Welcome to Fractal Forums

I've been thinking about this but I can't figure it out. :hmh:
Do fractals actually exist or is it just our mind's abillity to recognise patterns which makes us think that?

There has for example been some proof why we find them in nature but the the main reason behind it has not. Are the fractals in nature actually found there because they are useful? Or is it because we automatically see patterns that might not even be there?

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

I kind of forgot that fractals are ideal shapes compared to real things. But then why do we connect these imperfect shapes?

We're both Chemists, so if I ask you to consider The Ideal Gas Law, you know that it so very closely approximates real world situations that it is used very effectively to describe reality and make predictions.

But we know that it is merely a very close approximation.

Ideal situations simply do not exist in reality.

The True Nature of Reality, I suspect, will prove to be far more complex than anything we have proposed so far in our Philosophy.

More close, yes.

But still not there yet   :gum:

Yes, thats true. So in that case we use fractals to understand nature and other things. (I hope I don't sound like a non-fractalist but I haven't thought about it this way.) But finite and infinite are also just concepts even if they are assumptions. How do we come up with these things?


Edit: I hope this makes sense, I'm currently still confused about how I thought things were different. :confused: And I think I am confusing myself a bit more than anyone else. Ill rate some images and Ill come back once I can think again.

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 ;)

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-thoughts

yes. 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.. 88)

btw, in case you didn't know, our ability(or fault) to see patterns where no patterns are is called Parodeilia (https://en.wikipedia.org/wiki/Pareidolia) - which indeed pobably often happens when actively looking for fractals, I've had this quite often I guess.

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 88))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 there still is embedded (https://www.youtube.com/watch?v=Ojhgwq6t28Y).
EDIT: youtubelinks that are posted using the url= function should not be embedded. do you have an idea how to fix that?

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.

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..    :embarrass:

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?

hehe, until you are home I've written many more pages, I'll drown you with my  :canadian:  
;D

I just love it how the fractal perspective gives a simple and clean answer to this problem - that includes both views.
if the universe is fractal, then it is infinite, but in a confined space, just as the M-Set is infinite, but it takes place in a confined space of the coordinate system.

but as long as the scientific majority neglects the possibility that the universe is a fractal, solely based on exactly your argumentation this is the only way to show them this view might be wrong and that it is at least worth to really examine this instead of ignoring it.

man, this feels like the futurama episode about the missing missing missing link to prove evolution.
https://www.youtube.com/watch?v=TTOla3TyfqQ

and yes, I am well aware of the irony that I use creationism as argument against science trying to disprove a fractal theory that can never be totally disproven (which is essential for science).
and this deeply sucks, but I can't just ignore the obvious fractal nature of reality just because of that.
this situation honestly is very annoying for me, because I like to think of me as a rational, scientific person.
if anyone has a satisfying and logically consistent solution for me, I'd really love to close this chapter once and for all instead of chasing windmills..
It really bothers me.

nope doesn't help.
my answer would be no.
except if we limit the resolution and just take a snapshot, thus making it tangible, visible, coming into what we call reality.

I guess the formula itself is a mathematical concept, not an object.
the result - the picture - is the object.

maybe the universe as a whole is also "just a concept" and not an object.
the actual result, our reality & conscious experience is the object.

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.


Nevermind that. That post has a lack of logic.


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.


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. :)


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. :P

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)



But isn't infinite selfsimilarity only possible after an infinite amount of steps?



But what about simple euclidian shapes that make up a fractal(basic IFS fractals)?

(https://www.zeuscat.com/andrew/chaos/sierpinski.clear.gif)



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. :P I like how this thread has grown, thanks for participating and I hope we can keep it going.

Then I guess it is in no way possible to make a fractals as a picture as it does not exist under perfect conditions. The only thing of fractals that could be called infinite is the concept itself.

Lunch Break!

Time to point out one further observation - I don't even think MATH is real!

