The Mandelbrot set and Julia Sets

The boundary of the Mandelbrot set

The boundary of the Mandelbrot set is the collection of all points c for which every circle, no matter how small, centered at c encloses points in M and also points not in M.

Pictured below is the boundary of M. At this magnification, the boundary agrees with our intuition about boundaries.

However, at higher magnifications the boundary becomes very fuzzy.

For boundary points, microscopic movements of c can change whether the sequence z_i runs away to infinity.

The boundary points of the Mandelbrot set belong to the Mandelbrot set.

(This is not true for all sets. If S is the set of all points (x, y) with x² + y² < 1, the boundary of S is the set of all points (x, y) with x² + y² = 1. Certainly, these points do not belong to S.)

Return to Some features of the Mandelbrot set boundary.