Are black holes giant 3d Mandelbrots?

[Tglad]

Perhaps the reason we can't find a 3d Mandelbrot is because we assume the normal one is 2d. But maybe it is just a set of 1d numbers that include imaginary parts. What if we think of the Mandelbrot set as one dimensional?
Imaginary numbers can sometimes be used as the velocity of a point. For example, if a spring is oscillating then a recording of the position and velocity of the end of the spring can be plotted on the complex plane and will build up into the shape of an ellipse, or a circle if you scale one axis.
In fact, you could model where each point lands as Z = W * Z where W is e^(i*w) and w is the spring's angular velocity multiplied by a time step.
This is looking a little like a fractal formula.. ok so the real world is continuous, not separate steps, but fractals like the mandelbrot do not change however small you make the 'time step'.

So spring dynamics almost produces a fractal, only a spring never 'escapes' however hard you twang it. But you can escape gravity, if you reach escape velocity.

So what if we model the point as three complex numbers so it has a position and velocity in 3d? We can create a formula that updates this point each time step, adding the gravity force from a star at (0,0,0). We could assume all iterations start with no velocity, then plot the path of every possible start position and colour that voxel if it doesn't escape. This sounds a bit like the Mandelbrot... OK, there is no +c. Well we could add some stellar wind, which applies a force away from the sun and so with the right scale values just adds c each iteration.
And since points follow elliptical paths under gravity, the position vector 'rotates around' the velocity vector, based on the period of orbit. Rotation is conformal.

Even better would be to change W*Z + c to (W+kZ)*Z + c somehow, because this _is_ the Mandelbrot in each (complex) dimension.
It looks a lot like adding an extra non-linear term into the linear dynamics of Newton gravity. Einstein gravity is non-linear... maybe it boils down to modelling...

Black holes !    

[jehovajah]

Perhaps the reason we can't find a 3d Mandelbrot is because we assume the normal one is 2d. But maybe it is just a set of 1d numbers that include imaginary parts. What if we think of the Mandelbrot set as one dimensional?
Imaginary numbers can sometimes be used as the velocity of a point. For example, if a spring is oscillating then a recording of the position and velocity of the end of the spring can be plotted on the complex plane and will build up into the shape of an ellipse, or a circle if you scale one axis.
In fact, you could model where each point lands as Z = W * Z where W is e^(i*w) and w is the spring's angular velocity multiplied by a time step.
This is looking a little like a fractal formula.. ok so the real world is continuous, not separate steps, but fractals like the mandelbrot do not change however small you make the 'time step'.

So spring dynamics almost produces a fractal, only a spring never 'escapes' however hard you twang it. But you can escape gravity, if you reach escape velocity.

So what if we model the point as three complex numbers so it has a position and velocity in 3d? We can create a formula that updates this point each time step, adding the gravity force from a star at (0,0,0). We could assume all iterations start with no velocity, then plot the path of every possible start position and colour that voxel if it doesn't escape. This sounds a bit like the Mandelbrot... OK, there is no +c. Well we could add some stellar wind, which applies a force away from the sun and so with the right scale values just adds c each iteration.
And since points follow elliptical paths under gravity, the position vector 'rotates around' the velocity vector, based on the period of orbit. Rotation is conformal.

Even better would be to change W*Z + c to (W+kZ)*Z + c somehow, because this _is_ the Mandelbrot in each (complex) dimension.
It looks a lot like adding an extra non-linear term into the linear dynamics of Newton gravity. Einstein gravity is non-linear... maybe it boils down to modelling...

Black holes !    

Seems to me if you leave out the assumption that there is a continuous reality that you about have a workable model! This was a flash of inspiration right? I mean at 2 .55 in the morning it has got to be a flash of inspiration, right?

[twinbee]

Tglad, you must have read my mind in a time machine, because I had exactly the same idea today (without knowing you wrote it 10 days ago). And then I go and read this post today. Weird feeling!

I was left with a vague feeling that it would still be tricky to determine the phi and theta forces/velocities, because as you know, if left up to physics only the great circle would be traversed, and we want some other kind of force acted upon it.

Anyway, I wonder if the 2D Mandelbrot has been programmed this way. It would certainly be another way of achieving smooth colouring as well as the usual technique.

[Timeroot]

but fractals like the mandelbrot do not change however small you make the 'time step'.

I would like to point out that this is false. If I may redirect you to your own thread, about the Whirlwind fractal... when fully continuous, it has zero size and the shape of [what seems to be] a cardioid. That's not to say fractal phenomena don't occur in continuous systems: if you look at the work of Feigenbaum, you will see how the same period-doublings as the Logisitic map arise in many continuous situations. And since the Logistic map's bifurcation diagram is just the Mset along the real line, it could be argued that the Mset does appear in, for example, the stability of swirling fluid under heating from underneath. If somehow one constructed a system such that only cases with a periodic solution leave traces, and then very slowly changed heat, you would build up an Mset... or at least part of it. And hey, in quantum mechanics you can have complex probabilities, I'm sure there's some context for a complex temperature. 

[Tglad]

Jehovajah, I think I'm in a different time zone than the clock is saying.
Timeroot, the whirlwind had a bug on the first iteration, different step sizes on a mandelbrot correspond to different powers, and they all have the same fractal dimension (2).. but I think that this is a weak part of my last arguement. Your insight about Feigenbaum's work sounds fascinating, I still maintain that the i part will represent the velocity/derivative rather than a 2nd dimension in space.

