Search for an agreeable definition of fractals in nature

[Chillheimer]

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

[cKleinhuis]

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

[Sockratease]

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

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 :

the surface of a table is behaving as well "strange" when measuring the surface counting in the atoms as well

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.

[Chillheimer]

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.

hmm, to bring you down, quoting mandelbrot is fun: "fractal geometry of nature" everything is fractal

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.

but the surface of a table is behaving as well "strange" when measuring the surface counting in the atoms as
well

I like how you put that!

(I still maintain that fractals are purely mathematical constructs with no real manifestations in reality)

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.

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'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


or is the sierpinsky triangle not a fractal?

I'd say it clearly is identical selfsimilarity.

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

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'?

[kram1032]

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)

[Chillheimer]

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.

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)

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.

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*

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.

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

[kram1032]

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)

[kram1032]

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.

[Chillheimer]

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

[kram1032]

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.

[billtavis]

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)

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.

[eiffie]

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.

[youhn]

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.

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".

[billtavis]

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.

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.

Boundary is a important key word, but as you say not all boundaries are fractal. It's also about the shape of the boundary.

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).

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".

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.

[kram1032]

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:


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