n-parameter calculations in 2D

Any comments to this? :

First the notation used here:

A complex number:

Z = [ x.x, x.x ]

A complex number that has got a complex imaginary part:

ZiZ = [ x.x, i[ x.x, x.x ] ]

The complex imaginary part:

iZ = i[ x.x, x.x ]

Parts of Z:

[ x, y ]

Parts of ZiZ:

[ x, i[ y, z ] ]

Parts of iZ:

i[ y, z ]

Note: the iZ complex can use any number of dimensions if one likes.

To do maths on the ZiZ complex then first convert it to a normal 2D complex
where the imaginary part always is a positive real number - the absolute
value of the imaginary part of ZiZ.

Z = Z|iZ|
Or Z = [ x, |i[ y, z ]| ], same thing =)

Now you can perform normal 2D maths like Z² or sqrt(Z) and such. (Additions
and subtractions are done in ZiZ space parameter by parameter, as normally
done for n-dimensional numbers).

Then to convert the number back from Z to ZiZ then do this:

ZiZ = [ x', i[ y' * ( y / |iZ| ), y' * ( z / [iZ| ) ] ]

Note: Here x' and y' (x prim, y prim) are the x and y you did get after your
calculations in normal 2D-space . iZ, y and z are the numbers that was in
the ZiZ number before the calculations was performed.

Remember that if |ZiZ| == |x|, (if it is only a 1D real number and nothing
but) then exeptions must be made in some cases. For example in 3D the root
from a negative real number is any point on a circle in the y/z plane,
(radius = root of |x|), in 4D it is a sphear and so on.

I'm using this to create non-linear 3D fractals and it works just fine =)
Here a image of a reversed formula Julia-set IFS-fractal, (Z = sqrt( Z -
C )).

http://members.chello.se/solgrop/Set%20d%202x%20003a.jpg

Are there any errors in my system? Is this method to expand from 2-space to
n-space used before? Is this useful for something but creating stupid
fractals?

Kenneth.



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