Search for an agreeable definition of fractals in nature

[Sockratease]

...
And the 'inventor' of fractals, Benoit Mandelbrot himself would disagree as well. And all those countless studies that proved fractals..

Please correct me if I'm wrong, but Mssr. Mandelbrot only ever used the phrase Fractal Geometry In Nature and never claimed that Fractals themselves existed in nature.

It's been a very long time since I read "The Fractal Geometry of Nature" but as I recall he was careful to use the full phrase "Fractal Geometry" and never actually called anything in Nature "A Fractal"

Even I have admitted that Fractal Geometry and Iterative Processes are everywhere in nature - I just draw the line at calling these things Fractals.  Nit-Picking at subtleties?

Maybe.

But that's what Scientists do!

Please take a look at romanesco broccoli again:
<Quoted Image Removed>
and then tell me again that the base for the construction plan of it is not fractal.
or, as an alternative, tell us what word you would use to describe this.
you probably just use the word fractal differently than 'we' do. (even different to the man who came up with the word fractal)
only because you can't see it or don't have the means yet to examine it doesn't mean it isn't there.
500 years ago people couldn't see the earth is round. so it actually was a flat plate and suddenly turned into a sphere?

the question is who gets to decide what "true" is?
in a way you are turning around my point from the last post, that all perfect squares(not sure if i translate this correctly, "quadrat" in german) are always squares but not vice versa.  
when you say only 'true fractals are fractals', you could also say only a quadrat is a square. if it hasn't 90° in all corners and the same lenght for each side it can't be a square.
that's perfectly fine and welcome! I guess we need this sort of input (even if it's clearly wrong   ) to improve our reasoning, to question the own beliefs..
I'm not sure if anybody really has.
And if someone (preferrably the inventor of the word fractal, Mandelbrot himself) has, I'd love to see a quote including the source.
(And if that exists, I'd disagree and say that this is a mistake we have to correct.)

If you watch this from a purely mathematical standpoint, Sockratease is right.
But without the intend to insult, to say that all those smart guys like Mandelbrot and other scientists that prove fractal structures in nature are simply wrong - that seems to be very arrogant..

OK, take that broccoli, and zoom into it with a microscope.  Look right in between two of the smallest nubs you can find.

It becomes smooth, and there is no new structures that emerge which have any self-similarity to the large scale version.  It has finer details like cells and stuff, but there is no longer any self similarity at multiple scales.

The cells don't even spiral, all you get is "Normal" plant biology.  I see that as a great example of Nature making use of Fractal Geometry, but I don't see a Fractal.

In Chemistry we talk about things like The Ideal Gas Law.  It describes the behaviour of gasses and provides fantastically close estimates of real world behaviour!

But we fully recognize the differences between the Ideal State and Reality.  It is stressed that Ideal Conditions do not exist.

I see an analogy here - Your broccoli is close as possible to get, but being a real world physical object, it can only approximate a Fractal while never actually being one.

And again - I don't recall Benoit Mandelbrot ever taking the position that such things were anything other than nature making use of Fractal Geometry - I think he knew that idealized things simply don't exist and phrased himself to reflect that.

So in fact - I am agreeing with Benny, not calling him wrong!

-----kidnapping 'my own' thread-----
I just read this again after recently watching a video by Nassim Haramein, a strange guy, not accepted in the scientific world who has a nice theory regarding this issue, resolving the problem without this 'fake-correction'
<a href="http://www.youtube.com/v/TW1gl6_QK-M&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/TW1gl6_QK-M&rel=1&fs=1&hd=1</a>
The essence is that empty space/vacuum has "incredible density", which can only mean that we live inside a black hole. (he goes on that in each proton(?) there's also a miniblackhole)...

Too bad he also has to talk about ancient egyptians being helped by extraterrestrials to build the pyramids, which makes taking this guy serious even harder.

I actually loved Nassim Herriman's stuff!

I agree he takes it too far with the Mysticism, but he makes some valid arguments about the short comings of The Standard Model of Physics.