It's just Intellectual Gymnastics.

It provides us with ways to model real things, usually only approximately  (reflecting it's lack of reality) but math itself... 

It's nothing.

Just thought processes getting carried away with their own self-importance  (a state of mind which I have been known to indulge in myself every now and again).

I guess you are not the only one. I think we all got carried away.

(The ideal circumstances for this good discussion were not there anymore like we discussed earlier with other examples. :tease: )

It might be a strange question but would like to keep this discussion going.

Would other intelligent life-forms be able to understand fractals like we do? (In this case I am relating to second half of my question.)

Interesting question.

And it raises another interesting question - What would The Mandelbrot Set look like in a base 12 number system?  Base 3?  Other "alien" number systems?

Who's to say they would use base 10?

How would one even go about checking other base systems?

Let me quote to thoughts and show that they are closely related:

1. Time to point out one further observation - I don't even think MATH is real!

2. Would other intelligent life-forms be able to understand fractals like we do?

Preliminaries:

"Reality" - more than one definition of the concept exist.

For example, there is "physical reality", meaning the existence of things and/or events based on nature's laws (system states and state transitions); regardless of observers, regardless of observers' knowledge of nature's laws - but implying that such natural laws exist. Without natural laws, the universe would be chaotic, arbitrary, and reality itself would be so random as to be meaningless.

Then there is "objective reality", meaning existence of observables, that can be defined so precisely that all impartial observers would necessarily come to an agreement about what they observed (i.e. measured). IMHO, this is a lesser form of reality, because it is constrained by the existence of observers.

Furthermore, there is "informational reality", meaning the existence of information, created however indirectly by the universe, stored and processed by physical entities (either biological, mechanical, electronic, or otherwise), that is interpreted consistently the same by a group of individual entities. For example if some of you understand some of the text I posted here, then our agreement about the meaning of words implicitly lends "informational reality" to those words and the abstract concepts behind them. The thoughts that were "real" in my mind gain more "reality" if you chose to copy them to your mind, so to speak.


So, with this gradual continuum of reality, I can claim that math is somewhere between "objective reality" and "informational reality". To see this, take some piece of crystal and count the number of atoms in it. That number is an abstract informational concept, but it is an observable attribute of the physical object - the crystal piece we started with - and any diligent and capable observer would count the same number of atoms in that particular object.

Or, in other words, math is certainly less real than the piece of crystal itself. But if we use math to process information which originates from "objective reality", then those kinds of math that can make accurate and useful predictions about "objective reality" implicitly gain some of that reality, too (that kind of math is called physics). That's because math grounded in objective reality is not arbitrary, but constrained by nature's laws.


That leads us to question 2: would an alien intelligence understand fractals as humans do?

IMHO, if those aliens live in a universe with the same natural laws, and if they discovered and use physics to make use of their environment, then their math will most likely be very similar to our math, because both schools of thought were shaped by identical constraints.

So this leads to a two-pronged answer: if the aliens have some form of technology (that is sufficiently advanced) then they will be able to understand fractal geometry. But if their use of math is detached from "objective reality", if it is a purely philosophical hobby to the aliens, then they might not recognize the concepts underlying "our" fractal geometry.

I haven't thought about that. Nice idea.

Although I don't know whether what the result would be different with different bases. It would involve devision and multiplication. Which doesn't have an impact on the general shape of the m-set. (it would only increase or decrease in size. Considering there is a large enough bailout radius.)

My two cents: First, fractals exists as mathematical concepts and are as real as the fact that 2+2=4; and second, they often appear in the world, and in fact describe it better than euclidian geometry.

But they wouldn't be different in other bases. 2+2=4, 10+10=100. If you program the Mandelbrot set (I did in the past), the computer uses binary internally. It doesn't look different from the same set calculated purely in the decimal system.

I don't know if this is on the same subject, but sometimes when I'm using MB3D I wonder if I'm finding something or if I'm creating something, it something that puzzles me a bit.

