How to plot a Circles-and-Squares fractal

Question

The Circles-and-Squares fractal is produced by iteration of the equation

$\quad \quad z_{n+1}=z_n^2\ ({\rm mod}\; m)$

which results in a Moiré-like pattern:

Source: Wolfram MathWorld

In another place I found the same pattern, but the equation said ${\rm modulo}\ n$ instead of $m$. Either way, this equation is very similar to the one generating the Mandelbrot set. How can it be plotted?

Answer 1

The image taken from MathWorld appears to be something like this:

Image @ Rescale @ Table[Mod[x^2 + y^2, 100], {x, -50, 50}, {y, -50, 50}]

In my opinion this is not a fractal, and it certainly isn't produced by iteration. You described it as "Moiré-like", which is a far better description. It is simply a uniformly sampled 2D parabola, modulo some number. The pattern arises from the interaction between the fixed frequency of the sampling and the increasing frequency of the underlying function.

It's clearer in 1D:

Plot[Mod[x^2, 100], {x, -50, 50}, Exclusions -> None, PlotPoints -> 200, 
 PlotStyle -> Opacity[0.5], AspectRatio -> 0.3, 
 Epilog -> {PointSize[Medium], Point[{#, Mod[#^2, 100]} & /@ Range[-50, 50]]}]

So I don't really understand your comment about it being low-res and wanting a vectorial representation. The pattern appears because of the finite spatial sampling. Zooming into the function is very boring - you will just see the underlying wrapped parabola, not the never ending detail that a fractal provides:

Something prettier

Ignoring the picture in the question, and looking at the iteration $z_{n+1}=z_n^2\ ({\rm mod}\; m)$ gives something much nicer (in my opinion). Here it is with $m=2$ and 1 to 4 iterations:

z0 = Table[x + I y, {x, -3.001, 3, 0.01}, {y, -3.002, 3, 0.01}];
results = Rest @ NestList[Mod[#^2, 2] &, z0, 4];   
GraphicsGrid[Partition[Image @ Rescale @ Abs @ # & /@ results, 2]]