Welcome to Fractal Forums

Hello eveyone.

I was thinking about fractals and I saw some problems:

-If fractals are really patterns that repeat an infinity number of times it will take us an infinity amount of time to prove that. So there are two options:

Somewhere there is an end of the fractal, but that means the the fractal isnt eternal.

And if the fractal is eternal we cant prove it.


-There are people who say that if fractals repeat themselves an infinity amount of times, there will be an unlimmited ammount of versions with differences. But I dont think i will see a real circle in the sierpinski triangle.


So... on these problems dont really have answers, or do they?

What do you think about these problems?
 

imho this is false logic (when I understand your term "eternal" ight). There are infinitely many natural numbers and we can proove easiely. You only need to show, that regardless which number you create, you can allways count one up.

I think what you mean are undecidible points. The borderline of the mandelbrot set is undecidible, 'cause you need an infinite number of iterations to decide whether the point is inside or outside the set. The proof about the property of that borderline is something completely different and does not need to be eternal.

This is why I am constantly pointing out the difference between a Mathematical Construct, and Reality.

Infinity, and Zero, are - in my view - purely Mathematical Constructs with absolutely no corollary in The Real World.

It is easily proven that Fractals fit the Mathematical definition of being Infinite, however do not confuse infinite Extension with Infinite Encompassing of Things.  As an example, a straight line is, by definition, Infinite.  Yet one will never find any variation at all.  The set of all fractions between "Zero" and One is infinite, but will never include 3/2 or any ratio greater than One.  Infinite variations within set limits.

Then there's the idea that some Infinities are bigger or smaller than other Infinities.

So just because it's infinite does not mean it will contain every possible thing.  Infinite variations do not necessarily contain all possible outcomes.

While I would quarrel with the notion of zero having no real-world corollaries (there are zero pink unicorns in my office right now), as has been stated, there are many finite ways to show that a mathematical construct is infinite.  And, just because something is infinite doesn't mean it will contain every single thing--the decimal expansion of 1/3 is infinite but contains zero fours.

In the specific case of the Mandelbrot set, the boundary is infinite in depth and in detail, and while it looks like it repeats, it doesn't actually repeat--each mini Mandelbrot set is slightly different from every other.

Linguistic Trickery!  :clown:

Zero is not included in the set of "Natural Numbers" (or "Counting Numbers") so one can not use it in that fashion without bending the rules of Math at least a little.

Yes, there is a recent movement in Math to include Zero by changing the definition of Natural Numbers from it's original "All Positive Integers"  (which excludes zero) to "All Non-Negative Integers"  (which would change it to include zero)  but this movement is a direct result of me and my minions rubbing it in the faces of Mathematical Philosophers.

If the only way they can shut me up is to change definitions which have stood for centuries, then I must have a pretty Powerful argument!   :educated:


EDIT - Since this is the internet, and my "tone of voice can not be heard), I am compelled to point out that this comment is laden with Sarcasm and is only half serious, and half in jest.  I had hoped the smileys would convey that but it's best to say so directly, just in case of misunderstandings.  I think the point is complicated and not so easily dismissed due to the definition of Counting Numbers.

Infinity is a mathematical construct, yes. But it is a very powerful one which does have merits in "the real world".
All that infinity means is that something does not end, that to any set of examples you have, you can add another example.
In case of the natural numbers, that means, that if you have the set {0,1,2}, you can add 3 as another example of a natural number, to make that set {0,1,2,3}.
Now you could add 5 and your set becomes {0,1,2,3,5} - note that the order here doesn't play a role. You might as well add 100 or 456451692060887 as two more examples.

The natural numbers have an additional property of being enumerable, listable or countable (all synonyms).
You can order them 0,1,2,3,4,5,...
and there is a simple algorithm to get from one natural number to the next (namely "add 1").
Since this algorithm exists, and no matter what natural number you have, you can add 1 to it, you get a countable infinity.

If you try that with real numbers, you will fail. You cannot list all possible real numbers.
E.g. there is no way for any algorithm to create an exhaustive list of all reals.
The real numbers are uncountably infinite.
This notion of infinity is trickier.

However, your given example of a sierpinsky triangle is very rigid. It repeats in a simple, obvious manner. (It is strictly self-similar) In that example it is very clear that:
- you will NOT, in fact, get all shapes imaginable
- the repetition can be executed indefinitely and thus no matter how deep you zoom, you can see more of the same.

The Mandelbrot set, however, is only "pseudo-self-similar" - it "almost" repeats itself over and over again.
There are an infinite number of shapes in it.
This can also be easily proven: There are sequences where, if you zoom in, you get patterns that double their symmetry each order of magnitude. You have an (almost) bilaterally symmetric pattern which, if you zoom in, becomes an (almost) 4-way-symmetric pattern, which, as you zoom in, becomes an (almost) 8-way symmetric pattern, etc. You can see such paths in lots and lots of deep zooms out there.

