the ABSOLIENS: natural generalization of complex numbers at any dimensions ?

[kram1032]

Matrix and Tensor notations are both simply strong short-hands. Of course, you need to tell a computer what such notations means. But at some point, not using them makes things so convoluted and messy, that all the beauty behind it would get lost.
Seeing that is just a matter of getting used to them.
In a way, it's higher level language. One step further away from the raw machine code of pure sets but much more capable of defining quite complex concepts in quite a simple way.

[Yannis]

Hello, thanks for your interest
I see that you are dubious...

I am travelling and will be back the july 20. I'll try to translate my site in english (page: absoliens) for your understanding of my proposition of new (?) numbers I named absoliens. It's difficult to explain here in a few lines: they are new family of numbers as quaternions, multicomplex, etc and you have to reed my site for understanding.

But be sure that all fractals I present on my site are real 3D (ray tracing on Z axis as you suggest).
I know that my software MANDELMINE is note very performing for rendering...

The image you exposed (Mandelbrot big slice in green) is 3D but corresponds to 3D absoliens: not very interesting.
Others are 4D absoliens projections or slices of 4D projections.

bye, see you in a week

[LhoghoNurbs]

I'm eager to see more details. As long as I cannot read French, it is not possible to understand it. However, my observations are:

- a true 3D fractal should show fractalish details in all directions. The images on your site have only two fractalish directions. The third direction appears like extrusion.

- as a rule of thumb -- inventing new things in Math (and in the area of complex numbers) is very difficult. Most likely when you invent something it is not a new thing, but just some re-invention. Make sure it is really a new thing before claiming it is a new thing.

I may definitely change my opinion once I'm able to grasp your ideas.

[taurus]

- a true 3D fractal should show fractalish details in all directions. The images on your site have only two fractalish directions. The third direction appears like extrusion.

dito 

[Yannis]

To LhoghoNurbs:
- as a rule of thumb -- inventing new things in Math (and in the area of complex numbers) is very difficult. Most likely when you invent something it is not a new thing, but just some re-invention. Make sure it is really a new thing before claiming it is a new thing.
Naturally, I totally agree with what you wrote . When I evoke the possibility of the novelty it's always with a question mark. But ok I shall try to be even more careful in the future...
To verify is exactly what I am trying to do.

- a true 3D fractal should show fractalish details in all directions. The images on your site have only two fractalish directions. The third direction appears like extrusion
Yes, my images are for instance really fractal only on strata (slices). But strata in several directions and very intricated/mixed. And I did not investigate all the possibilities: 5D 6D... (my computer is not rather powerful).
But who said that the 3D complex generalization had to come along inevitably with complete fractality 3D for Mandelbrot set ?  May be you will be disappointed !

[Yannis]

To Asdam:
A tricomplex numbers?
http://www.fractalforums.com/general-discussion-b77/tricomplex-numbers/msg41182/

I know that this image (Mandelbrot 3D slice) has already been obtained by different methods (see for example ternary numbers proposed in 2006 par CHENG Jin et TAN Jian-rong of chineese university Zejiang:   http://www.zju.edu.cn/jzus/article.php?doi=10.1631/jzus.2007.A0134).
In fact, my "absoliens" sets,  in their not canonical/classic versions (degenerate shape), include tricomplex en all multicomplex sets.
Here I obtain that image with simple 3D canonical/classic absoliens: she is really 3D but boring as you said.

[David Makin]

But who said that the 3D complex generalization had to come along inevitably with complete fractality 3D for Mandelbrot set ?  May be you will be disappointed !

Because that's the Holy Grail

[kram1032]

And it's also what wasn't accomplished yet...

