||last updated: 2004-12-04
where f is a Mandelbrot lemniscate
The Mandelbrot set
is one of the most popular fractals, on many places on the
internet pictures (often colored) of the curve are to be seen.
The curve is a generalization of the bifurcation to the
The boundary of the set is very complex, and while enlarging and enlarging this part of
the picture, again and again different beautiful structures are conveyed. In some parts
miniature Mandelbrot set formed Mandelbrot baby's are to be seen.
Before Mandelbrot already the Hungarian mathematician Riesz (1952) posed
questions related tot the Mandelbrot set. In 1978 Brooks and Matelski
made the first computer graphics of the curve. Adrien Douady and John Hubbard
studied the curve, they proved that the set is connected, and named it to
In 1994 it was Shishikura who proved the (Hausdorff) dimension of the curve being
It is not yet known whether the curve is path-connected.
The main part of the curve, on the right, has the form of a cardioid; the part on the right has the form of a circle.