Welcome to Fractal Forums

Hello everyone!

I'm in search for a good and easy definition of fractals.
Not the pure mathematic and "perfect" fractals we generate here, but the fractals we see in nature.
I have my own thoughts on this and done a bit of research, but will share them just a little later - I'd love to have a fresh start. ;)

It'd be great to hear what you personally think:

What features/qualities does a thing in nature need to have to be called fractal?
What is essential, what is additional?



Regards,
Chilli

edit: I suggest you write your first answer without reading the answers of the others in order not to "distract" you with slightly (or massivly) different opinions..
I personally don't think there is a right answer, and I'm very much interested in your 'gut-feeling' even (or especially) if you're not a mathematician or just a normal-brained guy like me ;)

hmm, to bring you down, quoting mandelbrot is fun: "fractal geometry of nature" everything is fractal ;)
fractal is a term to describe processes in nature, classic euclidean aspects like lines, triangles or circles
where a former method to describe nature, mandelbrot showed it is not sufficient to describe a tree, a cloud
or a mountain, so, in my point of view, fractals as we use them nowadays are a more sufficient method
to describe our nature, he demonstrated it using the coastline problem, so, fractals are the most recent
method to describe nature because ... ehrm, yes, as flat as it sounds ... everything is fractal, at least until
we find another method for describing nature even better ;)

natural elements under scaling behave "strange" measuring the surface of a cloud, the border of an island,
but the surface of a table is behaving as well "strange" when measuring the surface counting in the atoms as
well

my five cents :)

Forgetting my apprehension about calling Anything in our universe a fractal  (I still maintain that fractals are purely mathematical constructs with no real manifestations in reality)  there are many things which use fractal geometry and obey iterative rules to define them.

My "simple" answer to What features/qualities does a thing in nature need to have to be called fractal would be Non-Identical Self Similarity at varying scales.  I add the Non-Identical bit because a simple line displays self-similarity at all scales, but I doubt anybody would call it a fractal.

I think fractal geometry can help define many things, but it does not make those things Fractal.

And to sidetrack to what Christian said :


I believe that one area of study that is largely ignored, but should not be, is related to the fact that atoms are mostly empty space.  We focus on the "matter" and try to understand it - but it's only 0.000001% or so of what makes up stuff.  Stuff is 99.99999% "Empty" Space.

That space is said to have an enormous energy, so much so that Physics had to correct for it when defining things mathematically or else it would overwhelm everything else.

We should ignore the statistically insignificant matter and focus much more study on that so-called "Empty" Space!

I think that's where the next big breakthroughs will be made in out understanding of The Universe.

Okay, has been some time, but then again, this is not small-talk ;) and my slow brain needs time to think such deep stuff through.


oh, how could this bring me down?! :)
this exactly what I think and feel.. but I try to find the smallest common denominator on what everybody can agree that it's a crucial part of fractal definition.

I like how you put that! :)

I disagree. I believe math is just one type of language to talk about fractals.
Would you not call romanesco broccoli a fractal? only because it doesn't show all the features of a purely theoretical mathematical formula?
Although it is obvious that it's based on a simple formula?
What would you call or use to describe that romanesco instead?

I'd say mathematical fractals are just one category of fractals, a very special and pure form.

I'd say that you can leave non-identical out. identical fractals are just a very special form. Like comparing a perfect square or a rectangle with a simple square
(http://www.chillheimer.de/downloads/imagehost/viereck01.gif)

or is the sierpinsky triangle not a fractal?
(http://upload.wikimedia.org/wikipedia/commons/3/38/Sierpinski-zoom4-ani.gif)
I'd say it clearly is identical selfsimilarity.

yeah, that's really totally mindblowing..
(and if you compare this to the distribution of matter in the universe especially the incredible distances between stars and galaxies and the vast empty space in between, I find it astonishingly self similar and in my definition.. fractal)
I'm really looking forward to these breakthroughs.
To the interested german-speaking folks here I recommend a youtube search for Hans Peter Dürr. Great stuff, very understandably explained.

But I also think that there will be huge breakthroughs when scientists start to acknowledge the fact(<--my personal opinion) that everything is fractal and start to use that as a base for research.


ok, here's my unfinished "definition":

I find that the only really 100% sure thing of fractals is self-similarity.
although, if (like in mandelbulb3d) you choose the parameters too extreme it will result in chaos. the similarity might still be there mathematically, but it isn't perceivable anymore..
(which in my personal opinion is an explanation why - if everything is fractal - we can't actually see fractals literally everywhere)

Every other aspect seems to be subordinated, secondary to self similarity, and doesn't have to occur inevery fractal thing.
-fractal branching (is only fractal if it's self similar, but an easily observable and typical feature for many fractals)
-fractal symetry (selfsimilar mirroring)
-'deepness' (zooming in endlessly, like in time or scale, but not in a romanesco brocolli)
...?
do you have more suggestionss for 'second level fractal features'?

