This is a question to which we will never be able to give a definitive answer. At least not until we can reach a universally agreed upon definition of "fractal."
So until then, my answer is a resounding No!
Fractals, in my view, are a mathematical construct with no real world corollary. We see examples of nature using Fractal Geometry, iterative processes, and strange attractors - but I believe all of reality is Finite and so long as the definition of Fractal involves anything infinite, the answer will always be no.
If we loosen the definitions, we allow things in which many people say are not fractal (and I am amused by how these are often the same people who say The Universe is a Fractal!) (If The Universe is a Fractal, then there is NOTHING that is not a Fractal and this discussion is pointless).
everything we do is a simplification of reality, fractals are a finer definition of what we see in our world, and hence the world is - in my point of view - fractal yes
the branching structure of a tree is something we observe, reality in general is far more complex than any of our abstractations, although fractals seem to be unbounded due to the infinity of their mathematical nature, fractals are no more and no less of a tool to describe what we see
as mandelbrot pointed out, the world is not made of pyramids, cubes or spheres, these euclidean objects are a too hars simplification of reality, and fractals help in describing our nature in a more close way
Ok, so the definition could be that fractals are a good way to describe things in nature. Simple geometric shapes simply would not be realistic.
I kind of forgot that fractals are ideal shapes compared to real things. But then why do we connect these imperfect shapes?
Nevermind that. That post has a lack of logic.
from where does come the infity, infinity is present in nature, e.g. the size of our universe is considered to be infinite
mathematically it is a theoretical construct that is rather observed than constructed, e.g. the + (addition) has no limit, whenever you add one to something it gets bigger, and for that you can add another one, you see it occurs by the way we try to grasp our surroundings and just simple things as adding incur strange stuff like infinity in the simplest operation
so, as sock statet nature is far more complex than our fractal view to it, fractals are an improvement over euclidean (triangles...) worldview but surely not sufficient
Of course, more complexity is needed to get results that come close to nature. Like if something is infinite it doesn't mean it is a fractal.
Well, I'd say it's both, real and how our minds work.
Fractals are an emergent phenomenon of recursive processes - mathematical but also physical processes.
The result of such a process is either always the same (as in zero or a fixed value or going to infinity), self identical (as in the koch snowflake) or quasi self similar, as in the mandelbrot set, with it's evolving forms.
let's use that as a (crude)model for reality:
always the same: black holes and infinity of space as the 2 extremes.
self identical: suns and planets, 'lifeless' matter that repeats the same thing over and over, unless an external force 'changes the formula'
self similar, evolving: life
It doesn't matter if fractals in nature are imperfect. Actually I wouldn't say they are - they just are distorted through external influences (which I would call multifractal)
Take the good old tree. It might grow approximately perfect under perfect conditions. But those conditions don''t exist. Weather and wind (fractals again) influence it's growth and make it gnarled and distorted form the ideal form. This applies to all fractals in nature and is the reason why many are hard to recognize on the first glance.
then your second part - are fractals part of how our mind works:
yes they are, at a very fundamental level (hierarchical buildup of the brainstructure and the connections) as well as the way we actually think:
https://www.psychologytoday.com/blog/the-chaotic-life/200909/fractal-brains-fractal-thoughtsyes. fractals are the optimal way to 'store' as much information and create the most complexity with the least amount of effort or 'storage capacity'. nature always tries to use the most efficient way possible, so it ends up with fractals. I had a good scientific paper about this but can't find the link right now..
btw, in case you didn't know, our ability(or fault) to see patterns where no patterns are is called
Parodeilia - which indeed pobably often happens when actively looking for fractals, I've had this quite often I guess.
That makes sense. Perfect things can only happen under perfect circumstances. Which results we could use to in general group things that deviate.
I've been looking for Paradeilia but I couldn't find it. Thanks for helping out there.
haha, sock our eternal 'fight' deserves it's own post..
In my opinion this is the heart of the annoying problems any such discussion faces.
Would you mind showing me this definition that explicitely says a fractal is only a fractal if it is infinite?
Because usually (as in the wikipedia definition) the term that is used is "fractals show self similarity
on all scales". But that is NOT the same as infinity!
I wonder, do you have the same bias against euclidean geometry?
I mean, if you look at a soccer ball or a planet and someone says, hey that is a sphere, do you say: nope, you're wrong, it's not! because in nature there are no perfect spheres? actually it's surface is rough if you zoom in close enough. which would make the surface fractal. at least for a (elliptical
)sphere like our planet.
if a sphere like the earth is neither fractal nor a perfect euclidean sphere - what is it in your opinion and how do you suggest we talk about
anything in our reality at all, if no way of perception is 100% true ever?!
also, please take into account, that even a mathematical fractal is only infinite in reality, if you keep calculating infinitely long.
the same is true for a fractal universe, in which recursion could be described as time passing. so we are currently at iteration 10^12436. and it looks finite - but not if time keeps ticking and iteration count goes up.
what you do is the same as looking at a rendered image of the mandelbrot set and then say "it's not a fractal, because it doesn't go on forever, see, theres only 1920*1080 pixels of it, so NOT infinite!)
First of all - as long as there is no agreement that the universe is a fractal, such a discussion is never pointless, at least as long as there is no reasonable explanation for the omnipresence of fractal patterns in nature.