I just, as always, disagree that his notions of infinite regression are possible in a Universe like ours.  Even in a Multiverse!  I believe all things have a beginning, a middle, and an end.  The whole concept of infinity is fine for mathmatics, yet I still remain unconvinced it has any place in reality.

But his use of Geometry to explain things is where science began, and I'm glad to see somebody returning to it as a means of exploring things.

[kram1032]

In a Romanesco, there is self similarity at 3-4 scales or something like that. Zooming in beyond those scales doesn't change that they are there. You can't stop "details at multiple scales" from existing just by zooming past them. All you do is making them invisible in your frame of reference. That's like saying the sun stops existing at night.

And if you want geometry to describe the universe, just translate all the laws into a Geometric Algebraic form and you have all the geometric details right in front of you, as clear as they can possibly get. - in the very most abstract fields of research and fundamental theory, that's exactly what's done anyway. They won't always use Geometric Algebra, but they will try to extract exactly that same geometric meaning out of what ever framework they happen to use for a given task. The details still are in there, no matter what framework you use. It's just that some are clearer for some things than others. And Geometric Algebra happens to be one that is particularly great at revealing geometric details.
The other great concern people have is the other side of topology: Homotopy. And that is so fundamental that there currently is a sizeable movement trying to rewrite the entirety of mathematics in Homotopy Type Theory which does to Set theory roughly what Geometric Algebra did to Vectors in that it reveals the underlying structures much better.
In case of HoTT, in fact, the revealed structures become so clear that a computer can handle them. Or at least, that's the goal: The entirety of mathematics. Auto-Computable. And by extension, all of physics too.

[Sockratease]

In a Romanesco, there is self similarity at 3-4 scales or something like that. Zooming in beyond those scales doesn't change that they are there. You can't stop "details at multiple scales" from existing just by zooming past them. All you do is making them invisible in your frame of reference. That's like saying the sun stops existing at night.

You misunderstand my point.

I did not mean to give the impression that "details at multiple scales" don't exist at all in that example - I even made clear that they are a fine example of that!

My point is that a mere "3-4 scales or something like that" is grossly insufficient to qualify as "A Fractal" - it's just an example of Nature using Fractal Geometry and Iterative Processes to mimic a Fractal.

We agree more than you seem to think.  I just think that limiting your definition of the range of scales for self-similarity is changing the definition to force your examples to meet the criteria you want  (cheating).

[kram1032]

Nah I understood what you said. Really, I'm just nit-picking back:

there is no longer any self similarity at multiple scales.

This is a little unclear language and I (very consciously) chose to interpret it as you claiming that zooming past the self-similar parts makes those parts go away, rather than what you meant; that  there is no additional self-similar detail beyond the previously observed zoom.

I'm fine with you calling what's in nature "Fractal Geometry" rather than outright "Fractals". - That seems like a personal choice if anything. Though when people in this topic say "Fractal", just keep in mind that they mean "Fractal Geometry as found in nature" and go from there. What would be your definition for that?

[Chillheimer]

OK, take that broccoli, and zoom into it with a microscope.  Look right in between two of the smallest nubs you can find.
It becomes smooth, and there is no new structures that emerge which have any self-similarity to the large scale version.  It has finer details like cells and stuff, but there is no longer any self similarity at multiple scales.

So for you scale-invariance is an absolute must-have for a (theoretical) fractal. Understood.
But I'm all about fractals in nature. Theory is boring, real life rules!

We agree more than you seem to think.  I just think that limiting your definition of the range of scales for self-similarity is changing the definition to force your examples to meet the criteria you want  (cheating).

As I expected this is more about the use of a word than the topic itself..
I understand and respect your viewpoint, but I think differently. I'd too say that what you do is "nitpicking" (me learned a new word ) and misses the core of what I intended this thread to find out together.

