Introduction of an n-dimensional multiplication

[Leonhard E.]

Sorry my english is bad (1. I'm german and 2.  ) but I'll try it.

If X,Y are elements of Rⁿ with X=(x₀,x₁,...,x_n-1) and Y=(y₀,y₁,...,y_n-1)
the mapping X**Y is defined by:

X**Y:=(z₀,z₁,...,z_n-1) with z_k:=SUM sign(i,j)*x_i*y_j 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 Z_n+1=Z_n**Z_n+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,..,c_m,0,...,0) cause thats nothing else than the complex numbers.

<a href="https://www.youtube.com/v/tVSqBk9QxvI&rel=1&fs=1&hd=1" target="_blank">https://www.youtube.com/v/tVSqBk9QxvI&rel=1&fs=1&hd=1</a>


Please stop me if I'm telling things known since thousands of years or if its simply shit.

Sorry next time less  

[M Benesi]

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...