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Generalized Mandelbrot-Julia
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2004-09-29
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I have found fractal sets within Hoop algebras, and demonstrate them in GenJulia.nb.
My first problem was to decide what generalizations to pursue. I finally settled on an analogy with (x+i y)^2 +b (x+i y) +c; G[n+1]=G[n]^p+k G[n] +C. This reproduces Mandelbrot if (p=2, k=1,C=0) and Julia if (p=2, k=0,C={real,imag}) and the numbers are complex, G[0]=x+i y. Mandelbar splits k into {1,-1}. The standard definition of Mandelbrot, M[n+1]=(M[n])^2+M[0], is a degenerate case - making the final term M[n] gives the same result but allows wider generalizations .
The parameters are flexible, but are basically those needed to give Dickau's arrays, or give a single image. I find that it is vital to generate plots with different iteration limits, as the published boundary sets are often wrong - e.g. the stable sets for Dickau's Mandelbar plots (shown in Mathworld as basins with fractal boundaries) are actually branched lines (skeletons); Penrose shows "filigrees" (my term) that are outside the stable set (see examples 14 & 17, and my endquote). Hence the iteration limit is an important input parameter. I also include Hue as an optional parameter, for coloured plots.
A second problem is to display so many degrees of freedom; Julia has 4, with G=x+i y and C=v+iw, and Dickau' presentation gives {x,y} slices at different v & w. Beyond that, a case-study approach is needed, giving fixed values to other parameters. 3D images of fractal surfaces are incomprehensible. Symmetries help to reduce computation time; I do a full array at coarse scales and a low iteration limit, and can often restrict detailed examinations to one quadrant.
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Fractals, algebras, Mandelbrot, Julia
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| GenJulia1.nb (166.6 KB) - Mathematica Notebook |
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