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.
Speaking of fractional dimension isn't good enough either, since the Mandelbrotset fails that one yet again, having a hausdorff dimension of 2.
With that, the most typical, naive candidates are already crossed off the list.

However, the Mandelbrotset certainly is pseudo-self-similar. So that might be a good startingpoint.
Though if you wish to exclude simple geometric structures like lines or hyperplanes in general from your definition of what is a fractal, it's not quite good enough. I myself, however, would be content to call those things "degenerate" cases of fractals. Though it's not quite clear in what sense they are degenerate, beyond the obvious visual aspect.

And when it comes to nature, I'd say things are natural fractals if they are pseudo-self-similar over multiple scales - not necessarily infinitely many.

(To be clear, things that are self-similar are also pseudo-self-similar. And then there are scale-free things that can't just be scaled up or down by a fixed size to be self-similar but by any size, so scale-free things are also pseudo-self-similar. As such, "pseudo-self-similarity" is the weakest of the three concepts)

Isn't that exactly what self-similarity is about? That it gets more and more distorted but stays similar so the starting point (or at least to its 'neighbours', if you repeat the proces long enough you won't be able to recognize the starting point, but it still got there through self similarity)

I wouldn't want to exlude it - I believe that lines or circles are also fractal but are very, very special cases. But again, that is just my instinct telling me this, I don't like math ;)
I don't know about hyperplanes *google*

I understand but can't think of a good example for this.. can you?

No, self-similarity is the exact form.
Two shapes are similar if they can be precisely transformed into each other with a scale, rotation and translation. http://en.wikipedia.org/wiki/Similarity_(geometry)
If that's possible for a shape with itself in more than just the trivial way (0° rotation, 0 translation, scale-factor 1), then it's self-similar.
It's essentially a form of symmetry. If things are rotationally self-similar, it's just rotational symmetry. The really interesting ones are those that are geometrically similar to a rescaled version of themselves. - Perhaps that is a good definition of fractal, actually:
In case of a line, you can achieve self-similarity already through a translation.
In case of a circle, this works for rotation.
In case of a fractal, rotation and translation alone are not enough. You need to stay (pseudo)similar even if you apply scale transformations.
(Or at least, translational and rotational transforms alone do not cover all the symmetries and scale transforms give additional ones)

hyperplanes are just higherdimensional lines. The 1-plane is a line, the 2-plane is a plane, the 3-plane is a space with volume...

As for an example, the Wiener Process is scale invariant* (free wasn't quite the technical term, but it's the same idea)
(http://upload.wikimedia.org/wikipedia/commons/2/2a/Wiener_process_animated.gif)

Turns out when I said "pseudo-self-similar", the right term is "quasi-self-similar"

So I'd say, that anything that
- is quasi-self-similar (or self-similar or scale-invariant) and
- strictly requires scaling in some of the  transformations that shows this quasi-self-similarity
is a fractal.

This definition would include logarithmic spirals which are scale-invariant. You can zoom in indefinitely and if you rotate your view at the right rate, you won't even notice that anything changes at all.
However, a logarithmic spiral does not have any translational symmetry.

ah, I see, translation is the problem:

english                 --->   german
quasi-self-similar    --->   Selbstähnlichkeit (direct translation: self similarity)
self similarity          --->  strikte Selbstähnlichkeit (translattion: strict self similarity)

phew.. as if all the different definitions weren't messy enough ;)

Huh. To me, Ähnlichkeit is exactly the same as Similarity in the geometric sense. - I'm from Austria so my native language is German too.
Though apparently, reflections are also allowed. I previously thought similarity differentiates between chiral objects but if reflections work too then that's not the case. However, that does not change my given definition of a fractal in any special way nor does it change my classifications of self-similar or quasi-self-similar.

Quasi-Ähnlichkeit requires some additional, small distortions to get to an exact match. This does not happen for Ähnlichkeit.

I think of fractals in nature in a different way, because I can recognize a fractal in nature without taking the time to quantify its amount of self-similarity. To me, a natural fractal is a boundary or interface. (Someone please correct me if this is not true, because I do not understand the science well enough to make such a claim... OP asked for personal/layman opinions though, so this is mine).

In other words - if you identify a naturally occurring boundary/interface, you have identified a natural fractal; and vice-verse if you identify a natural fractal you have also found a boundary. I don't know of any counter-examples (please share if you have one!) and thinking of it in this way is clearer in my artistic mind. Sometimes, self similarity on different scales is very easy to understand intuitively (trees)... however since identifying and understanding it in some other cases is difficult, I shy away from it as the defining characteristic for natural fractals. To explain what I mean, think about the most common examples of natural fractals: coastlines are the boundary between land and sea; the surface of a mountain is the boundary between land and sky; the fractal of a tree is the interface through which it breathes. Logarithmic spirals form on the boundaries of hurricanes and shells. Fractals are found on the boundary between solid and liquid, between chaos and order, etc.
 