I thought I was close, thanks for explaining this a bit further.

I'm sorry Sockratease, but it is a nice idea though. Maybe other intelligent life will have completely different way of handling numbers:
https://www.youtube.com/watch?v=l4bmZ1gRqCc

I guess thats still on-topic.


And a very interesting thing to think about. I guess if math was ''meant to be'' you are probably finding something. If its not and math is actually part of human mind you are finding a variation on something and thus  creating something new.

Yes. 2 + 2 = 4 even before there was a being which did maths consciously.

I wasn't suggesting that the meanings would change, but I do recall something from the long gone college days about graphs of things set up in a different base look different.  Similar, but skewed or stretched because the scale changed.  I could be mixing things up from two totally unrelated things - that's why I mentioned it here.  I wasn't sure.

But I am sure that when I say math is not real, I mean that it is an ideal system.  Ideal situations do not exist in reality.  Math approximates reality, but is a bit too perfect.  Like our equations in physics and chemistry.  The true results of direct measurements vary a little from the predicted values, but are often very very close.  Close enough that nobody but people who philosophize about math give a crap.

Like I said - I see math as a thought experiment or intellectual gymnastics more than anything that reflects reality exactly, despite it's incredible usefulness in modeling things.  It's a great tool, but gets taken a bit too seriously for what it is.

I'll avoid going too complex here.. but what is an too perfect approximation of
one apple plus one apple = two apples
?!?
to me math is a language. probably the most basic and universal there is.

it's neither black nor white - euclidean is 100% perfect. fractal geometry is much more real, with infinite "imperfect" variation. reality is 100% real.
and if you'd ask me what is closer to reality, it definitely is fractals.

How hard Benoit fought to make his insights and expertise into an acadrmic field of study.
But he was a latecomer to the scattered and marginalised explorations of a disparate collection of geometers around the world  it was his study of the work, ideas and results of many geometers which gave him the academic right to name the topic of his dissertation.

Academically he defined his subject as tightly as he could so that it could be distinctively marked. So he in fact had several distinguishing elements to his dissertation topic.

We know he published in French his original,dissertation, and through circumstances to do with the war eventually took up an opportunity in America to work on computer applications to real world problems.  It is during this time that he was able to give form to a set of forms that earlier researchers had only been able to sketch!

So the words  "real", fractal and iterative process , and set of results ( Julia sets) all were drawn together by the diverse patterns in human and natural behaviours he was studying!

Academically he struggled to gain credibility for this subject. It was either defined so tightly as to render it as practically useless, or dismissed as a ragbag collection of indulgent musings!

It was when a pilot who required a way of generating realistic backgrounds for his flight simulations read the translation of Benoits dissertation and recognised how to design an application based on these principles in a computer code that was used widely in industry that Bemoits non Academic reputation started to take off and give him opportunities that his academic colleagues envied!

As collection of enthusiasts fractal hobbyists have driven the developments in the field and have importantly helped broaden the definitions of the field  it is these same enthusiasts who have gone on into academia and industry and infused these ideas into the mainstream of many engineering and scientific topics that are founded on constructive disciplines.
Is it real or is it in the mind is a restatement of an old Socratic and Platonic " game" often called the Platonic theory of Form or Ideas

"Is it there all along"    88) 

1 + 1 = 2 can even get tricky.  Bertrand Russell and Alfred North Whitehead needed hundreds of pages in Principia Mathematica to *prove* that 1+1=2  :P

But then there's irrational numbers.  They met Pythagoras' definition of imperfect enough to have the man who first showed him a proof that the square root of 2 is irrational murdered!  {OK - not proven, some say it was his students who did in poor old Hippasus and others said it was Poseidon  (apparently even The Gods thought Math was Perfect) and others say Pythagoras never existed at all and only a "Pythagorean School" - but the theme of Hippasus being killed for his crime of proving Math to contain "flawed Numbers" is persistent}.  Granted, Pythagoras made a Religion out of Math and Worshiped it's Perfection, so when something imperfect was shown to exist in math, it was Heresy and Blasphemy punishable by death.

