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Hi All. I just enjoyed the video news sent out by the forum and did happen to see the breaking news on the still undiscovered 3D Mandelbrot set.
Well, I have an interpretation that sadly settles on the 3D version merely being the 2D set extruded from zero to negative two on the new axis.

From an algebra perspective the question
   What is the 3D product?
settles the problem. For the Mandelbrot formula
   z[n+1] = z[n] z[n] + z[0]
we merely need a square operation, and we do see definitions of the nth power described for the new systems. But these should merely be the general product with substitution.
People are taking freedom in defining this product, but I do believe that it would be helpful if they would produce a true product
   a b = c
so that then other formulae could be explored on them as well. I do not mean to diminish those efforts and do enjoy the results, but this is a stricter mathematical interpretation that  may help to expose the subject from a fundamental viewpoint.

So, if we do have a clean 3D product, then along with the standard sum (which is standard vector addition) we have enough to do computations. The properties of the algebra used to matter quite a bit. The complex plane is algebraically well behaved with commutative, associative, and distributive properties intact. Should the 3D  version keep these properties then this places restrictions on the freedom with which the product can be defined. Unfortunately this leaves the 3D space as a morph of
   R x C
where the product is merely the independent product of these two parts. Thus the third dimension is merely a real line which does not interact with the complex plane within either the product operation or the sum operation.  There are theorems which claim this RxC form from associative algebra which I do not understand. I have constructed my own multidimensional system (polysign numbers) which do happen to be compatible with the standard algebra, and they do indeed yield the MandelBrick, which I will nickname the disappointing version of the 3D Mandelbrot set that my own code yields. This extruded shape is because the Mandelbrot test carried out upon the real line yields a solid band from zero to negative two as the Mandelbrot set under the strictest interpretation.

How to get pretty graphics going in 3D: it boils down to how you will mix the dimensions within the product operation. Taking this freedom one must break with ordinary algebraic principles. Whether there is any pure theoretical means here is hopefully what you are after. Meanwhile, what makes the Mandelbrot function so pure in the first place is as problematic. I don't mean to be a stinker, but also I don't care to give up on some fundamental thinking forming the next breakthrough. So I share these thoughts in the hopes of bringing the problem back to the ground level.

 - Tim http://bandtechnology.com

You have to sacrifice some algebraic property when you construct an algebra in 3D - e.g. your example algebra will have zero-divisors, and you will not be able to define division.

The reals and the complex numbers are example of commutative division algebras. In 4D the Quaternions form a non-commutatative division algebra, and in 8D the Octonions form a non-commutative, and non-associative division algebra. There is a theorem stating that no division algebra exists in other dimensions.

There are several examples on people suggesting different multiplication operators in 3D on this forum, see for instance the orginal Mandelbulb triplex algebra: http://www.fractalforums.com/theory/triplex-algebra/

hello and welcome to the forums, the triplex algebra will be covered in an own issue, and the problems arising from a 3component math ist the source for this "running joke" ... because we will never find the real 3d mandelbrot, but we got really close using the triplex algebra....
and in method linked by syntopia even division is doable on a inverse multiplication base ;)

I hear you Syntopia on the division issue. Still, even the reals and the complex numbers have an exception.
Zero is a special element under division and the field laws.
It is an exception, and pure mathematics is not supposed to make exceptions.
Well, but it is zero...
Well, division is also an inverse operator.
So long as we regard sum and product as fundamental then the conflict doesn't come up.
For instance in the RxC format ( a0, a1 + a2i ) we can provide
   ( 0, 1 + 1 i ) ( 1, 0 + 0 i ) = ( 0, 0 + 0 i )
as an arithmetic product of two nonzero values which yield zero for what is actually an algebraically well behaved math.
This demonstrates the problem but the actual behavior that is causing the problem is a quality of dimensional collapse which
may be a key thing to understand for the higher dimensional systems.

The resolution of the field requirements is merely to add more exceptions, for generally division can be had, except where the image has lost dimension via the collapse.

While many are working on 3D here what about general dimension? I know there are few to none of projection algorithms with high grade lighting and so forth. I have my own code which does perform multiD projections but I have no occlusion feature in it yet.