Of course, this definition fails immediately for non-natural boundaries because not every mathematical boundary is fractal (most are not)... however it's worth noting that most (all?) mathematical fractals like the Mandelbrot set are boundaries.

A layman's counter example would be the boundary between oil and water. I do see your point that most boundaries in nature are fractal at the scale we see them. The boundary between atoms would not be nor the boundary between distant galaxies.

Based on my view on the world (nature) and my understanding of the Mandelbrot set, my feelings about fractals go in the same direction. Boundary is a important key word, but as you say not all boundaries are fractal. It's also about the shape of the boundary.

In nature it is very, very hard to find (exact) self-similarity. So I would define fractals as a quasi-self-similair thing to include nature, and for brevity and clarity skip the word "quasi".

Thank you for providing counter-examples, but I have to ask: under what situations is the boundary between oil and water not fractal? Because when I walk on wet ground with oil-soaked shoes, amazing fractal rainbows of incredible complexity rapidly spread out from around my feet. Scientifically, fractal analysis is used to identify oil spills in the ocean, because they have a characteristic fractal dimension. A very still very smooth very circular glass of water with oil resting in it will still be subject to thermal fluctuations... at the microscopic scale, water molecules exhibit Brownian motion, a fractal. In this contrived example, it's just a very low amount of fractal roughness because it is an artificial situation, but the fractal boundary is still present nonetheless.

As far as pairs of atoms and pairs of distant galaxies go, the boundaries are not well defined and are not (to my knowledge) observable... so how do you know they are not fractal? Certainly the boundary of a single galaxy is fractal.

Well, not all mathematical boundaries are fractal. To me this demonstrates the disconnect between traditional mathematics and nature, but it does not say much at all about natural boundaries. I agree it's the shape of the boundary that's important and natural boundaries have a fractal shape (always? I'm still not sure).

Certainly every fractal has some degree of self-similarity, but this inability to be precise is exactly why I'm not a big fan of using self-similarity as the catch-all defining characteristic for natural fractals. It works great in some situations, but not others.

For example, when I look at a tree, it's self similarity is obvious and relates directly to the process by which it grows. However, when I look at something like an average boulder, I often have a much harder time identifying one piece that I can break off and transform in my head to fit another smaller piece. Big questions always seem to stump me: How closely must it fit the other piece to count as a fractal? How would I even measure the error? Do I need to identify self-similarites for the entire surface, as with the collage-theorem, to characterize the boulder as a fractal? In contrast, when I think of the surface of the boulder as the boundary between stone and sky, I can immediately trust that it is in fact a fractal. Further analysis beyond this immediate observation will only serve to characterize the fractal, not identify it.

There is no ambiguity, no "quasi" needed when using boundaries/interfaces as the definition. I just simply don't know yet if it is truly a universal quality (in nature), though the more I think about it the more it seems it must be the case. Consider this: the coastline of Great Britain was found to be fractal, not from studying its self-similarity, but from attempting to measure the length of its boundary.

I think, when eiffie said oil on water, he didn't mean a thin film of oil which would indeed cause pretty rainbow effects, nor did he mean huge oil spills which are not contained in any form and thus are more like clouds in the sky which also are decidedly fractal.
Rather, he refered to a situation like this one:
(http://images.dpchallenge.com/images_challenge/0-999/310/800/Copyrighted_Image_Reuse_Prohibited_148035.jpg)

Liquids can stay atop each other in a very clearly separated manner and the boundary can be incredibly smooth.

Also, stones are not really an example of self-similarity in the strict mathematical sense but rather of scalefree-ness: If you keep zooming in (over a couple of orders of magnitude - obviously this fails once you are at an atomic level), you'll see the same kinds of patterns over and over, all the way from an entire mountain down to a(n unsmoothed) pebble.
There will be finer and finer cracks and ridges, bulges, pores...
The reason you can't see a clear scale at which the whole pattern repeats is, because there literally is none. That's the whole point of scalefreeness. It doesn't repeat in discrete jumps where you can say "when I zoom in this far, I have a good match once again" but any amount of zoom will basically show you the same thing (within a range of magnitudes)

Though (quasi-)self-similarity and scalefreeness are basically the same concept. The difference is just that one is discrete while the other is continuous. So both are important and both describe fractals.

I'd argue that a tree isn't a boundary fractal.
I suppose the boundary can be fractal too - bark typically has rather self-similar or scale-free patterns and textures in it - but that's not related to the fractalness you typically consider when talking about a plant: The branching pattern. That pattern can be described as a Lindenmayer-System and it's not a 2D situation but rather a 1D one.