To quote Wikipedia:
There is some disagreement amongst authorities about how the concept of a fractal should be formally defined. Mandelbrot himself summarized it as "beautiful, damn hard, increasingly useful. That's fractals."[11] The general consensus is that theoretical fractals are infinitely self-similar, iterated, and detailed mathematical constructs having fractal dimensions, of which many examples have been formulated and studied in great depth.

So you are right for theoretical fractals, and this is obviously consense. But we are talking about natural fractals, that really exist in our cosmos.

To solve this problem I suggest that for further discussion (in this thread, and maybe also in other discussions where this difference is an issue) we start naming theoretical fractals what you are talking about sockratease, and natural fractals, which this thread is about. (and the whole cosmos..  )

I can't recall where exactly I found it but I think I saw somebody actually debunking that guy.

yeah, me to, don't remember either - but the problem there was, it was also mostly about the true 'bullshit' like the pyramids and not about the parts that really seem to make sense.
I'd love to see the point with the energy of space itself that needs 'correction' (aka cheating) to work with our current model debunked in a clear way.

But back on topic:
Would you guys agree that 1. at the absolut basic 'element' of fractals (natural as well as theoretical) always is recursion?
and 2. every other telltale-sign like self similarity or scale-invariance is an optional result of recursion?

[kram1032]

To state it in mathematical terms:
Recursion of some way shape or form might be necessary, but it certainly isn't sufficient.
I can't think of a theoretical fractal off the top of my head that isn't constructed by some form of recursion.
And in case of naturally occurring fractal geometry, all that is built off physical laws which essentially can be thought of as a few simple laws (merely 4 in fact, as far as our current models go) being applied over and over again, starting from some initial configuration (what ever that first particle or infinitely condensed *something* was, that was before the big bang) and that caused all of reality. So in that sense, it's obvious that recursion is a requirement here.

In fact, since all our thoughts happen in us and we are the ones who built machines that come a LOT closer to "true" visual representations of fractals (within technological limitations of not having infinite time to compute stuff) and both us and those machines are made up from stardust and those machines basically work by continuously manipulating and measuring a physical system which once again is bound to work via recursion, in the end even all the examples of theoretical fractals are somehow bound to be built from the recursion of physical laws. But that's a meta concept of course. And I guess that's at the heart of Sockratease's gripe with real-life-infinities.
Sooo basically I just turned the topic around once again, lol. Sorry, so easy to go that path since it's barely even a different topic at all.

Anyway, so yeah: I do think it's a pretty save bet that all fractal systems need some form of recursion to arise. I'm not 100% convinced but as said I can't think of a counter-example.
But what I can definitely say is that recursive processes do not necessarily create fractal structures (in the slightly stricter sense of excluding smooth structures) at all times. It could hardly be more trivial than to just iterate the identity function over and over:
f(x)=x<-f(x)=x<-f(x)=x<-...
which clearly isn't fractal in that stricter sense. Though it is scale-free, since you get more of the same (perfect, flat, smooth, boring) no matter what scale you look at it with.

[Kalles Fraktaler]

self-similarity isn't quite enough: The Mandelbrotset isn't strictly selfsimilar. (Its Juliasets, however, are)
The minibrots you'll find as you zoom in become more and more distorted.

I am sorry I haven't read this interesting thread through yet, but I think this statement is wrong.
The minibrots are not distorted, they are strict self similar if you strip off the patterns accumulated from previous minibrots. Which is showed by this movie I made a while ago:
<a href="http://www.youtube.com/v/CcaW94VI56E&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/CcaW94VI56E&rel=1&fs=1&hd=1</a>

[Sockratease]

...But back on topic:
Would you guys agree that 1. at the absolute basic 'element' of fractals (natural as well as theoretical) always is recursion?
and 2. every other telltale-sign like self similarity or scale-invariance is an optional result of recursion?

OK, in the interest of mutual agreement, I'll state where we do agree completely.

First, you can call it whatever you want.  In this wacky world we live in, there will always be people who will take issue with any given definition of anything  (Artists still argue over whether Black and White are even Colours!).

