








Generalized MandelbrotJulia












20040929






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 casestudy 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.












Fractals, algebras, Mandelbrot, Julia












 GenJulia1.nb (166.6 KB)  Mathematica Notebook 







   
 
