Triplex algebra

[Paolo Bonzini]

<Quoted Image Removed>
was the equation I tried to calculate because it stands in the top of this thread - but its false, isn't it?

Once more: these formula are okay only as long as their input does not make them undefined.  If x=y=0 you get a division by zero, not (0,0,0).

Then, you have to solve the indetermination.  Solving it can be done by plugging spherical coordinates (z,0,pi/2) into the trig formula, and this gives spherical coordinates (z^2, 0, pi) which in cartesian form are (-z^2,0,0).

I also stumbled on this problem, but now that I have explained it (multiple times), it shouldn't be hard.  If you do not understand what I mean by 0/0 and indetermination, look up limits in a basic (high-school level) calculus book. (Note that limits do not help solving this, but they should help you understanding).

[zee]

I'm sorry, I didn't saw the magnitude of this problem, because I though I can use it like the trig equations (and my C# didn't said that its dividing by zero)

[David Makin]

Just to add the obvious - that using the trig version just hides the atan2(0,0) issue and the actual results will vary from one implimentation to another (hardware and software).
According to Wiki Intel's FPU should return +/-0 or +/-pi depending on the signs of the zeroes, but in fact when I check atan2(0+flip(0)) in Ultra Fractal it actually returns -pi/2 !!
I'd be interested if anyone doing FPU coding can confirm the Wiki - and is the case the same for ARM ?
Also it would be interesting to know what GPU/CUDA return for atan2(0,0)
Obviously even the trig version needs special casing for (0,0,z) if it's to be consistent.

[Paolo Bonzini]

Just to add the obvious - that using the trig version just hides the atan2(0,0) issue and the actual results will vary from one implimentation to another (hardware and software).
According to Wiki Intel's FPU should return +/-0 or +/-pi depending on the signs of the zeroes, but in fact when I check atan2(0+flip(0)) in Ultra Fractal it actually returns -pi/2 !!

That's true, and it's a bit ugly indeed.  It's also a division by zero problem, just like with non-trig versions.  By using the trig version for the power, you can just check zri though, and special case it as mentioned earlier.  For multiplication it's a bit worse, but I think normalizing to 0.0 should be good (signed zeros appear only if you do 1/-Inf which shouldn't happen for Mandelbulbs...).

[zee]

Did you take a look at the "Epsilon-neighborhood " of (0,0,z)^2 ?



with

So the singularity will look like:



This is really hard!

I hope you see the problem that (0,0, z)^2 is only one number if z=0! Else its always an other number, depending from which (x,y) direction you are coming!

[Paolo Bonzini]

Did you take a look at the "Epsilon-neighborhood " of (0,0,z)^2 ?

Uhm, no. :-) Or actually, I tried using limits but discarded them quickly...  Good idea tabulating it like that (though of course there are infinite directions, not just those eight).  It's, at the very least, interesting. ;-)  It's related more or less to the fact that atan2(0,0) is undefined, I guess.

I hope you see the problem that (0,0, z)^2 is only one number if z=0! Else its always an other number, depending from which (x,y) direction you are coming!

I am still not certain that it means that.  It may mean, simply, that (x,y,z)^2 is not continuous around the singularity, except on the y=0 line.  I don't know enough math to understand what this means, and it's hard to visualize it, too many dimensions!

(BTW, your table is inverted: (eps,0,z) denotes the horizontal direction, not the vertical direction. Once you fix that I'll delete this line!) :-)

Before going on, I suggest you take a look at http://www.fractalforums.com/theory/the-real-math-of-the-mandelbulb/

[Paolo Bonzini]

All,

I started working on a paper summing up the current state of the Mandelbulb theory.  I'm pretty sure it is a bit biased and collaboration is not easy, nevertheless I am open to contributions from anyone.  You can read the current PDF at http://github.com/bonzini/mbulb/raw/master/mbulb.pdf and, if you have or create a github account, you can fork the repository at http://github.com/bonzini/mbulb

The plan is:

1) triplex numbers: what is necessary to draw nice pictures (more or less done);

2) quaternion representation of the Mandelbrot set (WIP);

3) back to triplex numbers: see if they make more sense now :-) (planned);

4) extensions to 4D and beyond (maybe);

(I won't be doing any consulting on git and LaTeX, but there's plenty of material around on both).

