Response to any redefining of fractals

[Thunderwave]

...quote from http://www.fractalforums.com/index.php?topic=4040.msg21503#new...

There's a confusing thing going on about the definition of fractals. 

But as to everybody that wishes to make up their own definition of the term "fractal", that has gotten way out of control.  Seems like everybody and their dog thinks they can call just about any image as "fractal" and get away with it.

I would rather stick with a definition closer to what Benoît B. Mandelbrot intentionally meant when he coined the word years ago.

Hey, my cat said it was a fractal!   j/k

Couldn't it be whatever actions lead to developing a fractal dimension near self-similar art be considered a fractal?  Mandelbrot also said that the most interesting fractals have a chaotic attribute to them.

 I'm somewhat frustrated by (present company excluded) mathematicians only explaining fractals in mathematical language.  Then they say it can only be explained that way.  As my friend who laughs at me when I try to learn the language of math, does.  It's far beyond me some times.

 I have a book that explains fractals in music and explains fractals by scanning images, shrinking them down, making x number of copies, placing them back on the scanner, shrinking down the same way, making the same number of copies, and then repeating the process the same way at least 3 more times.  I get the idea of applying a pattern (idea) unto itself is a fractal.  It creates self-similarity and dimensions of itself at many areas of itself, like my photoshop fractals and Matty's fractals. So my simpler understanding of what fractals are, is an idea, theme, or action made up of itself and refers to itself. 

Now that is an equation right, but explained without the fancy numbers and symbols that take me forever to memorize and understand.  Believe me I love math and the concepts of number theory and all of it, but it is taking me forever to understand and if ever catch up with the latest mathematical symbols and concepts.

I admit I made many mistakes in referring to a thing as a fractal when it isn't.  Such as the Moire Effect and the Fibonacci Sequence.  These are all things made up of itself but the chaotic aspect is not there.   IFS is not necessarily chaotic either though, right?  So this is where I get a little confused.

Okay, if I am wrong about all of this and the concept of fractals, can someone please make it clearer, because I really intend to learn it all.  If only I had more time and more brain power.

I am currently reading: Godel, Escher, Bach: an Eternal Golden Braid by Hofstadter.
It's taking up all my free time to read it and research to understand it.  Plus I'm going through Gleick's Chaos, Madden's Fractals in Music, Chaos and Fractals: New Frontiers of Science, the famous Mandelbrot's book, The Fractal Geometry of Nature, DVD teaching company lessons on Geometry, Algebra, Calculus, Trigonometry, Science.... etc...

I feel I have lost all highschool knowledge since 1996 and my BA degree was only music.  Not one math class was taught at Cornish where I graduated (exactly how is that possible?).  So when I worked on fractal music, due to boredom of only classical music and jazz,  I decided to learn all I can.

Sorry for the long winded rant but I feel this is the only place where I can get support but yet, I feel out of place for not being up on my math skills.  I can't hold a mathematical discussion here with anybody, because it's beyond me. Which is why I hardly respond to some comments that don't make any sense to me for fear I may make myself look stupid as I may have already.

Thanks all for doing such great work on applications and new ideas of fractals and 3D, 4D(That blows my mind), and perhaps maybe soon we will transcend beyond time.  Time is the 4th dimension right?

[David Makin]

It's good to remember that even the great man himself is very dissatisfied with his long quoted definition of what is fractal.

There are really 2 or even 3 issues here, not just one - the main 2 are:

What is a fractal ? (in the strictly mathematical sense)
What should be referred to as Fractal Art ?

The other that I also consider to be a valid question is:

What is a fractal ? (in a general sense)

Now both the numerical and geometrical fractals of various types (escape-time, IFS, L-systems etc.) quite obviously fulfill the first and third of these - and the images of them satisfy the second...... or do they ? Some formulas or specific parameters do not e.g. the Julia for z^2+zero or an IFS consisting of two affine transforms with simply a scale of 0.5 and different translations (== a straight line), or the Cesaro Sweep, or a Peano Curve.

My take on this is different from most, I simply prefer to think of *everything* as fractal and straight lines, triangles and even circles/spheres etc. as special cases of fractals i.e. fractals aren't the exception, the platonics and differentiables are the exceptions and in fact just a special case of fractals. To me it's about time math/science was reworked from this starting point, but I guess it may take a while for such a thing to happen

The question of what constitutes fractal art to me is therefore moot, since my defdintion of "fractal" is effectively universal !

However in the context of what most people consider to be "fractal", I would say that any image (or animation, or indeed music) that is constituted primarily (over 50%) of fractal form to the average viewer/listener familiar with fractals should be considered as "Fractal Art".
If less than 50% of the appearance/sound can be attributed to fractal origin then the item should be categorised as either "mixed media" if no one constituent is >50% or the appropriate category such as digital manipulation or photo manipulation etc. depending on the source of the other elements in the object.

Edit: Just to emphasize why I suggest reworking everything from a fractal starting point - consider for instance the possible 1D curves joining two points, how much larger is the infinity of curves that are non-differentiable compared to the infinity of curves that are differentiable ? At the moment the best way of describing this larger set (maybe all of it or at least a large proportion of it) is by means of fractals.

[Thunderwave]

David, you are so kind.  I agree with you 100%. 

Just as I noticed many issues floating around this world, everybody keeps trying to categorize things as just one thing only and any mix is not allowed.  But as most scientists in this modern world realize is even the minutest thing effects the greatest thing (butterfly effect).  So when we try to classify things as one thing only; we miss the general aspect of it all. (I learned this from what I have read in GEB by Hofstadter).