So I'll yield that your broccoli is a fractal in the same way that an exported 3D Model of a Mandelbulb is a fractal  (which is passing little for me, but much more for you!).  In fact, I have been known to state that images in our, or even any, gallery are not Fractals - just images made with a Fractal Process.  Frcatals are - to me - an Abstract Thing which uses many principles that are useful for making things like broccoli, Pretty Pictures, 3D Models, and maybe even Universes!

But ultimately I tend to agree, at least partially, with the favorite student of my Namesake - Plato.  Remember his rant about how we don't really see anything real?  Ever.

He maintained that what we perceive is analogous to shadows on a cave wall cast from outside.  He felt that things were just projections into our perceptions of idealized versions of those things which exist outside of our ability to detect.

There is, in his view, an idealized Chair.  It exists outside our perception and every chair we ever see is just a shadow on a wall, cast by that idealized chair.

I don't agree fully with that, but it's principle applies here.  I feel all fractals are idealized things we can never see under any conditions.

We only see slices of representations of them.

So yes, I am saying no fractals exist anywhere, not even in images we post here!

But I agree fully that there is no denying the Fractallyness of many things.

Personally, I would not even bother trying to define a difference between theoretical fractals and natural ones because the natural ones are mere shadows of the real thing - which we can never experience.

See?

I can get all mystical too!   

[youhn]

... The minibrots are not distorted, they are strict self similar if you strip off the patterns accumulated from previous minibrots. ...

What about the deformed minibrots?



Clearly similar, but I wouldn't call this strict self similar.

[kram1032]

thanks for that example, youhn Yeah, that's what it means to be quasi-self-similar: If you were to superimpose the full M-Set on that Minibrot in as best a fit as you can achieve, you'll see the same shapes, the same content, but you won't be able to get an exact 1:1 match. And the deeper you zoom (in certain areas), the more those Minibrots will distort like this.
I wonder if anybody has ever done a zoom sequence highlighting how different those Minibrots can become versus all the usual videos you see, traditionally ending in a very symmetric Minibrot, similar in appearance to the full M-Set.

And thanks for that explanation of your views, Sockratease.

[youhn]

OK, in the interest of mutual agreement, I'll state where we do agree completely.
...
But ultimately I tend to agree, at least partially, with the favorite student of my Namesake - Plato.  Remember his rant about how we don't really see anything real?  Ever
...
So yes, I am saying no fractals exist anywhere, not even in images we post here!
...
See?

* youhn cherry picking

Yes, hence my avatar-subscript "shapes only exists in our heads". 

While I agree and think this is correct, it's not very usefull. We are in fact so used to our imaginary world-model, that we have learn to communicate within it. I think this is a small subset of the true universe, and therefore by definition part of reality. Only ... we call it virtual, imaginary, thoughworld, ficiton, etc. It's basically the same thing that we divide our living world. Us and them. Godlike creatures as we are versus all the beasts and animals. I think plastics, oils, gasoline, polished plates, computers and lasers are all natural things. In the same sense you could call a natural egg a chemical thing:


source: https://jameskennedymonash.files.wordpress.com/2014/01/ingredients-of-an-all-natural-egg1.pdf

Blurring the boundaries is great for things like openmindness, the everything-is-connected feeling and the biggest feeling of understanding. But back down-to-earth is makes things pretty vague and hard to talk about. Everything needs to be chopped up, divided into neat little packs, named, categorized and registered or remembered. All in order to make it thinkable with our little brains.

But we all like to train to expand it sometimes.

p.s. please do not eat to much eggs, since they contain carcinogens. 

 
https://jameskennedymonash.wordpress.com/2013/12/12/ingredients-of-an-all-natural-banana/

[billtavis]

So I leave for a day, and look at all the new responses! Most interesting was Sockratease taking up the age old question: is the mathematical ideal a crude approximation of the real world? Or is the real world a crude approximation of perfect mathematical ideals?

We can get clear images of Mandelbrot Zooms much smaller than the Planck Length - until we can do that in Nature...  we'll never have "True" fractals outside of mathematical constructs.