I will stand by my strict interpretation that the 3D Mandelbrot set does exist, but that it is merely the MandelBrick. Beyond this the term 'dimension' is tied to the real line, and the means of constructing a high complexity space out of several copies of the real line is not actually a construction of higher dimension; it is a representation. For some reason we observe that a three dimensional representation is sufficient for our local space, but as to why this is so... I suppose this is not a place to discuss physics. It might be a stimulating tangent for some.

 - Tim
 

> For some reason we observe that a three dimensional representation is sufficient for our local space, but as to why this is so..

I would limit that to "observably sufficient" as we can't necessarily rule out an "invisible" 4th dimension (or even more).

But of course if it's not "observable" then generally physicists aren't interested ;)

After much trial and error looking for a 3D Mandelbrot, the simple fact is that conformal transformations are the best generators of escape-time fractals. The Mandelbox is a realization of this fact, and it bests the Mandelbulb in terms of popularity and aesthetics. Since conformal transformations are so limited in 3D and above, there's really not much left to find, at least in the space of all possible analytic transformations. Perhaps there's some nice non-analytic continuous transformation left to be discovered and enjoyed.

In terms of David's challenge to the three dimensional interpretation of existence I do see one small area that may be overlooked by wee humans.

The rotational qualities of an object are not covered within the ordinary 3D representation.
For instance we presume that a coordinate string such as
   ( 2.3, 3.4, 5.6 )
could be a firm position as some particle or object's address, of course relative to an origin and some real line axes that within standard math would be orthogonal to one another.

The trouble comes when we witness that any object, even down to an electron (thanks to its spin) also carries two additional degrees of freedom, even upon exacting that coordinate above as its position. For instance the fundamental electron supposedly has a magnetic moment inherent to its being which can then point not just along the x, y, or z axes as some physics seems to do, but is free to have two more angular degrees of freedom. Thus ordinary space might be claimed to have five dimensions of freedom, though the latter two take a different format from the prior three.

Applying this back down in the complex plane we would get one additional angular freedom atop the original two cartesian dimensions. Maybe this will stimulate someone into a new interpretation.

Back to the standard 3D Mandelbrot... Isn't it apparent that to claim this name for fame that the algorithm must be general dimensional and that the 2D version must then be the original Mandelbrot set? It may be possible to take advantage of mirror symmetry to somehow morph the original Mandelbrot function. For instance in the complex plane there is a devout symmetry accross the real axis, which is to say that there is little difference between
   - 1 + 2 i
and
   - 1 - 2 i.
In effect one cannot really know the sign of the imaginary axis if no RHS/LHS coordinate system information was imposed. This is why the Mandelbrot set is symmetrical about that imaginary axis. In effect this means that we may freely throw in a sign reversal in the 2D case, and what this would translate into in the 3D case might be of interest. But here we need a general dimensional product defined. My own polysign numbers do work in general dimension Pn, though this sign trick becomes a multiplication by a polysign constant since i is no longer in use.(http://bandtechnology.com/polysigned) I haven't tried this yet and I don't really see how the P3 constant should map into P4.

I encourage all onward and with maximum freedom. Still, the hopes of a fundamental breakthrough via this freedom implies that something simple may lay unexposed still. This is like heading back down to the roots rather than up higher in complexity to the leaves of the tree. One of you might discover the next physics breakthrough even if you think of the art as graphics patterns. The simplicity of the code and the level of complexity that it can generate is another metric to consider. Simpler code will always be more impressive. Also the possibility that modern understanding is mistaken means going beneath the accumulation; back down to the roots.

 - Tim

This sounds like idea I had when used modulus function for generating mandelbrot.


* is traditionaly used as a mutiplication operator. So at first X - X*X looked like X minus X multiplied by X;)

Maybe you should made this whole 3D in Chaos Pro or Fragmentarium, they all have built in compiler.

Hiya Tim. Nice to read your thoughts on the forum again, after so long a while.
Get Kujonai back in train too and we may be able to pick the bones out of your ideas and implement some code. People may not know about your polysigns research and renders! :dink: