Title: Introduction of an n-dimensional multiplication
Post by: Leonhard E. on June 19, 2015, 07:17:31 PM
Sorry my english is bad (1. I'm german and 2. :beer:) but I'll try it.
If X,Y are elements of Rn with X=(x0,x1,...,xn-1) and Y=(y0,y1,...,yn-1)
the mapping X**Y is defined by:
X**Y:=(z0,z1,...,zn-1) with zk:=SUM sign(i,j)*xi*yj where k=(i+j) modulo n and sign(i,j):= -1 if i+j >= n otherwise 1
the sign(i,j) is only for giving the multiplication a complex character
It's obvious that
X**Y=Y**X and
X**(Y+Z)=X**Y+X**Z and
E**X=X ,E:=(1,0,...,0)
With this basic definition we can calculate Zn+1=Zn**Zn+C (or even more)
As an example (I posted on the german board before) a 3-dimensional cut into a Mandelbrot set in R6
This movie includes rotations (staying in the same 3D subspace and some rotating into a additional dimension)
and a translation along a 4th axis. At the end the set is cut down to the known 2D Mandelbrot.
We don't have a surface(no normals) like we have no boundery in 2-dimensions, so the quality may be not satisfying.
Even dimensions (n=2m) are more symmetric, here you will allways find the "Apfelmännchen" as a 2D cut on
the axis (c0,0,..,cm,0,...,0) cause thats nothing else than the complex numbers.
https://www.youtube.com/watch?feature=player_embedded&v=tVSqBk9QxvI (https://www.youtube.com/watch?feature=player_embedded&v=tVSqBk9QxvI)
Please stop me if I'm telling things known since thousands of years or if its simply shit. :beer:
Sorry next time less :beer:
If X,Y are elements of Rn with X=(x0,x1,...,xn-1) and Y=(y0,y1,...,yn-1)
the mapping X**Y is defined by:
X**Y:=(z0,z1,...,zn-1) with zk:=SUM sign(i,j)*xi*yj where k=(i+j) modulo n and sign(i,j):= -1 if i+j >= n otherwise 1
the sign(i,j) is only for giving the multiplication a complex character
It's obvious that
X**Y=Y**X and
X**(Y+Z)=X**Y+X**Z and
E**X=X ,E:=(1,0,...,0)
With this basic definition we can calculate Zn+1=Zn**Zn+C (or even more)
As an example (I posted on the german board before) a 3-dimensional cut into a Mandelbrot set in R6
This movie includes rotations (staying in the same 3D subspace and some rotating into a additional dimension)
and a translation along a 4th axis. At the end the set is cut down to the known 2D Mandelbrot.
We don't have a surface(no normals) like we have no boundery in 2-dimensions, so the quality may be not satisfying.
Even dimensions (n=2m) are more symmetric, here you will allways find the "Apfelmännchen" as a 2D cut on
the axis (c0,0,..,cm,0,...,0) cause thats nothing else than the complex numbers.
https://www.youtube.com/watch?feature=player_embedded&v=tVSqBk9QxvI (https://www.youtube.com/watch?feature=player_embedded&v=tVSqBk9QxvI)
Please stop me if I'm telling things known since thousands of years or if its simply shit. :beer:
Sorry next time less :beer:
Title: Re: Introduction of an n-dimensional multiplication
Post by: M Benesi on June 19, 2015, 08:06:00 PM
Please stop me if I'm telling things known since thousands of years ...
I think others have spoken of something similar in the past, however the way you said it provided a new perspective on similar, slightly different thoughts.It got me thinking about branch cuts of complex roots, using the following for a Mandelbrot iteration:
from which there are various viewpoints on extension to 3 dimensions...