Your later explanation involving Plato was much more eloquent and one of the best interpretations I have heard when arguing for the latter of the two above questions. But I do feel it necessary to respond to this particular example.  The well-known Buddhabrot is just as much a fractal as the classic style of rendering the M-set... yet zooming in very far at all is "futile" wouldn't you say? Does this make it any less of a fractal? Of course not!  It's clearly a fractal because it's boundary is highly complex across various scales.
I would also like you to consider the case of Diffusion Limited Aggregation (DLA), one of my very favorite fractals.  There is no existing mathematical formula for this fractal. The only way to create it in the computer is to mimic the natural process in a strictly analogous way (diffusion of particles). So in this case, wouldn't the computer/theoretical model be a "shadow" of the real life process? Certainly zooming in on a real-life particle close enough would reveal new and different fractals, whereas zooming in on a computer particle would reveal it for the Euclidean ideal it really is (sphere/box/pixel/whatever).
That said, I think I might see what you are getting at - that "fractal" is a way of describing the form of objects, and not the object itself?

I just read this again after recently watching a video by Nassim Haramein

Looks interesting! This is certainly now on my list of movies to watch, thanks! As far as recursion goes, I feel that sometimes recursion is difficult to see in real life. Trees are obvious, but where is the recursion in a boulder?

Also, seeing you guys argue over the "self-similarity" of broccoli has once again spurred me to stick my neck out and say that this is why I simply can't get on board with using that as the defining definition! Broccoli has a boundary that exhibits complexity, from the scale of the whole object down to well below the cellular level. There are different modes of self-similarity along the way, but at no time is it ever 'not' a fractal, because complexity is always present!
Also, IMO the word "roughness" is not a suitable replacement for complexity, though it's close. For a theoretical example, consider the Lorenz attractor. It has no roughness at all, yet it exhibits complexity at multiple scale and so is a fractal anyway. A real life example would be blood vessels. From far enough away to see a circulatory system, the blood vessels are very smooth, yet the fractalness is obvious in the complexity of the branching. Zooming in close enough to lose the branching reveals a whole new level of complexity (on the previously-thought-to-be-smooth surface) that is unrelated to the branching, yet is certainly fractal as well.

Again, my revised definition of fractals is a boundary that is complex on a multitude of scales. I feel this covers the "everything is fractal" view of nature, as well as theoretical models.

[laser blaster]

I don't agree fully with that, but it's principle applies here.  I feel all fractals are idealized things we can never see under any conditions.

We only see slices of representations of them.

So yes, I am saying no fractals exist anywhere, not even in images we post here!

But I agree fully that there is no denying the Fractallyness of many things.

Looking at it that way, I guess you could say that trees and snowflakes are just nature's interpretation of fractals. But if I were to show someone a painter's interpretation of a tree, and ask "is this a tree? "they would almost surely say say yes. So I don't see any problem with referring to fractal-like patterns in nature as fractals, even if they fall short of the mathematical ideal.

[Chillheimer]

haha, first of all, I really love you guys! I had to chuckle quite often and yet this is one of the most inspiring discussions I've had in a long time..

To state it in mathematical terms:
Recursion of some way shape or form might be necessary, but it certainly isn't sufficient.
I can't think of a theoretical fractal off the top of my head that isn't constructed by some form of recursion.
...a few simple laws (merely 4 in fact, as far as our current models go) being applied over and over again, starting from some initial configuration  and that caused all of reality. So in that sense, it's obvious that recursion is a requirement here.

Pretty obvious, that's what I think too, everything is based on recursion.
What I don't see is, why this isn't sufficient to 'form a fractal'? What else is needed in your opinion?
(More further below)

Sooo basically I just turned the topic around once again, lol. Sorry, so easy to go that path since it's barely even a different topic at all.

No problem keep doing this, it's all interesting and tickles my mind.  

