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Hello from France,

I have just opened my site dedicated to a new (?) family of Mandelbrot-Julia multidimensional fractals that I have discovered (or rediscovered ?) in 2011. It aims to promote my proposal for a new (I hope) generalization of the complex numbers to all dimensions baptized: absolien numbers.

It is impossible to explain here in a few lines what are absoliens, so if curious about it see my site and please tell me your opinion :   https://sites.google.com/site/yannispicart/
If you knows antecedent about these numbers or fractals, thank you for informing me so that I mention it. My site is available now in french and english.

Below two examples of 3D fractals, which are some of the projections of 4D Mandelbrot ensemble calculated with my software MANDELMINE that I developed specifically: art is not my purpose, first of all mathematical and philosophical, so excuse the bad definition that you can improve I am sure  :D:

(https://79ff4d30-a-62cb3a1a-s-sites.googlegroups.com/site/yannispicart/home/Mandelmine%20maitre%20carre%202012-05-28_193632.jpg)

(https://sites.google.com/site/yannispicart/_/rsrc/1341608102096/absoliens/2012-07-06_223414%20carre.jpg)

Here I report the code for squaring, taken from the source :evil1:

x' = x^2 + z^2 + 2 yw
y' = 2( xy + zw)
z' = y^2 + w^2 + 2 xz
w' = 2( wx + zy)
m = min(x',y',z',w')
x = x' - m, y = y' - m, z = z' - m, w = w' - m

x' = x + Cx, y' = y + Cy, z' = z + Cz, w' = w + Cw
m = min(x',y',z',w')
x = x' - m, y = y' - m, z = z' - m, w = w' - m

 :D

See you, thanks for sharing

hello and welcome to the forums,

the image you posted does not look fractalish at all, but it might happen that you used low iteration counts or is it the "blurry border" that shows fractal behaviour ?!?!?

Thanks for your quick answers,

To DarkBeam: I dont use at all that sort of formulas ! :evil1: Only addition and multiplication in my absoliens algebra. The multiplication of absoliens is discrete convolution upon nupplets of positive reals only, with at least one term equals to 0 (this is absoliens class).
Function min is used too for "reduction" to obtain canonic representation of the absolien class with a 0 (the only similarity with yours): see my site for complete explanation.
Do you really obtain the same images with complete different method ? Interesting ! may I see them ? :D


To cKleinhuis: Yes, it is the "blurry border" where you can recognize Mandelbrot ensemble but only in plane stratums.
In fact you are true, selfsimilarity is not complete and I have better to speak of "pseudo fractal", but this is already the case with other "fractals" well known in 4D projection such as quaternions, bicomplex/tetrabrot and even hypercomplex at low power... But Mandelbrot ensemble is very present as you can see in the 3D slice below (attention: of course "slices" in 4D world are note plane !) even if he is deformed.
I believe that the 4D complete fractal is really fractal and selfsimilar, but we can access only to 3D projections !
In fact each part of these "fractals" is similar but different. That are distortions of the initial plane motive of Mandelbrot.

(https://sites.google.com/site/yannispicart/qui/Mandelslice%20PC01%20P2%20S0%20du2011-11-08_074902%20reduit.jpg)

Completely different? No really, just replaced your mysterious matrix notation with a more readable one ;) - Images? Probably they are not coming for now :evil1: :dink:

To DarkBeam:  :'(
"Completely different? No really, just replaced your mysterious matrix notation with a more readable one  - Images? Probably they are not coming for now "

"Mysterious matrix notation" :  Discrete convolution (my multiplication on absoliens) is not mysterious at all and employed in many domains around the world, very simple and beautiful in matrix representation at contrary of yours !
My originality is to employ it in a new context different of usual R geometry and much more elegant than attempts on R (until proof of the contrary).
More, this multiplication is valid at any dimension and any power, the only limitation is power of computer: what about your solution ? :evil1:

I hope that numbers I named absoliens are really new concept (but I am not sure and need your advice and help about that).

I am going to travel a week, and i'll be back july the 19. I will be very pleased to discuss more about it if you have more argued criticism. :D

there has been featured a similar number set approach, that i can only compare by the visible results, the one i dont remember
recently posted here had as result as well this extruded mandelbrot .. need to find it, but i am unsure for what to search
...

You should learn to love matrix notation. It simplifies a lot of concepts.
Ideally, you'd learn Tensor notation which gives rise to even more complex ( :D ) things.

to kram1032
You should learn to love matrix notation. It simplifies a lot of concepts.
Ideally, you'd learn Tensor notation which gives rise to even more complex (  ) things.

See my site please: convolution is of course used in matrix form !   https://sites.google.com/site/yannispicart/

But complexity is not a guaranty of result... simplicity and elegance of concepts much more  :D

Okay but computers don't understand those elegant concepts, they need explicit expressions ;D

You don't know what I learned until now, so don't give advices. More, I hate both tensors and matrix notations :tease:

Statistics is the worst.

This looks perspective.
(https://sites.google.com/site/yannispicart/_/rsrc/1341609413263/galery-1/2012-07-06_075840%20carre.jpg?height=320&width=320)

3d mandelbrot is quite an elusive thing. This looks stetched (fractal in just 2 dimensions and not in 3rd):
(https://sites.google.com/site/yannispicart/_/rsrc/1332074753365/galery-1/Absoliens%204A3%20011%2048%20H%202011-08-25_180313%20carre.jpg?height=320&width=320)

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

Please show how you generalise to 3 dimensions (and no more).

although i'm not an expert here, i guess this would be quite "grailish"...

It's stretched in z axis.

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.

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

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

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.

dito  ;D

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 !

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.

Because that's the Holy Grail ;)

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

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   ;D

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 :D

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

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...