And the only reason we don't see any higher-dimensional fractals, in the literal sense of "see", is because we are limited to 3 dimensions in space and we can't really see volumes as a whole. We can only ever see surfaces, featureless (transparent) volumes or blurry (translucent/semi-transparent) volumes which always occlude part of the structure.

Though 3D or even 4D fractals could happen in nature. The 4D ones would have to incorporate time, so you'd have to watch time in timelapse while being able to slice through the 3D structure to potentially be able to actually "see" the fractalness inherent in this.´

As for how exact a match you need: Usually, if it visually seems like a fractal, which a stone surface does, it'll be close enough.

Reality is further complicated by the existence of Multifractals which describe most natural, typically considered fractal phenomena better than single-dimension-fractals.

Note, though, that some pretty darn smooth things are fractals by this definition. For instance, a logarithmic spiral, a straight line, a plane... Anything of which a piece can be matched up with a bigger part, and where that can be done multiple times, qualifies.
http://en.wikipedia.org/wiki/Multifractal
http://en.wikipedia.org/wiki/Multiplicative_cascade - note how the top three images look very much like structures you might find on stone surfaces.

All physical laws also are scale invariant. That basically means that physics don't care how big you are, they always work the same way.
http://en.wikipedia.org/wiki/Scale_invariance

Let me just start by saying thanks for the wonderful, thought provoking discussion we are having! I was nervous, being new to the forum and finding myself trying to defend an unpopular viewpoint, but I must say this is a great group!  :beer:

Well that line between the red and blue certainly looks straight! It's a great illustration of the counter-example, but unfortunately I do have a couple issues with it.
First, OP asked about nature but this situation would not occur in nature. Natural interactions of oil and water do not occur inside circular containers, they occur inside fractal containers. Trying to separate the interaction from the container is missing the point, because everything in nature is a fractal pushing up against another fractal!
Second, I think that fractals are present even in this particular situation, they are just hard to see. At room temperature, water molecules travel along fractal trajectories at ~1400 miles per hour in constant collision     https://www.youtube.com/watch?v=w1aVqKDAx6o

Thank you for explaining more thoroughly why the surface of a boulder is fractal, scalefreeness is an interesting way to think about self-similarity.

An L-system is an abstract mathematical model and a 1D tree is not found in nature. Hele-shaw experiments are a good way to understand branching as a boundary phenomenon. But lets consider a real tree. At a distant scale, the shape of the boundary(silhouette) is a trunk with branches. Walking up to the tree, the individual branches of the boundary also have their own branches, and on for a couple more generations. At closer scales, other fractals take over, like the bark you mentioned, or a leaf. Looking even closer through a microscope, cellular and eventually molecular fractals would be seen.  Notice that self-similarity does not succeed in capturing all of this at once but must be redefined at various levels of zoom and for different botanical features. However, I point out that the entire time we were talking about the same boundary. The boundary-based viewpoint continues to serve us well when we look at the spaces between trees in a forest, which form yet another fractal:
(http://www.roadschooled.com/wp-content/uploads/2009/06/tree_canopy_fractal-400x267.jpg)


Wait, you're saying that a perfectly smooth plane is self-similar so it counts as a fractal? Then by your own terms, even if we consider the oil-water interface to be perfectly flat it would still be fractal anyways, and not a counter-example to my boundary definition. :dink:
I tend to think, though, that perfectly flat planes are not fractal. Let's say you can see the whole plane on a certain scale.  You can identify all four corners and all four edges, and that is a certain amount of detail. You can zoom in forever and not find any more detail. You can zoom out forever and not find any more detail. Thus it is not a fractal (thinking of the scalefreeness of the boulder right now).

I noticed that problem too, already while writing about it. :)
There are precisely two ways I could see people define a fractal in a sensible manner:

1) You can identify some kind of scale-invariance be it discrete (you zoom in a fixed amount to find the same thing or, in case of quasi-selfsimilarity, almost the same thing) or continuous (no matter how much you zoom in, you find the same things over and over)

2) The object you are looking at can be assigned at least one non-integer Hausdorff-Dimension

Definition 1 is very inclusive. By that definition, pretty much anything is fractal. Smooth surfaces can be self-similar and thus fractal.

Definition 2 is very exclusive. The beautiful, intricate boundary of the Mandelbrot Set, for instance, has Hausdorff Dimension == 2, so it would not be a fractal in that case.

I haven't seen any other sensible definition. I don't feel like your particular definition that fractals always occur at boundaries and boundaries always cause fractals (e.g. there is no natural fractal that isn't a boundary process and there is no boundary which doesn't create a fractal) is satisfying, however a wide array of examples for this coincide with definition 1) above.

I also prefer definition 1) simply because I don't think having the Mandelbrot Set not be classified as fractal would make any sense at all.

