Hi and help!

[Melancholyman]

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

[David Makin]

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

[Melancholyman]

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

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?

[David Makin]

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

[Melancholyman]

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

[David Makin]

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

[Melancholyman]

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!

[Melancholyman]

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?

[David Makin]

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 ?

[Melancholyman]

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.. 

[David Makin]

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........

[Melancholyman]

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

[David Makin]

Interesting

> This is not really a "rational" philosophical argument

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

[Melancholyman]

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  

[David Makin]

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