Does the Mandelbrot set get smoother because of a rounding error

Question

On many fractal explorer programs, after a certain level of zoom, the shapes begin to become smoother, as is shown. Is this a rounding error or an actual representation of the computation?

Answer 1

The closer a point is to the boundary of the Mandelbrot set, the more iterations it takes to determine whether the forward orbit of that point remains bounded--i.e., whether the point is inside the set or outside. Since the forward orbit of a point that is inside the set always remains bounded, and some of these are nonperiodic, it is in general not possible to prove that an arbitrary point is inside the set with a finite number of numeric iterations.

Since most algorithms to visualize the Mandelbrot set employ some preset maximum number of iterations, the lower this maximum, the more error is present near the boundary of the set. The algorithm fails to detect that these points that are actually outside the set have an orbit that eventually escapes, because the maximum number of iterations are reached, at which point the algorithm assumes the point is inside the set. Increasing the maximum number of iterations increases the accuracy near the boundary, but at the expense of increasing computational time.

Sophisticated algorithms exploit periodicity of bounded orbits of points inside the set to reduce the computational cost of high iterations, but as I have stated, some points inside the set have nonperiodic orbits. Also, periodicity detection is itself not free.

To be clear, the phenomenon you are showing in the picture is not due to rounding error. It is error due to the algorithm using a finite maximum number of iterations to determine whether a point is inside or outside the set, and this maximum number selected is insufficient for the magnification of the image. The higher the magnification, the larger this maximum needs to be.