Although thinking about it right now, perhaps one could slightly refine definition 2:

The Mandelbrot Set itself is a 2-Dimensional set. The boundary of the M-Set thus is a 1D-line, which corresponds to the topological dimension of the M-Set.
Perhaps, if the Hausdorff-Dimension of the topology is greater than the topological dimension, you are looking at a fractal? - Since that definition is about the boundary, it would fit nicely with your model while still excluding lines, circles, spheres or planes.
Not quite sure whether that works though. The Hausdorff-Dimension is always set between the topological dimension (dimension of the hyper-surface) and the geometric dimension (dimension of the hyper-volume) which always differ by 1. - However, if you talk about the dimension of the boundary, you are technically calculating the topological dimension, right? I should read up on that to make sure. But if that's the case, the M-Set has that weird case where the Hausdorff-Dimension of the surface is equal to the geometric dimension of the volume. Not quite sure how that works... I have never actually read a proof on the fact that the M-Set's boundary has Hausdorff-Dimension 2 though. Perhaps that would clarify.

Though I think you gave a good counter example in your last post: fractal trajectories of individual particles in a fluid.
A fluid does not have a boundary within itself. Only as it touches a different fluid and the two fluids are not nicely soluble in each other, you get a boundary.
However, even without that boundary, the particles move in brownian motion.

Furthermore, once you get down to the molecular level, speaking of boundaries starts become a little fishy. At that scale, everything would just look like fuzzy balls with vacuous gaps in between. You wouldn't actually see a boundary. Instead of seeing a wall you bounce off of, you see similar-sized individuals you bounce off of.

And even a single free particle in vacuum, where you most definitely do not have a boundary: A quantum physical trajectory effectively acts like brownian motion due to the uncertainty in both velocity and position. That very natural phenomenon most certainly does not stem from some boundary.

Well I do grant you that the boundary definition does not directly capture the idea of "complexity at varying scales" which is an important feature of fractals, and I realized that this is what I relied upon when making my case against a simple plane being fractal.
While researching this topic some more I found this quote from a professor who said "Fractals have two related characteristics: they show complexity at every magnification; and their edges and interfaces are not smooth, but are either perforated or crinkled." This satisfied me because it relates self-similarity with boundaries. And I think 'complexity at every magnification' is much easier for me to swallow than 'statistical self-similarity' even if they are referring to the same thing, because thinking of it as complexity takes all the pressure off of trying to identify a matching pattern.

I disagree. The Brownian motion of a molecule is a direct result of the constant interactions that occur at the boundaries between that molecule and it's neighbors. All the molecules in water are moving in their own individual fractal paths, constantly bumping into eachother, and as a result they cause eachother to continue moving in fractal paths. Like I said before, everywhere in nature we see fractals pushing up against other fractals.
And I think that even on the macro scale, the water exhibits fractal boundaries within itself: imagine separating the water into two groups: one group with all the water hotter than the average temp, and one group with all the water cooler than the average temp. The boundary between these two groups, if it could somehow be made visible, would be fractal.

I don't comprehend much about quantum mechanics... so I can't say whether or not a boundary of some sort is involved in your example. But I do know that the conditions you described don't arise in nature, only in carefully controlled labs - again to find a counter example you resorted to isolating an element from its environment, which IMHO is missing the point.

And yeah I don't understand how the Mandelbrot set has an integer dimension either. Sometimes I think maybe it's such a perfect fractal that it transcended all the way to the next higher integer, as if it is infinitely "rough"?  But with much certainty, I can use the professor's two-pronged definition above and state that the Mandelbrot set is a fractal because its boundary is not smooth and it has complexity at all scales.

It's just roughness (with self-similarity on a multitude of scales).

- Roughness excludes all things smooth (flat plates, circles, spheres, emptyness, lines, points, solids, etc)
- Self-similarity includes the iterative/recursive property of fractals
- A multitude of scale, to wider the definition to include natural things like trees, lungs, (natural) surface roughness, clouds, rivers, etc.

The conditions I described happen pretty much all the time in nature. In space.
And once again, once you go down to single atoms and molecules, there isn't really a boundary to speak of. There is no hard limit those particles can't pass through. No sharply defined wall.
Particles are not like billiard balls. - Natural particles, that is. It's artificial, newtonian, simplified mathematical particles that might act exactly like billiard balls.

Though I guess you could just add roughness/detail at all scales to the definition to avoid flat, smooth things like lines or planes to be fractal.
The question then arises, what exactly is meant by that. It's an intuitive idea but it might be hard to, like, find an algorithm that can decide that, yes, this has details on all scales so yes, this is a fractal.

ah, this finally gets really interesting :)
I have to admit that I'm relieved that I'm not the only one bothered with a real lack of definition and you guys seem to be struggling with it as well..

welcome billtavis - very nice discussion indeed!   :beer:

phew, there is so much to reply to, I need to check the forum more frequently.

first: I like the idea of boundaries as a "main-feature" of fractals.
it might not be the one feature, but you can't have a fractal without a boundary - right?
-then again, if you had no boundary, you'd have just 'one thing' and if there is no other thing that one thing might as well be no-thing :alien: -
is there any non-theoretical thing that has no boundary?
I can't think of one. (this would fit well with the "everything is fractal" thought.)

