Mathematical backgrounds of 3D sets

[Jules Ruis]

I asked Roger Bagula and Robert Devaney for more mathematical background information about the construction of a 3D Mandelbrot set.

Jules Ruis e-mailed this two persons the next message:
Some people make remarks about my 3D Mandelbrot set. Attached I send you a screencopy of this 3D image, made with the Fractal Imaginator (Fi) programmed by Terry W. Gintz.
 
See: www.fractal.org/Beelden/Mandel-3D.jpg and www.fractal.org/fractal.doc
 
As you will see the image is spinning around the z-as. Some people ask me if that mathematically is correct. They expect bubbles on the whole surface. Can you confirm me if the showed constructions are correct?

Roger Bagula answered
Yes, old friend I got your email.
I just don't particularly need another egroup to service.
www.fractalforums.com

I've been keeping it alive for the last 10 years.
Let somebody else worry about that one.
Roger Bagula

prof. Bob Devaney answered:
I guess i don't know why anybody would plot the M-set in 3D when the image is naturally 2D.
I don't see that you get any more understanding of the set in higher dimensions.
Bob

Jules:
So, that answers are not very encouraging. I will do my best to get a better explanation. Who can help me?

Regards,
Jules Ruis.

[lkmitch]

prof. Bob Devaney answered:
I guess i don't know why anybody would plot the M-set in 3D when the image is naturally 2D.
I don't see that you get any more understanding of the set in higher dimensions.
Bob


Bob Devaney is right--the Mandelbrot set is fundamentally a 2D object, like a postage stamp.  There's not much to see when looking at it in 3D.

The next thing would be to perform the z=z²+c iteration in a number space that has more than two dimensions.  Typically, folks skip to four, because 2ⁿ dimensions generally works better.  Three dimensions has some inherent difficulties.  But the beauty of math is that you can define a new system and its rules.  For example, if you have a 3D number T = x + iy + jz, how is multiplication defined?  What properties extend from the real or complex numbers?  Once you have the basic arithmetic operations defined and a distance metric, constructing the 3D Mandelbrot set should be relatively straightforward.

Kerry

[Jules Ruis]

Thanks Kerry for the answer,

Of course we work with quaternions at the construction of the 3D Mandelbrot set.

See: http://www.mysticfractal.com/cquats.html

Jules.

[Jules Ruis]

We got an answer of Roger Bagula:

"On your 3d Mandelbrot : from the picture right off I can tell your escape  is a measure one and not the
 r=x^2+y^2 > limit that is what Mandelbrot incorrectly used , but is the "historical" M set border defintion.
Roger Bagula"

This brings us on the right track. I will contact Terry W. Gintz, a good programmer of 3D fractal software.

Jules.

[lycium]

perhaps the "thomas ludwig" set can be a grid of q -> q^2 + q_0 quaternionic julia sets, corresponding to the mandelbrot scheme of iterating the julia with the first term as constant

[Jules Ruis]

We got an answer of Roger Bagula:

"On your 3d Mandelbrot : from the picture right off I can tell your escape  is a measure one and not the
 r=x^2+y^2 > limit that is what Mandelbrot incorrectly used , but is the "historical" M set border defintion.
Roger Bagula"

This brings us on the right track. I will contact Terry W. Gintz, a good programmer of 3D fractal software.

Jules.

Terry W Gintz answered me:

I cant say what Roger is referring to -- he uses a mathematicians jargon that is often hard for a layman to grasp.  The quaternion construction in FI is correct, as far as quaternion math is able to depict.  It helps to increase iterations to 30-50 to see a more rounded (and closer to traditional) view of the 3D Mandelbrot set, though this makes the surface texture very grainy.  The problem is how quaternion math operates in 3D space.  It creates forms that tend to look as if they were turned on a lathe.  The loxodromic formula (L0) by Thomas Kromer gives results that are closer to what you might expect for a true 3D representation of Mandelbrot and Julia sets, but they are only realistic when viewed from one angle.  Sort of like an elephant that has been squashed lengthwise...  If you look at the elephant from the side, it looks right, but not from any other viewpoint.  Ive yet to find any 3D algorithm that extends all of the 2D fractal, without some distortion.  (If you notice, using a hypernion type the figure is squared off.)

Best Regards,
Terry W. Gintz

[Jules Ruis]

Also Evgeny Emidov tells something about 3d sets; see

http://www.ibiblio.org/e-notes/MSet/Quater.htm

Jules.

[Jules Ruis]

perhaps the "thomas ludwig" set can be a grid of q -> q^2 + q_0 quaternionic julia sets, corresponding to the mandelbrot scheme of iterating the julia with the first term as constant

Who can help us with this suggestion?

Jules.

[Jules Ruis]

For an interesting article of Alessandro Rosa for constructing fractal 3D sets see:

www.fractal.org/Bewustzijns-Besturings-Model/JR-Set/Quaternion-Julia-sets.pdf

Jules Ruis.