What is not guaranteed is, that the M-Set actually contains all possible shapes or, more precisely, that it may approximate all possible connected shapes arbitrarily well.
It certainly does approximate all possible Julia-sets. But you can classify Julia-Sets by overall appearance and there are only a few types that "feel different" from each other with smooth transitions in between.
Non-Julia-Set-like shapes are a bit harder to find but if you look for it, you can often find it anyway, given some experience. There are pictures of the Mandelbrot set approximating all letters in the alphabet for instance. And a bunch of faces too.

Come to think of it, I'd love to see the corresponding Julia sets. Technically, they *should* show those same patterns, right? The deeper you zoom into the M-Set, the more it should look like the corresponding Julia set.
But Julia sets are way less variable in their appearance, than the M-Set. Could you get Julia sets that look like letters?

That being said, there are so-called Compositional Pattern Producing Networks (CPPNs) which are like Iterated Function Fractals where you use a different function (from a predefined set of functions) each time rather than using the same function over and over.
There are some fractals that come very close to this, like randomly parametrized möbius transforms (a x + b)/(c x+ d) with a, b, c, d random complex numbers each iteration. Instead of only allowing that family of functions, you could allow any other set of functions to be used.

What's possible with this is demonstrated in projects like
http://picbreeder.org/
or
http://endlessforms.com/

It's very clear that you can, at least in principle, construct any shape you can possibly think of in this way.
It's not entirely clear, though, whether it is sufficient to only iterate z²+c to accomplish this amount of complexity.
Though if that is not the case, it certainly comes close.

There also is the Zeta-Function (and a whole family of functions to which it belongs) which, in its major strip, approximates a small circular cutout of any other function arbitrarily well.
It is an open problem if it also approximates itself arbitrarily well in that strip, though if it does, that the Riemann Hypothesis is true.

This means that the Zeta-function *should* also approximate (sections of) the M-Set arbitrarily well, and the Zeta-Function isn't even defined recursively.

Originality does not make the older definition of Natural Numbers inherently better. Mathematical definitions rarely spring into existence as perfect entities. As time goes by, and more people keep using a particular mathematical tool, a definition might turn out to be a little awkward, or it might begin to appear limited when people finally learn to extend a particular concept beyond the original uses.

So mathematical definitions are being adapted over time, just as all sciences keep being refined by us as we keep learning more. Mathematics does appear absolute because the truths proven by abstract logic will remain standing even when definitions of terms adapt. This is a little bit like translating a fact book from one language into another: the information does not really change, only its encoding.

Therefor, the choice of including zero in the set of Natural Numbers is, on the one hand, arbitrary, but on the other hand it leads to pragmatic advantages. Some theorems and methods become simpler, easier to apply and to understand. The mathematicians who push to include zero regard it as fixing a bug. The other group, who would rather start at number one, dislike unelegant side effects like this one: the first element of a countable set would be element #0, the second element would be element #1, ... so they perceive an inconsistency in ordering and enumeration.

IMHO this is not a bug of starting at zero, but a feature. In my experience, starting enumeration at one is a human tradition. Recall that humanity discovered numbers 1, 2, 3, ... quite a bit earlier before we recognized "nothing" not as a special case, but as a number 0. More recent uses of (non-negative in loose sense) integers, especially those inside computers, generally work best with zero as the first number.

What currently doesn't work is changing human language and thought to things like "the zeroth element" and so on. Traditions don't change quickly or easily, especially not when the cost of change is high and the benefit is small.

I'd love to see the alphabet and the faces! Do you have any links?

where are the banners? Currently, it shows me none. Is this a bug?

Anyway, I do not have a link, but one of the banners used for this very forum actually shows "Fractal Forums" spelled with pictures taken from the M-Set. And recurring letters (like the two Fs) are different too.


As far as enumeration goes, I think that there are some instances in which starting at 1 makes more sense and others where starting at 0 does.
And there even are weird, rarer circumstances where it makes more sense to start at, like, -1 or -2.

Overally, though, having the natural numbers start with 0 brings a lot of benefits with little to no downsides.
And you do not need to index things with precisely the natural numbers. You can define an arbitrary index set that suits you for a given concept.

In the end, belief in Zero and Infinity existing outside of Mathematical Constructs is a personal choice we all must make for ourselves.

Neither is subject to Empirical Proof.

That said, there actually are places where people choose to count beginning at Zero in the real world.  Our local State Lottery Tickets begin counting at Zero  (a book of 300 tickets at stores runs from 0 to 299), and as if to begin indoctrinating our children to the notion - certain Comic Books have Issue #0 in their runs.  We even technically begin our own age counts at Zero because our first full year of life we are "less than 1" and count our ages in years!