Ah, I love it to meet likeminded people! I keep wondering where they all are, as this topic seems so incredibly important, yet no one in my real life surrounding cares a bit.. cheers bill! :)
I actually think that fractals don't just appear in nature but also in man-made objects. if you e.g. see the tree of evolution as a fractal, you can also see the evolution of man-made design as a fractal.
you probably enjoy this 'little collection' I'm working on: http://www.pinterest.com/chillheimer/life-is-fractal/



But a perfectly flat plane probably doesn't exist in the universe. If you zoom deep enough, the single atoms will 'bump out of the plane. So I think this only is a theoretical thought and thus not so relevant.

and yet, if you zoom in close enough the boundary probably becomes fractal, even if it's the 'boundary of the forces active between the atoms'  (sorry for repeating myself, this post hat become too long...)
thank you kram for pointing out the difference between self similarity and scalefree-ness. I hadn't really considered this.


I'd counter that a tree is a boundary fractal, as it extends the boundary of a solid (earth) into a gas(sky) through self similar branching (as a nice addon)..
[/quote]
I personally believe that time is fractal. It's only a matter of perspective. There are so many time scales and different speeds happening, comparing femto-seconds to the age of the universe, in a tiny moment countless things can happen, like reactions single atoms. compared to thousands of years, were countless reactions between humans happen. or billions of years where countless interactions between stars happen..
--edt: of course it is, and it's been proven. I keep forgetting important stuff, this topic is so huge..
the heartbeat rate is proven to follow fractal variations in time, as well as the stock market, the frequency and strenght of earthquakes.......
a quote from this interesting article: http://www.psychologytoday.com/blog/the-chaotic-life/200909/fractal-brains-fractal-thoughts (http://www.psychologytoday.com/blog/the-chaotic-life/200909/fractal-brains-fractal-thoughts)
Other examples include the size of extinction events in animal species, numbers of academic publications (a few researchers do huge amounts of work and the rest of us do just a little), numbers of hits to web-sites, wait times in stop-and-go traffic, and word usage in literature (i.e., zipf's law).

the amount of roughness can be extremely high, try fiddling around in mandelbulb3d with extreme values. you'll see just noise or stuff that doesn't look like fractals at all. and yet it comes from a fractal formula.
I believe that this happens in nature a lot, which is why we don't see directly that absolutely everything is fractal (yes, there I go again ;))
exactly. as reality is a multifractal system of multifractals  :o  
there's no way we can easily see the overall connection. that everything is connected. scientifically speaking, like in the m-set.. and as a nice side-effect this fits to what most religions and esoterics say.
for me this 'scientific base' has huge implications. absolutely ground shaking, a different paradigm..
that deserves huge attention!

Is that truly so? Would be nice, but what about the quantum world?


If I understand the brownian motion correctly it doesn't happen for one particle in the vacuum and there is no uncertainity, as the one particle will keep flying in the same direction with a constant speed.
Ah, now I get what you mean - heisenbergs uncertainity principle and us not being able to measure where in time and place a particle is. I'm not sure if you can say with certainty ;) that this is the same thing as brownian motion.
maybe we are just not yet able to measure position/time as we're technologically not advanced enough at this point in time.


which all seem to be theoretical things that only exist in the mathematic realm.
and maybe even these smooth things like lines, spheres, planes are just very special cases of fractals with very special properties.
just like a perfect square always is a square, but not the other way round.
Afterall, I think we've seen 'perfect spheres' in mandelbulb3d.. but maybe we haven't zoomed close enough ;)

I tend to think that the iterative-part could be the one main feature. self-similarity comes naturally with it - not so vice-versa..

hm.. should it be 'so easy' ?

• recursion
• multitude of scale

and the only necessary 'adjustment screw' is the variable of roughness


seems plausible. and as I think of it, it's exactly what mandelbrot says.
probably my brain needed 9 months to really understand what he meant.
thanks for naming this here youhn!

hm. maybe the only absolutely basic thing needed for fractals is recursion and everything else results from it automatically (if the conditions are right)

I need to think more about this..


phew.. 1 hour later... why do discussions about fractals tend to branch out into so many sub-branches that it gets complex close to chaos?!
pun intended? ;)

Brownian Motion corresponds to a gaussian uncertainty in every single step of a path. And if you look at a free particle in space, you also have precisely this gaussian uncertainty in both impulse (impulse = velocity * mass, so you could have uncertainties in the mass of your particle, the speed of your particle and the direction of your particle) and position, so the end result is that such particles behave like a particle in Brownian motion would.
There are some caveats to this, but it's pretty much correct.