[Yannis]

To David Makin and DarkBeam
"Please show how you generalise to 3 dimensions (and no more)."
"Okay but computers don't understand those elegant concepts, they need explicit expressions"

Hello. You want clarified computer formulas. So I give you  
I simply use Mandelbrot formula An+1= An^2 + Ac  or An+1= (An*An*An...) +Ac  at any power and dimension. Or logistic equation.
The mutiplication * of absoliens A0*A1 is discrete circular convolution on positive coordinates.
Attention:  there are 4 positive coordinates for absoliens in 3D because we work on tetrahedron base (u, i, j, k) were u is "real" and i, j, k imaginary half-axes.
In 3D this function multiplication is below. In any dimension convolution stays very symmetric, commutative and associative (that's one of the main interests of my proposal):

// absoliens 4A3 multiplication A0 * A1, discrete circular convolution equivalents to complex multiplication
vector<double> operator* (vector<double>A0, vector<double>A1)  
{
vector<double> AC(4,0);
AC[0]=A0[0]*A1[0]+A0[3]*A1[1]+A0[2]*A1[2]+A0[1]*A1[3];
AC[1]=A0[1]*A1[0]+A0[0]*A1[1]+A0[3]*A1[2]+A0[2]*A1[3];
AC[2]=A0[2]*A1[0]+A0[1]*A1[1]+A0[0]*A1[2]+A0[3]*A1[3];
AC[3]=A0[3]*A1[0]+A0[2]*A1[1]+A0[1]*A1[2]+A0[0]*A1[3];
double m=mini(AC,4);
for (int i=0;i<4;i++) AC=AC-m; // canonic reduction from symmetric equation in absolien base: u+i+j+k=0
return AC;      
}

At the begening before calculation, we need to convert the sample vector from classic euclidian base R3 (screen+Oz) to tetrahedron one (u, i , j, k), that is equivalent to transform usual R3 and vectors space to absoliens space. The function is:

// gives absoliens coordinates in 4A3 base (u, i, j ,k) from V in R3 base (U, V,W)
vector<double> coordR3to4A3(vector<double> V)
{
vector<double> A(4,0);
double coe=-1.0/(2.*RA3);
A[0]=0.; // we choice u coordinate arbitrary null
A[1]=coe*(V[0]+V[1]);
A[2]=coe*(V[0]+V[2]);
A[3]=coe*(V[1]+V[2]);
double m=mini(A,4); // canonic reduction relative to axis symmetry, all resulting coordinates are positive and one null at least
for (int i=0;i<4;i++) A=A-m; // if m<0, Ai becomes positive or null
return A;
}

We obtain 3D big slice of Mandelbrot set: boring but the elementary stratum that we find again in 4D dimension but in several different orientation very mixed.
Not so complicated, computers also like elegance !
I am trying to translate my site for details and to show you the change of paradigm that implies (R cube geometry is not the world of absoliens. Absolien universe is geometry of simplexes: regular triangle, tetrahedron, pentachore etc) .

My purpose is not to find the grail of fractals (sorry) if it exists. My grail is the true/natural generalization of complex numbers and finally to improve the philosophical comprehention of numbers and space geometry associated  

[Apophyster]

Because that's the Holy Grail

The one Parsifal sought?

I forgot did he complete his quest?
The quest being the one in the book?

? Fred ?

[cKleinhuis]

The one Parsifal sought?

I forgot did he complete his quest?
The quest being the one in the book?

? Fred ?

we refer to the "true 3d mandelbrot" as the "Holy grail" here, although this does not mean it is not the same as parsifal meant

[Apophyster]

Yes I know the reference is to a 3d Mandelbrot (and I also think using the term "Holy Grail" exhibits some cultural bias).

Apart from that, if your search is for the same grail which Parsifal sought, does this mean Jesus drank from a 3D Mandelbrot cup?  Is there a 3D Mandelbrot Loaf too?

Fred

[Yannis]

Hi,

As information, after researches I found that:
-my absolien numbers describes spaces probably equivalents to multicomplexes MCn or quotient ring algebras.
-my formalism is equivalent to "polysign numbers" of Tim Golden e.g. : see his posts of 2007 anterior of mine on Fractal Forum: Fractal maths/the 3D mandelbulb/theories/simple algebra

However my formalism (vectors and matrix of only positives values) seems more simple, and my site presents pseudos-fractals probably original (even if they are a bit disappointing).

I continue the quest of the grail...