which can be further divided into the head and the tail. Eventually how many iterations this head/tail breaks can go indicate the scaling hierarchy of far more small things than large ones, or scaling hierarchy of numerous smallest, a very few largest, and some in between the smallest and the largest.

>If that number sequence is a fractal, then what stops a simple straight line from being fractal?
That number sequence contains far more small numbers than large ones, while a simple straight line does NOT contain far more small things than large ones, no matter what perspective one takes.

>It contains many more line segments than lines, right?  And they come in many lengths, thus it is self similar, right?

Note that the notion of far more small things than large ones must recur at least twice to be quantified for being fractal. You are right that "it contains many more line segments than lines", BUT not recursively. In other words, the notion of far more small things than large ones needs to occur again and again.  

>And if a straight line is a fractal, then what is there that is *not* a fractal?
>I like this definition least of any definition I have seen so far.
>It's a good start, but it needs a whole lot more work to be useful!
Many thanks for your discussion!

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?

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

A line is a straight line I presumed, but a curved line does contain far more small bends than large ones. Note that a bend is defined recursively by three vertices, and small bends are embedded recursively in large bends. Sorry I have to refer you to slide 26 of this presentation https://www.researchgate.net/publication/317615403_Why_Should_Spatial_Heterogeneity_Be_Formulated_as_a_Scaling_Law (https://www.researchgate.net/publication/317615403_Why_Should_Spatial_Heterogeneity_Be_Formulated_as_a_Scaling_Law)

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

You are right that a straight line with many vertices in it may contain many line segments than the whole line, BUT according to my definition, the notion of far more small things than large ones recurs at least twice with ht-index being at least three.

> 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.
See the immediate above comment, the scaling or notion of far more small things than large ones must recur at least twice. So a straight line does not meet my definition.

> In what way are line segments and points not things?
You consider line segments and points to be things, BUT they are more or less similar, instead of far more small things than large ones.

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.

>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.

Sorry I have some difficult to fully understand your point here. Could you help clarify? I will answer your the last question on "I really don't see any reasons why this definition is not too broad to be practical."

To honest, I do not think it is too broad, but I must admit that the third definition is relaxed, just as Mandelbrot relaxed the first definition. For convenience, I would have to insert a recent presentation, in order to put my answer into a context https://www.researchgate.net/publication/317615403_Why_Should_Spatial_Heterogeneity_Be_Formulated_as_a_Scaling_Law (https://www.researchgate.net/publication/317615403_Why_Should_Spatial_Heterogeneity_Be_Formulated_as_a_Scaling_Law) 

The third definition has been used practically for various mappings such as classification, map generalization, cognitive mapping, and perception of beauty; see some specific examples shown in this presentation: https://www.researchgate.net/publication/280732530_Scaling_As_a_Design_Principle_for_Cartography_in_the_Era_of_BIG_Data (https://www.researchgate.net/publication/280732530_Scaling_As_a_Design_Principle_for_Cartography_in_the_Era_of_BIG_Data)

Sorry I have to insert some links to put my discussion in a context.

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:
(http://fractalfoundation.org/wp-content/uploads/2010/05/kochprog440.jpg)
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!

Thanks for bringing the discussion to a higher level!

I believe if something is power law, then it is fractal for sure. My contribution to the literature or the third definition of fractal has moved away from this power law-based fractal, towards a more relaxed definition. A set or pattern is fractal if there are far more small thing than large ones, or more precisely, the scaling or notion of far more small things than large ones recurs twice with ht-index being at least 3. See slides 25 and 26 and those nearby https://www.researchgate.net/publication/317615403_Why_Should_Spatial_Heterogeneity_Be_Formulated_as_a_Scaling_Law (https://www.researchgate.net/publication/317615403_Why_Should_Spatial_Heterogeneity_Be_Formulated_as_a_Scaling_Law)

The first definition of fractal is strictly power law, and all points are within the power law trend line. The second definition of fractal is still power law based, and all points are just around the power law trend line, rather than exactly on the trend line. I think the first two definitions are a bit too restrict, since they need power law fit either strictly or statistically.

Under the third definition, no geographic features are not fractal, given a right scope and perspective. For example, a street network perceived from street segments and junctions perspective is not fractal, but it is fractal seen from individual streets, since there are far more less-connected than well-connected; see the figure in this post https://www.researchgate.net/publication/316845391_Why_Topology_Matters_in_Spatial_Cognition_and_Analysis (https://www.researchgate.net/publication/316845391_Why_Topology_Matters_in_Spatial_Cognition_and_Analysis)

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

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

I suggest you use a site like https://de.imgbb.com/ (https://de.imgbb.com/)
there choose bb-code and paste into your post.
or upload elsewhere and paste the link into the image-function in red here:
(https://image.ibb.co/fbseAF/attach.jpg) (https://imgbb.com/)
upload image (https://de.imgbb.com/)
alternatively you can use the attachment function. it uploads the image when you actually post the text. then you rightclick and get the image-link, edit your post and put it into the img-brackets. this one is the most inconvenient, but then the image is hosted here at fractalforums. I'm looking for a better solution for the planned big update of fractalforums



I used the road example as example for a power law.
I see no contradiction here - except that you only watch a single iteration, and of course without context it is not fractal, as my example with the koch curve..
It really seems to me that if you just say "there are more small things than large things" it is a bit too general and won't be accepted in the mainstream.
even though it's very close to the truth. the problem with fractal research is, that the opinions are just as fractured. everyone defines it differently. and we would need something that at least a large majority can agree upon.
(and no, sockratease, you will probably never be part of this majority. And I'll happily ignore that! ;D)

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 (http://www.fractalforums.com/index.php?action=gallery;su=user;sa=userlist) for future use.

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

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?


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..

> 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

@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.
 
(http://www.fractalforums.com/index.php?action-gallery;sa=view;id=20485)

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.

(https://ibb.co/maGZUa)

@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.

(https://ibb.co/maGZUa)

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

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


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