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Information regarding the Bristorbrot and Doug Bristor's complex rotational maths

(http://4.bp.blogspot.com/_RtmvIFH1wRs/Sw8a4t8knHI/AAAAAAAAAEE/Jm0tnxd_hj4/s1600/doughowto.jpg)

http://www.fractaldimension.org.uk/voldsitemirror/ (http://www.fractaldimension.org.uk/voldsitemirror/)

Different Complex Numbers
With 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.

(http://www.fractaldimension.org.uk/voldsitemirror/ij.gif)
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
(http://www.fractaldimension.org.uk/voldsitemirror/ji.gif)
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/ (http://code.google.com/p/fractaldimension/)

The java code includes 4D and 5D functions

but here is my original 3D algorithm

Quote

i.j = -j
j.i = i

Code:

	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;
}

(http://lh6.ggpht.com/_RtmvIFH1wRs/SxFMCxhlZqI/AAAAAAAAAFs/mg-wsPwr33M/s640/Outlook.jpg)
Bristorbrot rendered by Jos Leys

http://www.youtube.com/watch?v=AdmZ7ypglIU
Bristorbrot rotated. Rendered by Jos Leys

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Fractal Math, Chaos Theory & Research => Theory => Topic started by: dougfractal on November 28, 2009, 06:08:15 PM