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

[Yannis]

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  :

[DarkBeam]

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

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

 

See you, thanks for sharing

[cKleinhuis]

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 ?!?!?

[Yannis]

Thanks for your quick answers,

To DarkBeam: I dont use at all that sort of formulas ! 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 ?


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.

[DarkBeam]

Thanks for your quick answers,

To DarkBeam: I dont use at all that sort of formulas ! 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 ?

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

[Yannis]

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 ?

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.

[cKleinhuis]

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
...

[kram1032]

mysterious matrix notation

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.

[Yannis]

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  

[DarkBeam]

"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, the only limitation is power of computer: what about your solution ?

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

[DarkBeam]

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.

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

[Alef]

Statistics is the worst.

This looks perspective.


3d mandelbrot is quite an elusive thing. This looks stetched (fractal in just 2 dimensions and not in 3rd):

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
...

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

[David Makin]

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

[taurus]

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"...

[Alef]

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

It's stretched in z axis.