True 3D mandelbrot fractal (search for the holy grail continues)

[David Makin]

Oh yes, I did...
I kinda used the dotproduct instead of the crossproduct :S

It's <Quoted Image Removed>

That makes more sense

I guess that's correct because that's the result I got too after looking on Wiki

[kram1032]

Pretty simple

To get the area of any triangle in R³, simply do the crossproduct of two side vectors and half the result.
The sign afaik also tells something about the direction...

and for a tetraeder's volume, take a sixth of the dotproduct of one vector and the crossproduct of the two others.
here also the sign tells something about the direction... I read about it but kinda forgot....

[David Makin]

So, we have the areas of the sides x*y/2, x*z/2 and y*z/2 with the area of the base as sqrt(x^2*y^2+x^2*z^2+y^2*z^2)/2 which gives us 3 ratios equivalent to sine and cosine in 2D:

r1 = x*y/sqrt(x^2*y^2+x^2*z^2+y^2*z^2)
r2 = x*z/sqrt(x^2*y^2+x^2*z^2+y^2*z^2)
r3 = y*z/sqrt(x^2*y^2+x^2*z^2+y^2*z^2)

I started going round in circles trying to see if this helps get a "correct" method for a 3D Mandelbrot and finally hit upon the following idea:

Consider the 2D complex iteration as follows:

1. Take a 2D coordinate = (x, y)
2. Convert to complex = (x + i*y)
3. Raise it to a power (e.g. 2) using complex algebra = (x^2-y^2 + 2*i*x*y)
4. Convert back to 2D coordinates = (x^2-y^2, 2*x*y)
5. Translate by a 2D offset = (x^2-y^2+cx, 2*x*y+cy)

So for the 3D version (using r1,r2,r3 above) we could try:

1. Take a 3D coordinate = (x,y,z)
2. Convert to the new 3D format = (r1, r2, r3) (edit. i.e. to the area ratios using the formulas above for r1,r2,r3)
3. Convert to an algebra format - e.g. cut down quaternion = (r1 + i*r2 + j*r3)
4. Raise to a power in the relevant algebra and convert back to new 3D format = (newr1, newr2, newr3)
5. Convert back to 3D coordinates = (newx, newy, newz) !!
6. Translate by a 3D offset = (newx+cx, newy+cy, newz+cz)

Obviously step 5 in the above is problematic since it needs newx, newy and newz in terms of newr1, newr2 and newr3 (with the signs correct).
Also the translation could be done after step 4 but I'm betting on the above producing more interesting results.

[kram1032]

I don't really get your point.... what's the difference to the current method?

Btw, sin/cos/tan also are defined on the unit-circle, so it might help to look at the unit-sphere
tan would be 2D (defiing a plane)
and sin and cos would either also be 2D or some extra values would get added to that.
- or there would not only be tan and cot but a third thing, making it 1D again...
or something like that^^

I think, a 3D variation should both work in a rightangle tetrahedron and in a sphere, as both the right triangle and the unit circle help to add acuracy (arg()-function which is quadrant-dependent... - similarly the sphere's arg would be octant-dependend, so to say... [and probably 2D rather than 1D as spherical coordinate systems show anyway]) and simplify formulae (sin²(x)+cos²(x)=1 - for the 3D equivalent it would be sin²(x)+cos²(x)+___²(x)=1 )

[David Makin]

I don't really get your point.... what's the difference to the current method?

Btw, sin/cos/tan also are defined on the unit-circle, so it might help to look at the unit-sphere
tan would be 2D (defiing a plane)
and sin and cos would either also be 2D or some extra values would get added to that.
- or there would not only be tan and cot but a third thing, making it 1D again...
or something like that^^

I think, a 3D variation should both work in a rightangle tetrahedron and in a sphere, as both the right triangle and the unit circle help to add acuracy (arg()-function which is quadrant-dependent... - similarly the sphere's arg would be octant-dependend, so to say... [and probably 2D rather than 1D as spherical coordinate systems show anyway]) and simplify formulae (sin²(x)+cos²(x)=1 - for the 3D equivalent it would be sin²(x)+cos²(x)+___²(x)=1 )

Apologies I guess I wasn't clear enough - in the step "Convert to the new 3D format" I should have explained I meant convert to the area ratios using the formulas for r1,r2,r3 in terms of x,y,z.
This way the 3D transformation is performed on the areas rather than the distances.
Also if you look at r1,r2,r3 then r1^2+r2^2+r3^2 = 1.

[kram1032]

ah, got it
Really a nice idea...
Any results so far?

[David Makin]

ah, got it
Really a nice idea...
Any results so far?

Just about to try it - at least ignoring problems with the signs when converting back from newr1,newr2,newr3 to newx,newy,newz - if anyone has any bright ideas on how to get the correct octant for newx,newy,newz please let me know....

[David Makin]

OK, it's pretty useless, at least it is without handling the octant problem and divide by zeroes correctly.

[kram1032]

so... we gotta figure it out
that's probably a bit more tough...

the octantproblem most likely needs the definition on the unit-sphere to be solved...
(as the quadrant problem of argument needs the unit circle)

and getting along with zeros.... maybe the rule of l'hospital or how it's called can help here?
which solutions turn out just before or just after 0?
(I hope it's clear what I mean^^)

[jehovajah]

Actually i have just worked out that this strand is where i should post my musings for the  math suggestions toward a 3d mandelbrot. A mandelbulb as it is now called is a beautiful thing but not what i have been thinking about.

http://www.fractalforums.com/theory/transform-for-r3/  

This is my transform. I wonder if you could roughly render it David.

http://www.fractalforums.com/theory/formula/

And this is my exploration guide. Be sure to look at the following posts as i complete the guide with regard to a handedness term which affects all 3 cartesian components.

