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Author Topic: Third definition of fractal - a relaxed definition  (Read 1512 times)
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binjiang
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« on: July 10, 2017, 10:49:25 PM »

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.




* Figure0Y.jpg (21.81 KB, 657x264 - viewed 212 times.)
« Last Edit: July 18, 2017, 11:25:58 PM by binjiang, Reason: fine-tone » Logged
Sockratease
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« Reply #1 on: July 10, 2017, 11:05:00 PM »

Hi, greetings from me - a new comer!
I re-defined fractal as a set or pattern in which there are far more small things than large ones
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   afro

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...
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binjiang
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« Reply #2 on: July 10, 2017, 11:14:59 PM »

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.
« Last Edit: July 10, 2017, 11:54:06 PM by binjiang, Reason: Sorry adding one more point on Mandelbrot's set » Logged
Sockratease
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« Reply #3 on: July 10, 2017, 11:49:29 PM »

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

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!
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« Reply #4 on: July 10, 2017, 11:57:29 PM »

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.
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binjiang
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« Reply #5 on: July 11, 2017, 12:10:34 AM »

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!
« Last Edit: July 11, 2017, 12:45:20 AM by binjiang, Reason: A small typo » Logged
Sockratease
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« Reply #6 on: July 11, 2017, 10:09:24 AM »

>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.

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?
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binjiang
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« Reply #7 on: July 11, 2017, 10:47:08 AM »

>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

>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.
« Last Edit: July 11, 2017, 11:25:25 AM by binjiang, Reason: Typo » Logged
Sockratease
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« Reply #8 on: July 11, 2017, 11:34:30 AM »

>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

>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.
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binjiang
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« Reply #9 on: July 11, 2017, 11:53:20 AM »

>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 

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

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



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« Reply #10 on: July 11, 2017, 12:06:10 PM »

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.

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.

@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!
« Last Edit: July 11, 2017, 12:18:49 PM by Chillheimer » Logged

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binjiang
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« Reply #11 on: July 11, 2017, 01:28:03 PM »

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

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
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binjiang
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« Reply #12 on: July 11, 2017, 01:37:10 PM »

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

Is it possible to insert a figure from my computer? Thanks!
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« Reply #13 on: July 11, 2017, 02:04:39 PM »

I suggest you use a site like 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:

upload image
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



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
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! grin)
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« Reply #14 on: July 11, 2017, 02:51:34 PM »

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

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.
« Last Edit: July 15, 2017, 09:42:10 AM by 0Encrypted0 » Logged
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