Here's a more general thought-
In continuous dynamical systems, I think the boundary between escaping and not escaping an attractor will always be fractal... provided that attractor is chaotic.
A single star with Newton gravity isn't chaotic, but a black hole with Einstein gravity and spinning (as all black holes do) would almost certainly be chaotic dynamics, it will be non-linear after-all.

So to me it seems inevitable that the border is fractal, ie the event horizon, and that it is 6d, so the event horizon looks different depending on how fast you are going.

This also solves the black hole information paradox http://en.wikipedia.org/wiki/Black_hole_information_paradox without requiring us to be living in a giant hologram http://en.wikipedia.org/wiki/Holographic_principle, since the 3d volume of space that gets sucked into a black hole can indeed be encoded in its surface if its surface is also 3 dimensional ie a fully space filling fractal, like the mandelbrot.

The maths of an ideal spinning black hole should produce an exact mathematical shape. I think people currently think it a perfect sphere but what if they're wrong? It might be an amazing fractal to behold.

[This is complete arm chair physics so don't take it with any grain of authority ]

[Timeroot]

Well if you take it from that point of view, then the sphere could still retain a 2-dimensional boundary, because of the equal cardinality of 2-D continuum and 3-D continuum. If there was some function like the alternating-digit one which actually made physical sense (sort of like an external angle), one particle on the boundary could encode the information for all of them.

In fact, this has always bugged me: how can one measure the dimension of an object if a function cleanly encodes two into one? How can the border of the Mset be 2D if each point has exactly one external angle, telling us everything about the so-called 2D point? 

[jehovajah]

Yes min is about an hour advanced to.

Picking up on timeroots point,my experience of step size is the increased visualisation of detail, so are you relating that effect to increased logarithmic index? That is to say z^2+c at a finer step size is equivalent to say (z^2+c)^n for example at the original step size,where n is determined by inspection at this stage.

As you may know my basic assumption is all is "iteration", That is the fundamental processes are iterative by default. I have not made a full study of dimensionality because everything has its own order, but i do not trust it as a concept to define reality. For example i have this thought experiment in which an abstract euclidian line is crumpled up into a point, or more on topic with my current thinking: spun into a point. Both these euclidian abstracts allow this to be possible, consequently so called 3d space "compressing " into a so called 2d surface is analogous.
I made the point about continuity as it is also an abstract concept, which i think has obscured the iterative nature of  some of the basic mathematical concepts. A circle or a sphere for that matter may be familiar but that does not make it non iterative in nature, and therefore non fractal in the general sense. We were scared by "monsters" in the 17th century, but now we have processors we can look at the gorgons face of fractals and not be frozen into terrified stone! 

[jehovajah]

Perhaps the reason we can't find a 3d Mandelbrot is because we assume the normal one is 2d. But maybe it is just a set of 1d numbers that include imaginary parts. What if we think of the Mandelbrot set as one dimensional?
Imaginary numbers can sometimes be used as the velocity of a point. For example, if a spring is oscillating then a recording of the position and velocity of the end of the spring can be plotted on the complex plane and will build up into the shape of an ellipse, or a circle if you scale one axis.
In fact, you could model where each point lands as Z = W * Z where W is e^(i*w) and w is the spring's angular velocity multiplied by a time step.
This is looking a little like a fractal formula.. ok so the real world is continuous, not separate steps, but fractals like the mandelbrot do not change however small you make the 'time step'.

So spring dynamics almost produces a fractal, only a spring never 'escapes' however hard you twang it. But you can escape gravity, if you reach escape velocity.

So what if we model the point as three complex numbers so it has a position and velocity in 3d? We can create a formula that updates this point each time step, adding the gravity force from a star at (0,0,0). We could assume all iterations start with no velocity, then plot the path of every possible start position and colour that voxel if it doesn't escape. This sounds a bit like the Mandelbrot... OK, there is no +c. Well we could add some stellar wind, which applies a force away from the sun and so with the right scale values just adds c each iteration.
And since points follow elliptical paths under gravity, the position vector 'rotates around' the velocity vector, based on the period of orbit. Rotation is conformal.

Even better would be to change W*Z + c to (W+kZ)*Z + c somehow, because this _is_ the Mandelbrot in each (complex) dimension.
It looks a lot like adding an extra non-linear term into the linear dynamics of Newton gravity. Einstein gravity is non-linear... maybe it boils down to modelling...

Black holes !    

As you may know i have derived these forms in my exploration and attempted to sculpt from them using Quasz.

I only now realise and acknowledge your possible influence. I come back after a brief exploration of fractalwoman and acknowledge the connectedness of thinking here. But after all this is why i am on this forum because the crown jewels are here by the bucketload!

[jehovajah]

A more detailed possible connection between and mandelbrots

[Kali]

I'm reading some old posts, and I found this very intriguing. After reading tglad's thoughts, an idea came to my mind... what if all that we see, and what we are, are the black hole's fractal boundary? Instead of solving the "information paradox"... what if there is no paradox, as everything that came to existence IS this information, being each massive b-hole at the center of the galaxies, kind of a minibrot... and the whole connected continous space-time being a giant fractal generated by some simple formula iterated to infinity? I think space-time it's an interpretation of our senses that happens to give us -as animals- more chances to survive... but what if space and time don't really exists in the way we think, and are only the interpretation that our brain and senses make of the values & variables that comes from this universal fractal?

Sorry if I'm flying too high, but I had to write this.