[David Makin]

All,

I started working on a paper summing up the current state of the Mandelbulb theory.  I'm pretty sure it is a bit biased and collaboration is not easy, nevertheless I am open to contributions from anyone.  You can read the current PDF at http://github.com/bonzini/mbulb/raw/master/mbulb.pdf and, if you have or create a github account, you can fork the repository at http://github.com/bonzini/mbulb

The plan is:

1) triplex numbers: what is necessary to draw nice pictures (more or less done);

2) quaternion representation of the Mandelbrot set (WIP);

3) back to triplex numbers: see if they make more sense now :-) (planned);

4) extensions to 4D and beyond (maybe);

(I won't be doing any consulting on git and LaTeX, but there's plenty of material around on both).

Hi Paolo, did you see Paul's (bugman's) earlier version of the Mandelbulb extended to 4D ? (I think it's somewhere in the original "true 3D" Mandelbrot thread).

[Paolo Bonzini]

Hi Paolo, did you see Paul's (bugman's) earlier version of the Mandelbulb extended to 4D ? (I think it's somewhere in the original "true 3D" Mandelbrot thread).

Hmm, no, I saw the Hopf map though.

[David Makin]

Hi Paolo, did you see Paul's (bugman's) earlier version of the Mandelbulb extended to 4D ? (I think it's somewhere in the original "true 3D" Mandelbrot thread).

Hmm, no, I saw the Hopf map though.

I didn't realise how far back it was, anyway it's here:

http://www.fractalforums.com/3d-fractal-generation/true-3d-mandlebrot-type-fractal/msg7711/#msg7711

[cbuchner1]

Hi,

rather than opening a new thread for this trivial idea, I thought I could post it into this thread and ask whether or not this has been considered before. This is using euclidean coordinates.

Let us pick an arbitrary unit vector (0,0,1), let's call it the pole vector (north pole vector, as it points upwards in my coordinate system). The coordinate origin (0,0,0), the end point of the pole vector and the point z span up a plane in space.

Define the squaring of z such that it entirely takes place in this plane, in total analogy to the 2D complex multiplication.
This would also work nicely with higher exponents for z.

In other words (not necessarily the most efficient for an actual implementation): Rotate the world space such that above mentioned plane comes to rest in the x,y plane, perform a complex 2D multiplication in the x,y plane, and rotate the world space back.

If I am not mistaken, the original Mandelbrot set is contained in the x,y plane.

This is so trivial, it's probably been thought of already. Would the resulting fractals look fractal at all?

Christian

[Paolo Bonzini]

My wild guess is that it "lathes" the Mandelbrot set around its axis...

[fractalrebel]

Hi everyone,

I may have missed it somewhere in this long thread, but is there a treatment for derivatives (both 1st and 2nd) somewhere in the thread?

[David Makin]

Hi everyone,

I may have missed it somewhere in this long thread, but is there a treatment for derivatives (both 1st and 2nd) somewhere in the thread?

The first derivative as works for the analytical DE is given in several places in the forums...

The second was mentioned around here:

http://www.fractalforums.com/3d-fractal-generation/true-3d-mandlebrot-type-fractal/msg8694/#msg8694

[cbuchner1]

My wild guess is that it "lathes" the Mandelbrot set around its axis...

I am so glad that you are wrong. The 4 hours I invested paid off. I modified a CUDA based voxel renderer, however the computation of the iterative formula is fully CPU based - and slow.

I have to make a correction to my method description above. The pole vector must point in the direction of the x axis, then you get the Mandelbrot in the x,y plane. The results when you point it someplace else are also interesting, but I wanted to see a 3D mandelbrot first.

I am in fact getting the familiar mandelbrot shape, but in three dimensions. It's probably not the extraordinary sensational thing that you've all been waiting for because the fractal detail is concentrated mostly in the x,y plane as far as I can tell. I am currently trying to get a voxel render of 512x512x512 resolution done and may record a youtube clip or something.

Here is a 3D voxelbrot previews.



Christian