How is it possible to get an accurate representation of the Mandelbrot set (and similar)?

Question

Since these sets are ordered patterns that emerge from chaos, it seems that even the slightest roundoff error would distort the output, making the generated set different from the actual Mandelbrot set.

Answer 1

It's only chaotic at all near the boundaries of the escape/prisoner sets, so if the resolution has a cell which is in both the escape and the prisoner sets, then we can re-resolve that cell so that we get more information out of it.

I don't believe that the iterations will ever tend towards the boundary, so in the actual computation, we just go out far enough so that we don't have any critical error.

In other news, the actual pictures you see of the escape/prisoner sets do have a certain resolution.  If necessary, we can resolve it much finer than we need and then downscale the resolution in the end.

Answer 2

Saying that something "emerges from chaos" might sound profound, but in reality the Mandelbrot set emerges from a very well-defined, well-understood mathematical equation, which is not at all chaotic (in the sense of dynamical systems).

It's the limiting boundary of the Set that forms a fractal, which is all-too-easily associated with "chaos."

There is no "roundoff error" when calculating or plotting the Set. The more precise your calculations, the sharper the limiting boundary of the Set will be.