And yes, scale-invariance of physical laws actually is an important property of them. There is an axiom which states that even new laws we might discover (for instance what ever exactly dark matter and dark energy are) should be scale invariant.
Of course, this invariance does break down at some scale: Nothing can be smaller than the Planck Volume. Or, more accurately, while you can think of scales smaller than that, physics (actual, real life physics, at least as they are found to be today) cannot distinguish between two things that are closer to each other than this quantity.

Btw, you shouldn't imagine this to be like pixels or voxels or something like that: There is, to our knowledge, no grid to the structure of the universe. It's all about the difference. If two precise positions happen to be closer to each other than a Planck length, no physical process exists - none even could exist in principle - that could differentiate the two locations.

But above that scale, all the way, in principle, to infinitely many Planck lengths, all scales are treated the same.

Btw, I think the reason we find so little research into fractals actually is the same as the reason this topic came to be:
People are unsure what a fractal even is or, more importantly, what isn't a fractal. Thus, as far as research goes, it's kind of a vague, useless term.
If you want to look for research about fractals, you'll probably rather need to look into "complex systems" or "emergent properties" or "non-equilibrium thermodynamics" or "strange attractors" all of which often have fractal properties. It is only when people directly do research on already well known fractals like variations of the M-Set, L-systems and other geometric substitution fractals, Indra's Perls style fractals or cantor dust style fractals (I guess those all are either geometric substitution or functional iteration which in turn also usually have geometric concepts attached to them), only when those are directly researched you'll find a new paper on fractals pop up.
When you guys say you thought you are alone, the research landscape on all those topics really does proof that you are very clearly not.

I was keeping out of this discussion because I didn't want to spoil the fun   :gum:

But I do believe the reason we can not agree upon a definition of fractals in nature is because there are none.

Just things that resemble fractals and stuff made by some sort of iterative process.

But I doubt we will ever find anything fractal outside of mathematical constructs, and from that point of view the difficulty of reaching a definition becomes clear : One cannot define something that does not exist!

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.

Sorry to raise my unpopular views again, but it's a possibility that should not be left out when discussing the problems of reaching an agreeable definition of fractals in nature.

Who or what has made the definition for fractal infinite and therefore stricly mathematical?! :fiery:

I think this was plain stupidity.

That would be the same person who included "scale invariant" in the definition   :evil1:

And I agree - it is - as you say - "plain stupidity" to expect anything infinite to appear in reality, but we try not to call it that because plain stupid people take offense to such words.

Let's just call it "futile" to avoid offending anybody.

thank you kram for taking the time to explain in detail.

hihi, nope, it would take much more to spoil the fun.
first of all, I started the thread deliberatly asking for a definition for fractals in nature. not for the pure mathematical ones.

I couldn't agree less.
And the 'inventor' of fractals, Benoit Mandelbrot himself would disagree as well. And all those countless studies that proved fractals..
I don't even see a point in explaining why, isn't it obvious?
Please take a look at romanesco broccoli again:
(http://upload.wikimedia.org/wikipedia/commons/1/1b/Romanesco_Broccoli_detail_-_(1).jpg)
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  ;D :dink: ) 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..

I for myself believe that math isn't at the root of fractals but that (as everything is fractal) they also can be seen in a mathematical way.
for me, at the moment, math is probably just another language that speaks/describes fractals.

- Infinite isn't technically part of the definition
- The M-Set is not embedded in a physical space but rather in the space of Complex numbers which is isomorphic to R² which does not feature a minimal length scale and thus has no limit in detail scale. Pictures of the M-Set do not have an inherently minimal scale and thus you can't say that any particular resolution goes beyond the Planck Scale.
- All the definitions of scalefreeness and selfsimilarity can easily be altered to not go over all scales (from infinitely small to infinitely large) but only over all scales (from planck scale to infinitely large) or over a (wide) range of scales (from minimal_scale to maximal_scale with min << max) without altering them in any substantial way.

That's exactly how natural fractals are handled. If you look at a river or the shoreline of an island or a mountain or something, the things you can say about it will be limited between two scales after which the whole definition won't make any sense anymore. That'll typically be from the scale of the whole object - say, measuring the "height" of Great Britain with a ruler, to the molecular scale, realistically earlier than that because individual molecules - even individuals pebbles and grains of sand and what not - move way too quickly for this amount of detail to be of any use.
And for a fundamental law of physics, it'll be from Planck-scale to infinity.
This still spans several orders of magnitude and that's already truly mind-bending scales our minds can't really handle anyway.

Also, infinity, for the most part, unless you are trying to talk about the entirety of the universe beyond the observable parts, is just a convenient mathematical approximation to the truth.
And in case of the entire universe, we simply do not know yet. We could live in a spherical universe which is finite in total volume, or in a flat or hyperbolic universe in case of both of which our universe would be infinite. Evidence currently supports a flat universe best, but the other possibilities are not yet ruled out.