I'd say that fractals tell us there is more to a thing then we can see.  We have to examine every aspect to truly understand one thing and each aspect to one thing is infinite.  Unless we can live forever, we will never be able to understand it all.

[Nahee_Enterprises]

I agree with over 50% of what David has stated, but definitely not 100%.   
 

[Thunderwave]

I agree with over 50% of what David has stated, but definitely not 100%.   
 

Understood!

Nahee, you represent knowledge but you don't reveal it, per se.  Interesting "per se" means of itself, that is a fractal huh?

[jehovajah]

Do not sweat the hard stuff!

Words and symbols and marks and logos only have a meaning in you not outside of your experience. So take your current understanding and let it grow naturally.

Even the President/ Queen has to pull his/her  pants down to poo! to misquote Dylan.

Do not be overmuch righteous nor overmuch wicked for 'h¶h will determine whats right for you in the end: to misquote the bible.

Paper tigers.. one and all!

So chill out man, and play. One day you will wake to find peace of mind down here below, to misquote Coldplay.

Nothing is fractal! Everything is FRACTAl, man!

Keep op the good work, and maybe share insights from what you find out in them thar books, eh?

[Thunderwave]

Paper tigers.. one and all!

My favorite Quote!

Thanks, I guess I made it out to be too big of a deal.  Just like the square-root of -1, it's not that hard to understand.  It just takes me longer to figure that out.

[hobold]

I'm somewhat frustrated by (present company excluded) mathematicians only explaining fractals in mathematical language.  Then they say it can only be explained that way.

There is a common misconception about mathematics. Mathematics is absolute, detached, in principle, from the material world. Any connections are there solely because we are much more interested in formulas that model, or emulate, some qualities of the material world. Mathematics is only really limited by the imagination of those who develop and employ it (more fundamental limits of logic notwithstanding).

The misconception is that most people, many mathematicians included, believe that "absolute" implies "objective". But the surprising truth is that mathematics is the most subjective of sciences. Modern mathematics, all of it with no exception, is based on a set of axioms. Those axioms are true by definition, and have been chosen very very carefully, but they are still just an agreement. A subjective choice, ultimately. Other sentient entities in other universes with other natural laws could very well choose their own, different set of axioms, and derive an equally useful mathematics from there.

(Both resulting theoretical systems of thought would be compatible in a fundamental sense. But the different factions of mathematicians would have no concrete incentive to explore the respective "alien" branch of mathematics.)


So, at long last, here is the reason why mathematicians are unable to explain the meaning of the word "fractal" in colloquial terms: they made up that word, in a language that has only coincidental connections with reality. :-)


(This explanation was exaggerated, but you get the idea.)

[M Benesi]

  Fractals are fractals man.  It's that simple.  You know, even the word makes different patterns in different minds, so it's fractaling the hell outa a bunch of gonzo freakazoid tube spazzers.  Not that it's any different from any other word out there....

http://www.etymonline.com/index.php?search=fractal&searchmode=none

  But then again, it's a newish word, with not a lot of history.  Does there need to be a definition?  Well, look at all the bullshit arguments that the dual definition of "Free Will" has caused, one side arguing that people do stuff of their own volition (true: I am writing this willingly) so free will exists (as per the pertinent definition), and the other side arguing that there are prior factors that determine our decisions (true: natural law, existence of stuff, etc.) so free will doesn't exist (as per the other definition pertinent to this argument).   Blehhh...

Thanks all for doing such great work on applications and new ideas of fractals and 3D, 4D(That blows my mind), and perhaps maybe soon we will transcend beyond time.  Time is the 4th dimension right?

  Hahaha... well, this is a 5d video, although the 4th dimension is static, the 5th varies:
<a href="http://www.youtube.com/v/XtXyUOXWZfw&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/XtXyUOXWZfw&rel=1&fs=1&hd=1</a>

[Sockratease]

I choose to define Fractals as "Funny Ridiculous And Crazy Totally Asinine Little Sh*ts"

All the best definitions are acronyms...

This definition is subject to modification if I think of a Sillier one!

[Thunderwave]

I choose to define Fractals as "Funny Ridiculous And Crazy Totally Asinine Little Sh*ts"

All the best definitions are acronyms...

This definition is subject to modification if I think of a Sillier one!

F is for all the friends I made here.
R is for all the responses I have gotten.
A is for the apple computer I use.
C is for the care there is here.
T is for all the tea I drink
A is for the A's I got in college
L is for all the ... guess...


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LIMA BEANS!

[M Benesi]

Hilarious image Sockratease.     

[Sockratease]

Hilarious image Sockratease.    

Thanks!

I've been redefining fractals with Lissa's images for years now  (even if it's a bit hard to tell those are her feet!)  (I mean...  just because *I'd* know them anywhere, doesn't mean others would   ).

But that is a favorite of mine - if only because it took 5 Different Programs to make happen!  (Mutatorkamer, Chaos Pro, Bring It In, Particle Illusion, and ULead gif Aimator)

[Thunderwave]

Back to my really serious topic that is life and death to me.....  

I think, fractals need to be realized by my own understandings of life in general!
 
I need a description of fractals as a general concept outside of the Mathematical system, sort of like generalizing the concepts for daily life to relate!

So I am going to write one!  Maybe it's been done but I need to write an essay for getting in to a Masters program anyways so I might as well get to my books faster and deeper.  If I find any other similar ideas, I will incorporate them with my own.

I'll post more soon. 

[Sockratease]

There was an episode of NOVA called Hunting The Hidden Dimension which put it all together quite well in layman's terms  (even has interview segments ith Benoit Mandelbrot!).

http://www.pbs.org/wgbh/nova/fractals/

That may help