But to the main point of discussion, I'm talking about using Math to model reality.

It never works.

My example was the theorems of Chemistry and Physics  (there are many who would agree that 1+1=2 it not Math - it is merely Arithmetic and that is a noteworthy distinction).  The Ideal Gas Law is a fine example.  It provides a darn near perfect predictive tool and is treated as a means of measuring and determining things, while at the same time accepting that it does not reflect reality entirely accurately.

It is, in the words I used which seem to bother some, too perfect.

I don't deny it's usefulness, but it is a tool - not a religion.  It has flaws.  Either that, or reality has flaws.  But either way Math does not, and never will, reflect Reality perfectly.  Fractals may be closer, but we may never be able to model reality in math due to many factors including things like Quantum Uncertainty, Godel's Incompleteness Theorem, and many other stuffs which expose the limitations of trying to define things in this manner.

However - if all you ever want to do is count apples, then I suppose it is good enough   :-*

Phrasing it so absolutely seems unnecessarily confrontational to me. Math has its limits (and you don't really need to dig as deep as Gödel's theorem of incompleteness). On the other hand, behind almost every technology we use, there was some mathematical thought performed by some mind during the invention/development. The math may not have been explicit, or not formulated in traditional ways, perhaps it was only a vague imagination at the time.

I don't mean to claim that math is the source of all good ideas - it isn't. Surprisingly often, random events (observations thereof) were the starting point of something that became an important invention later. But technological advances are fairly often created in a somewhat targeted manner from an already existing strange observation: experiments try to repeat the original random event in a controlled manner, then try to modify the event to shape the effect in desired ways. Mathematical models are the main and primary way to guide this exploration.

(But admittedly random chance is still the main and primary way to prompt the starting of exploration.)

If the words bother, maybe different words are understood better? Perhaps math isn't too perfect, but too smooth, too tidy, too clean and (in the sense of intentionally abstracting from detail) too simple?

After all, the infinite detail in the Mandelbrot set does not directly arise from z = z² + c, but from the iteration of that formula under certain conditions. I mean, the formula alone does not in fact describe all the detail - you really need a program, an algorithm, to fully specify the Mandelbrot set; and then you need a substantial amount of numerical computation to realize a picture that displays an approximation of the infinite detail.

In other words, the clean and simple formula z = z² + c is math, but it is only a small part of the much larger machinery required to actually show the Mandelbrot set around.

Well, I truly didn't mean to sound confrontational.  That's why I clarified it was just my opinion.

I don't expect others to take it as anything more or less than that.

But I do apologize if it came off as harshly phrased.  I always figured the one place such absolutes are appropriate is in matters of personal belief and opinions.

With small simple calculations you could easily say that. Imagine doing that with molecules...


You can never get the exact number of molecules. Because it is an approximation. There a lot of other things that also affect it. (Equilibrium for example)



An other example:

With precipitation reactions you use more of one of the salts because you can't be sure you have the exact number of ions you need. With a solution with silver ions you always use more sodium chloride than you actually calculate (the calculation is just to prevent that you use a lot too much). You will keep adding it untill all silver ions have reacted with the chloride ions to form a salt that isn't soluble in water.

http://m.youtube.com/watch?v=6oWLIVNI6VA
This very interesting talk on the interface between maths and physics .
Gauss famously through Riemnn appealedbo physicists to rescue mathmatics fom it's demise!
In fact prior to Riemann many philosophers already saw the death of mathmatics despite Kant's valiant assertions. In point of interest The Grassmnns were right at the forefront of this next stage in human combinatorial thought.

