       A better Implicit Mandelbrot

• To: mathgroup at smc.vnet.net
• Subject: [mg51045] A better Implicit Mandelbrot
• From: Roger Bagula <tftn at earthlink.net>
• Date: Sat, 2 Oct 2004 03:18:59 -0400 (EDT)
• Sender: owner-wri-mathgroup at wolfram.com

```  Suppose we use both sets of polynomials at once in a rational like
function:

Rn(c)=Sum[Pcn(0),{n,1,Infinity}]/Sum[D[Pcn(0),c]),{n,1,Infinity}]
At n=7 I'm no where near Infinity, but I'm gretting pretty good results.
Doing this in any language but Mathematica would be much harder I think,
so developing software from it might take some effort
and the polynomials still have to be found for each case.
After n=7 they become really big.
It has been several years since a new visualization approach
that is entirely different has been found.
There is a scaling problem when doing close ups in the ContourPlot
which can almost be handled by making the plot point level
the same but not quite ( Don Taylor has been helping me on this
as ContourPlot has been crashing on my machine).

Mandelbrot Mathematica rational implicit:
<< Graphics`ImplicitPlot`;
g[z_]=z2+c;
nl=NestList[g,c,7];
nr=D[nl,c];
c=z;
q[z_]=Apply[Times,nl];
p[z_]=Apply[Times,nr];
z=x+I*y;
f[x_,y_]=Re[q[z]/(p[z])];
ImplicitPlot[f[x,y]==0,{x, -2.5, 1}, {y, -1.75,1.75},, PlotPoints -> {300, 300}]
Plot3D[f[x,y], {x, -2.5, 1}, {y, -1.75,1.75}, PlotPoints -> {145,
145},Mesh->False,Boxed->False,Axes->False]
ContourPlot[ f[x,y],{x, -2.5, 1}, {y, -1.75,1.75},PlotPoints -> {300,
300},ImageSize -> 600,ColorFunction->(Hue[2#]&)]
DensityPlot[f[x,y],  {x, -2.5, 1}, {y, -1.75,1.75},  PlotPoints -> {145,
145},Mesh->False,Axes->False]

Respectfully, Roger L. Bagula