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Hi, first I want to say hello to everyone on this forum. I just discovered the Mandelbrot phenomenon in the last few weeks, and it has had a tremendous impact on me. I have been trying to understand the Mandelbrot set, and I've been reading about it, seen a movie and an introduction, both from links on this forum. What is puzzling me is this : "The Mandelbrot set is defined as all the numbers that DO NOT grow exponentially within the function", and this is defined as the circle with a radius of two on the complex plane. In the introduction video the complex number 1 + 1i is iterated, and falls outside of the circle. Does this mean that this number DO grow exponentially? Isn't it obvious that any complex number will fall outside of the radius after just enough iterations? It seems as there are NO numbers whatsoever in the Mandelbrot set? I just need some clarification that I haven't got from reading the web, I hope someone is willing to enlighten me here! Thanks in advance all!

Best Regards
Erik

Although all numbers outside a radius of 2 of the origin will escape the standard Mandelbrot Set, the converse is not the case i.e. some points inside that radius also escape the Set - hence the Set is not defined by a circle, rather we get the classic "beetle" :)
In fact it has been proved that if at any time during the iterated orbit z(n)=z(n-1)^2+c the magnitude of z(n) is >2.0 then the point c is outside the Mandelbrot Set.
As an example the point c = (-2,0) is considered as being "inside" the Set since that coordinate gives a point attractor of (2,0) as follows:

  0^2 - 2 = -2
 -2^2 - 2 = 2
  2^2 - 2 = 2 etc.

Whereas the point c = (2,0) is outside the Set:

  0^2 + 2 = 2
  2^2 + 2 = 6
  6^2 + 2 = 38 etc.

For more on orbits and attractors see:

http://www.fractalgallery.co.uk/orbits.html (http://www.fractalgallery.co.uk/orbits.html)

Thx alot! I will try to read up my mathematical knowledge and other stuff. But when do we say we have a complete Mandelbrot set? After how many iterations? The "beetle" is just the starting point we have chosen then? Since it could very well be the circle of radius two? It seems to me that the important thing is not which numbers that stays within the set, because it seems as no number does after enough iterations, but the structures that are created when the points shoot off to infinity?

No there are many points that "obviously" stay "inside" - such as (-2,0) as I mentioned, or more obviously (0,0) !
The beetle is the defining shape but the resolution required to show the true boundary between "inside" and "outside" would have to be infinite.

One can only render any fractal to a given resolution/mathematical accuracy, if you render z^2+c in a window 640 pixels by 480 pixels with complex coordinates from say top-left (-2.5,1.5) to bottom-right (1.5,-1.5) and render it such that any points where the magnitude of z(n) exceeds 2.0 are "outside" allowing a maximum iteration of say 1000, then change the max iteration to say 1000000 you will find no change in the image - the larger the iteration count you use, the closer to the true fractal you get but at that viewing resolution using 1000 iterations is already as close to the true fractal as you can get at that image resolution.

It is also possible to have bailout tests that confirm points as being "inside" the Set in addition to the standard test for confirming that points are "outside" - this can be done by testing for point attractors or periodic attractors, testing for other types of attractors is decidedly more tricky.

I should add that to understand how the boundary of the Set can continue shrinking at higher iterations but will never get smaller than a particular size then you need to understand the principle of mathematical/geometric "limits" for which this may help a little:

http://www.fractalgallery.co.uk/cantor-set.html (http://www.fractalgallery.co.uk/cantor-set.html)

Ok thx! I will read on a lot before asking any more beginner questions. Thx again! It's a bit clearer now though! :)

Don't let the reading stop you asking questions - there are lots of folks here who can provide answers - many can provide better answers than mine :)
Actually I find that more reading usually prompts more questions - I think there's a quote "The more we know, the more we know that we don't know" or something like that ;)

Oh yes, knowledge is like a fractal. You know one branch, but that branch divides into two, and that one into two others, and so on ....ad inifitum   :) I am more interested in the idea of the fractal and Mandelbrot rather than the mathematical properties. I am a philosopher you see with limited mathematical knowledge :) Cheers!

I've been reading some more. If I have understood it right, on the Mandelbrot the "black area", the "beetle" are orbits where the attractor is not infinity. It's either a finite number, or a sequence of numbers. And these attractors must be smaller than any complex number "c" outside of the circle, with radius 2?

