So what do the Julia sets look like? Here is the one we know. The small image on the left of the Mandelbrot set has a cross at c=0 and the circle on the right is the Julia set corresponding to the function f(z)=z*z+0.
1.
We now show some Julia sets for a simple set of points in and outside the Mandelbrot set. We start wandering toward the cusp of the large cardioid and then go beyond it. For each value of c, we give the value of c, an image of the Mandelbrot set with a cross at that c value, and an image of the corresponding Julia set. The image of the Julia set is clickable to give a larger image of the Julia set. The larger images of the Julia sets on this page and the next two pages are from 2000 to 6000 bytes.
With one exception later on (where we show a detail of a Julia set), all the images of Julia sets that we show are in squares that cover real and imaginary values from -2 to 2.
c=.15+0i.
The circle starts to distort and become dimpled.
2.
c=.25+0i.
At the cusp of the cardioid, the dimples have deepened
and have become cusps themselves.
3.
c=.26+0i.
Just beyond the cusp and outside Mandelbrot set, the
dimples do not deepen further, but split. The Julia set has no
continuous thread. Each part that looks like a thread is cut
infinitely often by spiral pairs similar to the main ones at the
sides, top and bottom.
4.
c=.3+0i.
Farther from the Mandelbrot set, the pieces of the
Julia set spread farther apart.
5.
c=.5+0i.
The Julia set spreads farther...
6.
c=1+0i.
...and farther.
7.
Forward to What is the
Mandelbrot set - Page III (10 embedded images).
Back to What is the
Mandelbrot set - Page I (1 embedded image).
Back to The Mandelbrot Set and Julia Sets.