Maths is dead!
But we love it so much we have preserved it in Wax!
Here the physicists are guiding the so called Mathematicins.
So without labouring the point nature and natural Philoophy gives the human mind the contexts to develop topology and topological relationships that express our experience of magnitudes,,extnive and intensive.

What have natural behaviours and phenomena to do with the circle ? Fundamental perspective is based on sphericity! We as relative observers can only perceive spherically.

The sphere underlies all ancient wisdom, but the sphere is clearly a pragmatic ideal!  This was always known!
But we have been misled or obscured from this simplicity.

The fundamental topological fractal is the sphere . It is totally in the mind in my opinion , but it comes from our spheroidal perspective.
Note that sphericity is a concept that we develop from observing closed objects , while vorticular topologies come from imagining dynamic processes that seem to have a form or pattern: almost!!

We can not escape the concepts of Fractal Topology and our perceptions can not find ultimate cause . It can choose and does choose to elicit distinctions but distinctions do not discount the things distinguished. Thus the sphere does not discount spheroidal objects. And we can say that the inter relationship between the sphere and spheroidal objects is tautological, they are both interdependent!

So I believe we can expect to see versions of fractals in nature and perfected versions of natural fractions in our minds/ mathematical displays.

http://m.youtube.com/watch?v=LYNOGk3ZjFM
So when you were told convergence of series is crucial to proving certain infinite series equations , and that Euler was wrong to use infinite series to solve so many problems , you were being misled by someone who could not understand Euler or indeed what is proof and how to prove in the real world!

Each art or term of a perturbation series represents a term at a different scale ! And assymptotic solutions ar perforce almost similar solutions. These are 2 of the major characteristics of a fractal process!
Cyclical processes are perturbation processes , circles again!
Oh by the way circles and probability are models of combinatorial outcomes for finite or truncated possibilities!
DeMoivre knew this as dud Newton

"Maths is dead!
But we love it so much we have preserved it in Wax!"

WTH is that supposed to mean? That sounds unserious.

@Max Sinister
It is. Poetic reference Madame Tussaud s famous wax museum! In such a place wax copies are made of once living things.
I make no bones about the state of Mathematics . NJWildberger is seriously tackling the issues within the subject, but the more he solves the issues the closer he comes to melding the subject into a type of computational science .

However the question about fractals is really independent of Mathematics . It is a question regarding the philosophy and psychology of perception .

http://m.youtube.com/watch?v=L45Q1_psDqk
All you can do is Accept!

Or Reject!   :tease:

https://youtu.be/3MRHcYtZjFY

Here is a quite detailed description how we look at things. The video itself is not related to fractals but is to the way we use fractals to describe things.

At the fundamental level I think my choice is limited to supnsion( and thus limbo) or acceptance of a chosen experience. Perforce this means rejection in some instances if I am in the superpositional state of suspension . But active rejection seems to me to only crystallise after I have accepted some state and am defending or modifying it.
I may be wrong, of course  :dink:

http://m.youtube.com/watch?v=vZ5ItJkfLy4
As Norman rebuilds naive concepts such as nothing and something only make sense in relation to a something called a page!
Similarly the concepts only make sense in relation to a one thing called a region!
They make n sense at all in relation to an infinite space! We literally have no conception of an infinite space unless we mean a process we sense we can never complete!
Such a region exists only with the process of measurement inherent within, and the sense of our mortality, and exhaustion!
Such a region therefore exists as inherently fractionable that is as a fractal topology .
The possibilities are indeed endless!

I agree.

But I guess it is different for everyone, we can not confirm we are talking about the same thing:

https://www.youtube.com/watch?v=evQsOFQju08

http://m.youtube.com/watch?v=8OvltlOA8XE
Any philosopher has to tackle the use of the word "time" . But many will ignore the Psychological basis to the discussion . Culturally to assert something is true is to challenge or assent to a cultural or social norm

I am not sure whether this is related but it might add something to the discussion.