Actually the attractors are smaller than or equal to 2, but essentially correct :)

Of course the 2 is specifically proven for z^2+c, for other fractal formulae a similar principle holds but I don't know how many others have a similar proof for the boundary limit.

On a philosophical note - would you agree with the concept of existence as a multi-fractal multiverse ?

Could you elaborate a little bit please, I am not sure I quite understand what you mean by a multi-fractal multiverse..  :'(

Built of fractals that are made of fractals and dividing into multiple universes at every instant these being all the possible results of the "now" - in some cases for example I wrote this and you are reading it, in others I died before I finished it or you died before reading it or Trifox deleted fractalforums, or Trifox banned one or both of us from fractalforums or........

I just wanted to say that I am in the beginning of my philosophical education, and that I am rather young (25), so everything I say should be taken with a big load of skepticism. :) Regarding your question, intuitively I don't really accept it, although I embrace the idea of universes within universes, or universes within fingernails and so on. In regards to what I just said, I would have to accept your notion of the multi-fractal multiverse.

 Think of perception. When you look at something, you NEVER see a whole object at once, in fact, your gaze is centered at an infinitely small area, a point. It's easy to figure this out. Start by looking at some rather big object, like a bottle of vine or something like that. You have to move your gaze around before you actually see the whole object even though the object is within your perceptual range. Try to see where your perception "really" is centered when you are looking at an object. It seems to me as the gaze is locked at an infinitely small area, and from that point everything you see gets more and more blurred. In fact the only thing you see clearly is that point, and this point is "empty" or like the "black area" within a Mandelbrot set. This point can be seen as any point within the Mandelbrot , where when you zoom in endless structures unfold. But, we are living in a physical world, and our brains and  our physiological attributes "fixes" us at a certain distance (but I believe we are able to see, infinitely many iterations)  from this fractal. So, from this point where your gaze is fixed, originates fractal like structures, which creates the physical world we are all experiencing, all we see are "orbits" that shoot off to infinity, but it appears solid since we cannot zoom in, but it is constantly renewed, but since our perception is locked at a certain distance from this fractal, and there seems to be rules applied to what color objects have (just as we color fractals) the physical world "seems" fixed, but it is not. We are so to speak "locked at a certain distance within this fractal", and it would be possible to zoom in or zoom out indefinitely, and this would represent different universes.
I want to say that I myself do not take this seriously, it's just an idea that popped in my mind, and most often they are false. But I thought it was an interesting idea. It is obviously flawed, but there might be some truth in it. This is not really a "rational" philosophical argument :)

Interesting :)

> This is not really a "rational" philosophical argument :)

Give me irrational over rational every time - apart from anything else it gives more choice ;)

Also, it is possible to travel through this endless fractal. Since our imagination is not bounded by any physiological attributes (well it seems not to be). So close your eyes, put on some psytrance and just shoot off to infinity  O0

Is "Melancholyman" from the track by the Moody Blues ?

Yes it is indeed  ;D

David Makin: Your idea of the nature of the universe (and beyond :) ) is very close to mine :)

Half comes from relativistic and quantum-phyiscs and the other half from the fact that each time "step"'s state is the result of applying the whole set of rules (physical formulae, many of them not yet found) to the state of the previous time step.

However, time seems to be a creation of our perception.
There must be something more general, outside of time.

The trick would be to get fractals in one step; to find the closed form of the sum. :)

(Hmm... that might be one of the shortest tries to explain this, I ever wrote :)

@ Mandelbrot set definition:

There are three kinds of points:
Those that converge against a fixed point, those that converge against infinity and cycles.

Afaik, the "beetle" is all that converges against a fixed point (like 0), the outside is what goes to infinity and the border is basically all the cycles.
Though I'm not sure about the cycle-part right now...

The circle of radius >2 is just where it becomes obvious that the points escape. There are certain geometric propperties which could be used to define the shape of the M-set more clearly but they are all way more complex and most likely would take more time to calculate than the additional information would shorten the needed time.

For instance (dunno if that helps for the outside but it could for the inside) the main "circle" of the Mset is indeed a circle (while all the other copies are slightly distorted) and the thing the circle is attached to is a true cardioid.

