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