And why would it be pointless to search for other patterns and to find the connections between these patterns and the 'full view'? you see a deep zoom image of the mandelbrot set and it doesn't look anything like the zoomed out full view of the set. And yet there is the deepest connection.
If you count me to the people who say something is not really a fractal while saying the universe is a fractal, please let me clarify:
some things might not be visibly fractal but still have a fractal nature or be the emergent result of recursion of a basic formula.
fractals can become so rough and chaotic and so distorted that it is impossible do recognize the fractal nature of it. especially if we are talking about multifractals. just combine 6 formulas in mandelbulb3d and if you'Re lucky you get at least some noise that you would never recognize as fractal.
just because we don't have the means to recognize something as fractal doesn't mean it isn't.
I know this is a dead end in science and just as bad as explaining the existence of god with god being powerful enough to create himself.
but then again, I'm just some guy on the internets and if I had the perfect answer and proof for all this we wouldn't have this conversation.
But at least I'm trying to find answers that are in my reach.
Also: If the universe is a fractal, it is clearly not a "simple" fractal like the koch curve or the mandelbrot set. And it has different levels of complexity on countless but limited scales (like the m-set is limited to -2 and +1).
In the mset you have shapestacking that gives rise to new complexity, while the basic information that lead
<![CDATA[<a href="https://www.youtube.com/v/Ojhgwq6t28Y&rel=1&fs=1&hd=1" target="_blank">https://www.youtube.com/v/Ojhgwq6t28Y&rel=1&fs=1&hd=1</a>]]>.
EDIT: youtubelinks that are posted using the url= function should not be embedded. do you have an idea how to fix that?
Yep, fractals≠infinity. Although they are related. I guess the same could be said with all euclidian shapes:
Shapes we find aren't perfect.
About the infinite calculations, I found something related:
http://www.fractalforums.com/new-theories-and-research/we-will-never-know-the-real-answers/If we wouldn't know we couldn't reach infinity.
Well, I would say the universe is an ''inaccurate fractal''. It does not have the ideal circumstances to be a fractal but it has fractal properties that at some point aren't accurate anymore.
The last thing simply is the difference between ''strict'' fractals and quasi-fractals which have more variation (and which are in my opinion closer related to nature)
This is not part of the definition because it is a consequence of self similarity.
Consider a tree, with two branches splitting off from the trunk. For the branches to be similar to the whole tree, each branch needs to have two twigs splitting off of it. In order for the twigs to be similar to the whole tree, each needs two ... and so on and so forth, forever. There will be an infinite number of ultimate branch tips ("leaves" if you will).
And there actually must be an infinity of them, because infinity is the one (non-zero) magnitude that fits twice into itself. That's the only way in which one of the two biggest branches can be truly alike the whole tree: one such branch has exactly as many leaves as the whole tree, despite being just half the tree. Infinity delivers on that seemingly impossible constraint.
But isn't infinite selfsimilarity only possible after an infinite amount of steps?
well explained, that helped, thank you!
but then again, just as the "inventor" of the actual word "fractal" said: he was talking about fractals describing actual nature and reality. not hypothetical infinities.
In his book "the geometry of nature" he talks about these fractal patterns in nature and he calls the actual 'objects' fractals, to him the coastline IS a fractal.
I find by always just fixing only on the theoretical concept of infinity we rob mandelbrots discoveries of their actual, practical use.
"not infinite, so not fractal, so no need to think about using them as a way to explain our reality" (of which they are perfectly capable, much more than euclidean mathematics)
that in my eyes is ignorant. (with absolutely no intention to insult anyone, especially not you hobold!)
sorry, but who are all these mathematicians who think they can re-define what Mandelbrot actually said and explicitely meant by only fixing on a small detail of his definition?
(it might be that I have misinterpreted or remember the details of his book false. but even if that is the case, I still find it worse to look at a tree or romanesco brocoli and say: not infinite, not fractal.
what is it then, if not fractal?? as in my previos post - it's all about the limited resolution of real life objects, as any snapshot of the mandelbrot set is not infinite, it's pixels.)
man, this topic always gets me going..
But what about simple euclidian shapes that make up a fractal(basic IFS fractals)?
The magnitude of the universe (finite or infinite) is absoultely undecided! It is a subject of constant debate amongst cosmologists because it makes all the difference in many of our grander theories.
Whenever infinity is encountered in any explanation of things, it is usually considered a failure of that model. Black holes almost did in The Standard Model of Physics because many physicists felt that infinitely dense objects cannot exist. That is still debated hotly to this day!
I'm at work, so will try to be brief - hopefully more details later! But in short; If the Entire Universe is a fractal, then everything in it is also a fractal. Therefore, at that point, it becomes meaningless to even ask if any individual part of it is a fractal since it all must be. By definition. Specifically to this discussion, if The Universe is a fractal then they must be both real and a part of our thought processes. No further debate is possible. Even a straight line becomes a fractal since it is part of a larger fractal.
Consider a parallel question that may help illustrate my point : Is any part of The Mandelbrot Set NOT a fractal?
I agree with Chillheimer although I must say that that is a very good point.
Edit: I hope I do understand thing again and that my answers/questions aren't too silly.
I like how this thread has grown, thanks for participating and I hope we can keep it going.
February 19, 2011, 03:21:21 AM
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