That's at least what I've read :)

But the border around the "beetle", isn't that border getting smaller and smaller after more iterations? One thing also that is boggling my mind, is the fact that what I've heard the Mandelbrot was "discovered". Isn't our reality totally anthropomorphistic? Isn't math a human invention, what if we created fractals, and we are creating our future, and the Mandelbrot wasn't discovered, it was simply invented by ourselves?

Yes the iteration bands get progressively thinner as the true fractal boundary is approached.

IMHO maths itself was "discovered" rather than "invented" - the rules of maths are absolute, like the rules of logic :)
Similarly fractals themselves were "discovered" rather than invented and in this case it's straightforward since no-one can deny the fractal nature of a coastline, or clouds or broccoli or ferns or lungs etc.
Also I'm pretty sure that the same applies to the Mandelbrot Set because, although maybe the exact equivalent of z^2+c hasn't yet been found in nature, studies of the variation of magnetism in materials at different temperatures and pressures show fractal behaviour which when mathematically modelled results in fractals that contain z^2+c type "minibrots" - just look up "Magnet fractals", or simply try them in Fractint or Ultra Fractal :)

Ok sure, there is obviously a connection between fractals and nature. And we are able to describe nature in a more elaborate way with fractals. But how do we know that fractal geometry isn't just that, an elaborate way to imitate and describe reality. The true essence of the universe might not have anything to do with numbers, math, or fractals, it's just happens it's an improved way of describing it. I am just trying to countermeasure, I'm not saying the laws of logic aren't absolutes, all I am saying is, how do we know that when we are discovering we are in fact inventing it?

Aha - I take it on faith that we've just "discovered" such things as logic, maths, entropy.....whereas things like philosophy or spiritual beliefs are things we've invented - I do believe there is a "God" but that's purely a belief and a faith and until proven otherwise to me this "God" is an invention of Man just like philosophy and other spiritual beliefs, but even if just an invention that doesn't make such a "God" an impossibility.
Having said that I also think spiritual beliefs and such are on the whole far more important to the well-being of humankind in the long run than logic, maths, entropy etc. In other words I'm not really a materialist ;) I should also state that that doesn't alter my extreme suspicion of all organised religions (plus dislike of some and of all fundamentalism==intolerance) (though I am "officially" Church of England).

Should not this discussion be moved to the new Philosophy Board within the Forum ??     :evil1:

Ok, thx for your opinions. I myself am trying to make up my own mind about such matters. To take a radical example, when we discovered America, ofc the land didn't just appear out of nothing, so it cannot have been invented. But there are other areas of reality, where the border between invented and discovered seems to erode a bit. It is also quite common among known philosophers, to believe that the world is in some way, subtle or radically, "made" by us. I would like to add one thing aswell, you mention the laws of logic are absolutes, but isn't it true, that in modern quantum physics the law (or principle) of the excluded third has been altered to fit empirical data?




 :police:  Well it has strayed away from it's original topic.  :-\

I wouldn't know - but Physics is not Logic.
IMHO if Physics defies the logic or maths behind the modelling then there are more variables to consider or the model is incorrect in some other way.
In the same way as Maths, to me Logic is about the actuality whereas Physics is restricted to the merely apparent (observable) ;)
Some things that are actually true may never be observable directly or provable with satisfactory certainty even indirectly and hence can never be part of accepted Physics.
Also IMHO logic/pure maths are not materialistic in nature but applied maths/physics/chemistry/biology are.


I guess any further such discussion should now be be posted on the Philosophy board :)

My 2 cents...

There's a theory called Bohm theory, which defines a deterministic quantum world. It states there are "pilot waves", and whenever a superposed particle is observer and must collapse into one of its states - something that would otherwise be random, an idea which many reject - the pilot waves value at the specific space-time location determines which state it will collapse into. I don't know how well substantiated this theory is, but I like it a lot, and I've seen it mentioned in several respectable science magazines (not tabloids). I like it because it allows a scientific explanation of basically whatever religion you want: For instance, one could argue "God" is actually a collection of pilot waves working in unison to make minute changes everywhere and make apparent magic happen; to create the human race by changing some particles in some faraway galaxy; to spread the will they describe, the will which we say is the will of God. I'm not saying I'm Christian, I'm mostly agnostic, but that's just an example. Also, things such as Karma or whatever you want can be described with this idea. Beyond it, I don't have anything to say on religion/philosophy... :-D