Other

While using a broken distance estimator on the M-Set... these were the midnight results. Interesting enough to explore some more at a later time.

The center of the 5 point star is (0,0i).
The x and y limits are -2 to 2.

The x and y limits are -6 to 6.
The vertical asymmetry is still apparent.

The basic idea is this:
Take a point and do the first three iterations on it to get Z1, Z2, Z3.
Explore any relationship that might exist between these three points and escape time or being bound.
Does the first few iterations tell a story about how the point will ultimately behave?

My code was adapted from a program from an article titled "Fractal Geometry and the Escape-time algorithm as Graphic Art Tools" by Daniel E. Lanier for a presentation at SDSU Mathematics Summer Seminar in July 2008.

MATLab code used

h=400;
w=400;

ETMAX = 32;
MAXDIST=2;
Xaxis = linspace(-2,2,w);
Yaxis = linspace(-2*h/w,2*h/w,h)*i;

% matlab functions define colormap matrix (ETMAX rows x 3 columns)
mp = colormap( hot(ETMAX) );
mp = cast(mp*255,'uint8');

% allocate the matrix ␣> use unsigned 8␣bit integers for images
A = uint8( zeros(h,w,3) );    % think three pages of h by w pixels

for m = 1:h
for n=1:w
Z=0;
c = Xaxis(n) + Yaxis(m);

Z1= Z^2 + c;
Z2= (Z^2 + c)^2 + c;
Z3= ((Z^2 + c)^2 + c)^2 + c;
Mv=   ((Z2-Z1).^2 - (Z3-Z2).^2).^0.5 ;
esctime=30;
Ma=1;
while Mv >.01*Ma *Ma && esctime >1
Ma=Ma+1;
    esctime = esctime - 1;
end

A(m,n,:) = mp(esctime,:);
end
end

imshow(A);    colorbar;    impixelinfo;