But I will never be convinced of the possibility of the Infinite - either infinitely large or infinitely small - as anything other than a thought experiment.

One with value, but not one we can ever experience.


Not sure!

They do disappear for me on occasion but I always wrote it off as my security, ad blocking, and java script inhibiting software.

They always return, so I never raised any questions about it with the Admin.

If you ever want to see the collection of banners we use, they are right in the gallery!  http://www.fractalforums.com/index.php?action=gallery;cat=27 (http://www.fractalforums.com/index.php?action=gallery;cat=27)  You can see them all there, or even contribute your own creations which will be added to the rotation   O0

Thanks. Unfortunately I cannot find any Mandelbrot letters...

The question of fractals being fractal is something I've been thinking about as well. How can we be sure fractals are fractal? It seems unlikely the mandelbrot set will ever stop, but how can we be absolutely certain?

My avatar contains a few fractal faces if you like.

The corresponding julia sets will also look like letters at the same zoom level. Julia sets are not less variable in their appearance. Actually the mandelbrot set is a map of all julia sets. Consider that the set of julia sets represented by the mandelbrot set is much "larger" in a way than the mandelbrot set itself (although bot contain an uncountable infinitude of elements).

I know that. What I'm saying is, that each individual Julia-Set is less variable in appearance than the entire Mandelbrot set.
However, if you take the set of all Julia-Sets, you'll get the same things, since the Mandelbrot set directly represents that set. In a way, the Mandelbrot Set IS the Set of all Julia-Sets.
Well, to be precise, it is the set of all Julia-Sets that are connected.

As far as the banner goes, I can not find it either. But I'm absolutely sure I saw an image like that. Maybe it was no banner after all? The gallery is so huge by now...
Can anyone else remember anything about an image containing letters found in the Mandelbrot set? Even if I'm not so sure anymore that it actually ever got to be a banner, I am certain that it was on this forum.

I don't recall the image, and it sounds like one I'd notice!

Granted, I don't spend much time in the gallery, so maybe that's why.

If I see it, I'll be sure to post here   O0


EDIT - A quick web search turned up this : http://tararoys.deviantart.com/art/The-Fractal-Alphabet-37524134  A big version of the image can be seen via the "Download" link on the upper right.  Not all Mandelbrot, but it's a start.

Maybe it was this video?
https://www.youtube.com/watch?v=Bv6Snj9czLs

Well Infinity is the tip of the iceberg.  Real numbers: pure abstraction.  But wait abstraction is pretty much the whole point of the exercise(s) isn't it.

That video is a good start - it contains three different letters, two different words and a true sentence.
But S, I and X ought to be amongst the easiest to find in the M-Set. It's absolutely littered with those.
Letters like F or Q or R should be a bit harder to find.

Though what I saw was most definitely a still image.

So let's collaborate and create the alphabet then!!
Wouldn't it cool to be able to write your name in Mandelbrot letters?
http://www.fractalforums.com/mandelbrot-and-julia-set/mandelbrot-alphabet/

Hmm, I'm not quite sure if it would even be possible to find a well-defined F, Q, or R in the mandelbrot set. I say that because it seems like virtually every shape in the Mandelbrot set exhibits some sort of radial or spirally symmetry. And F, Q, and R are not radially symmetric. But I'm just speculating.

That's kinda the challenge.
The less symmetric the letter, the harder it would be.
Though what I remember, and the longer this discussion is going, the more it looks like I remember wrong, was just a bunch of square cutouts from a Mandelbrot Zoom, spelling out FRACTAL FORUMS.
No whole sentences.
Of those, if I had to guess, R would be the hardest to find. C, L, S, U should be really easy. T and A are at least bilaterally symmetric. They seem doable.

Going for a full alphabet would be interesting too.
It would certainly put the claim that the M-Set contains pretty much all shapes to the test.

But if you are allowed to clip the view where you want, maybe it is possible to create also asymmetrical letter...

I read most of the posts. I think I can change the way i think about fractals. Some of the answers were actually very simple but i didnt think about the fact that there are several kinds of fractals with different properties.

Glad you came back to this thread   O0

I was worried we scared you off...   :nastyteeth:

It is only natural for our points of view and opinions to change  as we learn and experience more stuffs.

Change is, after all, the basic process of all life   :educated:

Just look at how we sidetracked it it into talk of finding the alphabet in The Mandelbrot Set!

It is always funny to see people change subject, it could give other people other ideas to solve things.

This was my first real post so I didnt expect so many replies. And I was actually scared of the long posts.

Btw the the letters in the mandelbrot could actually be a nice example in one of my other posts, maybe im going to change that.