But what I can definitely say is that recursive processes do not necessarily create fractal structures (in the slightly stricter sense of excluding smooth structures) at all times. It could hardly be more trivial than to just iterate the identity function over and over:
f(x)=x<-f(x)=x<-f(x)=x<-...
which clearly isn't fractal in that stricter sense. Though it is scale-free, since you get more of the same (perfect, flat, smooth, boring) no matter what scale you look at it with.

And I say (sorry for repeating myself) that this is just a very special case of a fractal.
we have the m-set-formula for 2d, we have mandelbulb formulas for 3d, (maybe other formulas involving time for 4d), I would see the formula of your example as one for 1d..

The minibrots are not distorted, they are strict self similar if you strip off the patterns accumulated from previous minibrots.

I totally agree. I've zoomed deep into the mset for the last year now, with at least 50 very deep zooms, and I've only encountered these deformed minibrots at low magnifications surrounding the mainbrot.

What about the deformed minibrots?
<Quoted Image Removed>
Clearly similar, but I wouldn't call this strict self similar.

all deeeep minibrots I've visited were pretty much perfect. ( I have to admit, I didn't actively search for deformed ones near the perfect deep-brots, might be there as well

In this wacky world we live in, there will always be people who will take issue with any given definition of anything

I'd call it wacky to follow your path of thinking. (No offense intended)

...Plato.  Remember his rant about how we don't really see anything real?  Ever.

He maintained that what we perceive is analogous to shadows on a cave wall cast from outside.  He felt that things were just projections into our perceptions of idealized versions of those things which exist outside of our ability to detect.

There is, in his view, an idealized Chair.  It exists outside our perception and every chair we ever see is just a shadow on a wall, cast by that idealized chair.

I don't agree fully with that, but it's principle applies here.  I feel all fractals are idealized things we can never see under any conditions.

We only see slices of representations of them.

So yes, I am saying no fractals exist anywhere, not even in images we post here!

This is truly the best way you described the problem you have with our assumptions yet, I finally see where you're going/coming from.
And I agree.
With a huge BUT:     *I - like - big - BUTs andIcan'tde-ny.. youotherbrothers.......     ermmm..sorry  )
If we assume that Everything is just 'fake' we can't talk about the 'being' of anything.
And that might be cherrypicking-ish perfectly correct but then what's the point of any talking ever? how can you discuss anything if nothing is real?
we just have to assume some 'reality' we share to live in, to behave and, to discuss on wacky internet-forums
.. or as lazer blaster already said..

But I agree fully that there is no denying the Fractallyness of many things.

phew, now I'm relieved..

Personally, I would not even bother trying to define a difference between theoretical fractals and natural ones because the natural ones are mere shadows of the real thing - which we can never experience.
See?
I can get all mystical too!  

   

* youhn cherry picking

I think plastics, oils, gasoline, polished plates, computers and lasers are all natural things. In the same sense you could call a natural egg a chemical thing:

very well said. and a funny example too..

Blurring the boundaries is great for things like openmindness, the everything-is-connected feeling and the biggest feeling of understanding. But back down-to-earth is makes things pretty vague and hard to talk about. Everything needs to be chopped up, divided into neat little packs, named, categorized and registered or remembered. All in order to make it thinkable with our little brains.

Again, my revised definition of fractals is a boundary that is complex on a multitude of scales. I feel this covers the "everything is fractal" view of nature, as well as theoretical models.

I nearly agree with everything, although the roughness part stays a little unclear, as I haven'T had the time yet to try to understand the lorenz attractor.



Anyways, nice to talk to you guys!

[billtavis]

I nearly agree with everything, although the roughness part stays a little unclear, as I haven'T had the time yet to try to understand the lorenz attractor.

Cool! It's really just my subjective opinion that 'complexity' does a better job than 'roughness' at defining fractals in a general way. To me, 'roughness' conveys the mental image of something like sandpaper, something that would feel abrasive if I ran my fingers over it. And certainly many fractals fall into this category, but not all of them. For example, a fern is 'complex' but it doesn't seem 'rough' to me. Or consider this turbulent flow I found on wikipedia, it's very complex but everywhere smooth: