Information regarding the Bristorbrot and Doug Bristor's complex rotational maths
http://www.fractaldimension.org.uk/voldsitemirror/Different Complex NumbersWith the understanding that [imaginary number i] can be represented by rotation of 90° about the origin. I defined [imaginary number j] as a rotation of 90° about the origin and a rotation of 90° with respect to
.
So with this in mind I picked up a cube, to help me visualise the rotations, and then recorded the results.
I rotated it upwards 90° , then following it by turning it a different 90° backwards [j], I defined the result:
i × j = -j
Then starting again
I rotated 90° backwards [j], followed by 90° upwards . This gave me the result:
j × i = i
It is the use of these results when applied into the [Mandelbrot algorithm], that produce the [Bristor set]. Imaginary numbers in additional dimensions can be derived by substituting in the new number in the above equations.
http://code.google.com/p/fractaldimension/
The java code includes 4D and 5D functions
but here is my original 3D algorithm
i.j = -j
j.i = i
public int iterate3D(double rec,double imc, double jmc,int max)
{ double re,im,re2,im2,jm,jm2,ij,tmp;
int itr=0;
double mag=.0;
im=re=jm=re2=im2=jm2=.0;
do {
ij=im*jm;
tmp=re2-im2-jm2+rec;
im=2.0f*im*re-ij+imc;
jm=2.0f*jm*re+ij+jmc;
re=tmp;
re2=re*re;
im2=im*im;
jm2=jm*jm;
itr++;
if (itr>max) break;
mag=re2+im2+jm2;
} while (4.0f>mag );
return itr;
}
Bristorbrot
rendered by Jos Leys
<![CDATA[<a href="http://www.youtube.com/v/AdmZ7ypglIU&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/AdmZ7ypglIU&rel=1&fs=1&hd=1</a>]]>Bristorbrot rotated.
Rendered by Jos Leys
August 05, 2012, 03:44:41 PM
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