Within several weeks I will start my study for analytical chemistry. One of the things I am going to learn is to correctly measure and observe things. I can of course start making measurements but that does not mean that the measurements are accurate enough to make up a correct conclusion out of the observations. I will learn how to analyse the measurements in such a way that they can be considered to be good enough.

I have seen that for my education I have to assume some generally accepted standards are right. And from there I have to start making my own conclusions. That way there actually are for everyone, two types of data:
The observations someone makes and the observations of others that are observed by that person.

In the case of fractals we could say that for instance the evolution of nature caused fractals to be there. Or it could be that our own observations different than we think they are. If they would be then we would not have to worry but observations are different of everyone. If there were no different observations we would not have this discussion! :)


This is just an observation. Feel free to correct me or add you own thoughts. ;)

The scientific evidence says they are real.

Take the general study of turbulence from which the term of self-similarity originated.
Virtually everyone who has studied turbulence has independently noted the many levels of self-similarity that are fundamental to this exceedingly common physical phenomenon.

You could say the fractal self-similarity is real, but how could you explain the fractal properties of turbulence as some kind of mental illusion?
I doubt that one could make a convincing, or even sensible, case for the latter.

RLO
http;//www3.amherst.edu/~rloldershaw

The question at the head of this thread is a labelling question.
Any dispute would be in defining what characteisics merit the given label. Most scientists are trained in visual characterisation, and indeed a fractal topology impies space.,but it is space to experience the 8 senses; the 8th sense is a sense of time. That is a sense of process sequence and stage duration.
It is the synaesthesia of all these sensory experiences that I learn to call reality, and that is totally subjective. Later I learn to perceive a part of my experiential continuum as objective. It helps if it stays fixed as I move relative to it.
The collection of these perceptions I learn to call inner and outer space, and because they are learned  labels I am subjectively unsure if I agree with the others who use the terms, but social interactions enable me to perceive a range of semantically similar applications .

As a subjective entity I learn to anchor perceptions against verbal and non verbal agreements with others but it is my acceptance of a fixed set of relations that transforms my experiential continuum into a rigid model of the continual flux of experiences. I learn to develop confidence and certainty as experiences of my model along with the perceptions of identity and independent individuality.
 I learn to protect this certainty by many internal referrents, these experiences are either fixed or dynamically stable like a movie loop for example or a song set on replay.

Later I apprehend the role this collective plays in developing my  identity, understanding, personality, intuition and social adjustment and acceptance. Around teenage years my brain is developmentally streamlined and in a process kin to lobotomy certain connections are lost others strengthened , all I know is maturity tends to establish a fixed model of my experiences which I have learned to call normal.

But I notice some have differing expressions beliefs personality systems etc and if I am lucky I learn to apprehend these differences as structural outcomes of the same model of development.

So now I apprehend that my reality is unique to me, the other reality is a convention I adopt first from my parents and family, then peers and schools and eventually from a larger collective labelled as society. .
I learn if I meditate that the "I" is not constant and so every day it must refresh itself , but if I refuse to acknowledge these refreshes I become stuck in a mind warp. In that case I believe certain things are objective and independent of a me that is subjective and I therefore have to decide if I have created the objective world or if it has created me!

But if I do not get stuck like this I can accept that a process has resulted in an experiential continuum such as I have described which is influx and can be expressed in any way it chooses to accept a perception collection as fixed.
Thus an experience of the cyclicity and spirality in perceptions underpins the apprehension of "almost similarity.". For me this is the heart of fractal topology a perception not of self similarity or evn similarity, duality trilingual etc but of almost similarity.

The question about labelling is in my opinion counter productive if it has a fixed answer! But very productive if it stimulates thought meditation and growth in awareness .


These considerations are not necessarily my own. Classical Philosophy has pondered these things from of old . The challenge is not to forget or ignore these deliberations of the enquiring mind for the latest technological innovation or the next scientific advance!