In any case, whether it's infinite scales both ways or it's a finite but wide number of ranges, it doesn't really matter. Both are valid descriptions of a fractal.
If you prefer, and I really think this is nitpicky at best, you could call what we find in nature "approximate fractals".

Not even then, for you could actually define spaces in which a minimal scale does exist and in doing so, you would basically get much more natural fractals which feature a minimal scale equal to the scale you choose for your space.
The real numbers can be extended to so-called real closed fields which behave just like real numbers in all the important ways (they are dense, ordered and closed under subtraction and division) but they can have differing properties. One of which, in principle, could be such a minimal length scale.

-----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'
http://www.youtube.com/watch?v=TW1gl6_QK-M
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.

But besides that, my problem is that my math-&physicsknowledge by far isn't enough to dis-/prove him.  :hurt: And googling finds esoteric pro guys and no real scientists backing the theory, but also none that disprove him, only comments like  "this is such nonsense, not even worth of disproving" -which reminds me of the reaction of people when galileo wanted to prove and show the disbelievers by a look at saturn through his telescope: "this is so stupid, we won't even look through your telescope"
I'm unsure what to think of this whole thing, although it sounds so logically.

Maybe one of the math-heads here takes  a little time to look through his telescope and tell me what he sees?
------threadnappig*off-----------

I can't recall where exactly I found it but I think I saw somebody actually debunking that guy.
He's been around for years with his pretty ridiculous claims.
What he does sounds nice but it comes out of nowhere. He pretty much sells pretty pictures with no true background.

And people very much study empty space. It's an important topic. Just look at all the research, calculations and thought experiments that happen around the topic of vacuum. There are tons each year.

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!


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!


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.

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.

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).

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

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! ;)

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..  ;D)

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?

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.

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:
http://www.youtube.com/watch?v=CcaW94VI56E

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!   O0 :toast:

What about the deformed minibrots?

(http://i.imgur.com/2f6r2yS.jpg)

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

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.

/me cherry picking

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

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:

(http://i.imgur.com/FAvF9rE.png)
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.  :police:

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

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?
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?

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.

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.

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.. :)

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)

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

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..

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.
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

;D

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

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.......  O0   ermmm..sorry  :embarrass:)
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..
phew, now I'm relieved.. :)

:o  ;D  :toast:

very well said. and a funny example too.. :)


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! :)

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:
(http://upload.wikimedia.org/wikipedia/commons/f/fb/Jet.jpg)

Quasi-Self-Similarity which isn't strict Self-Similarity doesn't require this to be the case at all scales. It suffices to find few examples.

And I can think of two reasons why you might not have encountered a distorted Minibrot at deeper zooms.
The first is the one you gave yourself: you never actively searched.
The second, however, would mean that you actually saw numberous examples at all scales:
Attractors have the property that things not *quite* but nearly in the attractor rapidly converge towards the attractor but - and this is the catch - never actually quite land directly on the attractor.
This might mean that, as you zoom deeper and reveal Minibrots which require more iterations to form, the distortions do indeed diminish - and quickly so - but they never quite actually go away. So while they visually do not seem like they are distorted anymore, they still most definitely are.
If I had to guess, the only Minibrots that are not distorted at all are probably those on the real axis.
Oh and those which require infinite zooming.

Given how the M-Set seems to tend to work, though, to find maximally distorted Minibrots, you probably need to find the maximally distorted Minibrot at a given scale and then "follow the same path" you got to here from the full Mandelbrot. That, I'd think (but I don't know) would lead to Minibrots which still are distorted. And if they are, that could be used as a good first test (but not a proof) for my hypothesis: If each time you "follow that path" your Minibrots becomes less and less distorted (but they keep being slightly distorted in the beginning) what I guessed above just got some evidence.
It can't be considered perfectly disproven if you don't find any distorted Minibrots right after the next scale jump though: It might be that those distortions go away really quickly.
In that case, to really prove it (really the only way to be dead sure about this) is to go a more mathematical route: The resolutions and contrasts of our screens are finite and if the distortions drop off too quickly, there might no longer be a visual difference while there actually is one if you compare precise values.
However, if you actually find Minibrots that are more and more distorted as you go down, both of our thoughts on the matter can be considered disproven.

Hi, I have a very simple definition of fractal as follows:

A set or pattern is fractal if there are far more small things than large ones in it. I called it the third definition of fractal https://www.researchgate.net/publication/309428627_The_third_definition_of_fractal (https://www.researchgate.net/publication/309428627_The_third_definition_of_fractal)

Let me know if it makes sense to you. Thanks!

I think a fractal is completely characterized by its high complexity, and must be generated by some kind of algorithm.