[jehovajah]

I do not mean to distract your thinking but you guys who are writing about conformal maps and covers from could explain what you are thinking about. Say i do read wikipedia i still do not know what you are thinking about here. There is in fact no loss of generality issues here where we are being specific to one method or another.

In a fractal i would expect discontinuities at boundaries where one region juxtaposes with another. The mathematical horror which called these things monsters must now give way to the realisation that their beauty and artistic worth allied with computers means the computational swamp no longer need fill us with dread.

[LesPaul]

In a fractal i would expect discontinuities at boundaries where one region juxtaposes with another. The mathematical horror which called these things monsters must now give way to the realisation that their beauty and artistic worth allied with computers means the computational swamp no longer need fill us with dread.

I like your train of thought.   

Regarding your "vx" formula, I have also wandered down rabbit holes thinking about similar approaches.  If we boil the 2D Mandelbrot down to its core, it is a process where do this, over and over:

z := z²
z += z₀

I tend to think about things like "what does it mean, conceptually, to 'square' a number?"

It's simple enough with reals:

1) Take a (1D) line of length x.
2) Take another line of length x and lay it perpendicular to the original line.
3) Take those two lines as sides of a square, and the area of that geometric shape is the "square" of x.

Extending that concept to 3D is quite natural (just form a "cube" instead of a "square"), but complex numbers aren't quite as easy to visually "square."

Squaring a complex number amounts to a rotation about the origin and a scaling of magnitude.  The term "square" doesn't have such a nice geometric analog any more.  Looking at the number being squared in polar form is sometimes more revealing...



In two dimensions, what it's saying is:

1) Take the distance to the point from the origin and square it (using the word 'square' here in the real sense).
2) Take the angle to the point measured from the +x axis and double it.
3) At the point described by this new distance and angle is your new, "squared," complex point.

Now we immediately run into the big question...  How do we extend that concept into three dimensions?

Things like this keep my mind very busy. 

[David Makin]

In a fractal i would expect discontinuities at boundaries where one region juxtaposes with another. The mathematical horror which called these things monsters must now give way to the realisation that their beauty and artistic worth allied with computers means the computational swamp no longer need fill us with dread.

I like your train of thought.   

Regarding your "vx" formula, I have also wandered down rabbit holes thinking about similar approaches.  If we boil the 2D Mandelbrot down to its core, it is a process where do this, over and over:

z := z²
z += z₀

I tend to think about things like "what does it mean, conceptually, to 'square' a number?"

It's simple enough with reals:

1) Take a (1D) line of length x.
2) Take another line of length x and lay it perpendicular to the original line.
3) Take those two lines as sides of a square, and the area of that geometric shape is the "square" of x.

Extending that concept to 3D is quite natural (just form a "cube" instead of a "square"), but complex numbers aren't quite as easy to visually "square."

Squaring a complex number amounts to a rotation about the origin and a scaling of magnitude.  The term "square" doesn't have such a nice geometric analog any more.  Looking at the number being squared in polar form is sometimes more revealing...



In two dimensions, what it's saying is:

1) Take the distance to the point from the origin and square it (using the word 'square' here in the real sense).
2) Take the angle to the point measured from the +x axis and double it.
3) At the point described by this new distance and angle is your new, "squared," complex point.

Now we immediately run into the big question...  How do we extend that concept into three dimensions?

Things like this keep my mind very busy. 

I've often found it annoyingly inconstistent that squaring is applied when dealing with both 2 dimensions and 3 dimensions - it makes more sense to me that to extend complex z^2+c to 3D triplex equivalent we should actually consider z^3+c rather than z^2+c

[cKleinhuis]

I've often found it annoyingly inconstistent that squaring is applied when dealing with both 2 dimensions and 3 dimensions - it makes more sense to me that to extend complex z^2+c to 3D triplex equivalent we should actually consider z^3+c rather than z^2+c

to bad we are not dealing with areas/volumes, if it would be a volume i would suggest cubing for squaring in 3 dimensions ...   

[kram1032]

I tried around with the area-based sine a bit but I don't have a CAD-program which would allow me to actually define the whole thing for areas...

I at best have two programs:

blender - not a CAD program, so if you don't know python well enough (which is the case for me), probably pretty useless and
GeoGebra - limited to 2D...

operating on areas is not too easy it seems...

one thing that isn't clear yet is, how does the "rightangles tetrahedron"-angle look like: it's neither a plane angle nor is it an opening like with cones, which could easily be meassured one way or the other...
how to define that double-90° angle and the surrounding 3 more double-angles?

btw: the definition of area-based sine also uses standard norm sqrt(x²+y²+z²)=1, so it wont work to use the third power either...

though, you could in general experiment with odd formulae... here for instance the real and imaginary part of z in a base-i norm unit sphere, if x and y vary between -2pi and pi...

http://www.wolframalpha.com/input/?i=3Dplot+real%28%28-x^i-y^i%2B1%29^%28-i%29%29%2Cx%3D-2pi...2pi%2Cy%3D-2pi...2pi
http://www.wolframalpha.com/input/?i=3Dplot+imag%28%28-x^i-y^i%2B1%29^%28-i%29%29%2Cx%3D-2pi...2pi%2Cy%3D-2pi...2pi