Mandlebrot Set Generator

I was playing around with the Mandlebrot set yesterday in Matlab, and eventually after iterating on my implementation of it, decided to take a few photos and screencaps of deep zooms on particular points. I have been having a bit of a hard time identifying the right points to zoom in on but was still able to arrive at a few pretty nice ones. At the end of this gallery I will also provide my very short code that can generate these.

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First, the famous Mandlebrot set in all it’s glory!
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I began playing around with the recursive function that generates the set by adding polynomial terms of higher order.
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I also played around with the various thresholds I could use to assess convergence of the set at all the points in the image.
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This one is particularly surprising in the minimal amount of chaotic behavior and the elegant structuring of the self-similar pattern.
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I find this one to be very lovely with its soothing colors and faded aesthetic.
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A highly structured higher-order polynomial cousin of the Mandlebrot family
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A low-res version with many different thresholding layers to achieve the coloring pattern you see there.
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More varieties, this is getting really fun!

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Finally, I decided to try and see how far I could zoom in on the images. I created a function that would be supplied by a point to zoom on and a number of zooming operations, and it then runs the set generator and appends each image to a video file. Below are the results of four points I pseudorandomly selected to zoom in on.

 

Finally, my MATLAB implementation that I used to generate the set. Feel free to use and test this function out as you please, but give me a link if you redistribute it.

function mset_layers = proceduralMandlebrot(dim,iters,bnds)
% proceduralMandlebrot(dim,iters,seeds,bnds)
%
% a function for visualizing the mandlebrot set
%
% dim: the delta_x we discretize by, default 1000
% iters: number of iterations, default 30
% bnds: [x1 x2 y1 y2] limits in the complex plane, default [-2 2 -1 1]
%
% by Majed Samad 2016
t0 = tic;
if ~exist('dim','var')||isempty(dim), dim = 1000; end
if ~exist('iters','var')||isempty(iters), iters = 30; end
if ~exist('bnds','var')||isempty(bnds), bnds = [-2 2 -1 1]; end
[a,b] = meshgrid(linspace(bnds(1),bnds(2),dim),linspace(bnds(3),bnds(4),dim));
x = 0;
c = complex(a,b);
for i = 1:iters
   x = x.^2 + c;
   mset = abs(x)>2;
   if i==1,mset_layers = mset;else mset_layers = mset_layers + mset/i;end
end
imagesc(mset_layers)%,colormap(autumn)
fprintf('it took %f minutes to execute\n',toc(t0)/60);