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This section summons some properties of the Mandelbrot set first without proof,
then some statements are proved.
Theorem 3 (Symmetry)
The Mandelbrot set is symmetric with respect to the real axis.
This means, if a complex number
belongs to the mandelbrot set
then this is also true for the conjigate complex number
.
(You can see this symmetry in Figure
3)
Theorem 4 (Boundary)
The Mandelbrot set is bounded. You can easily proove, thet the set
must lie in the interior of the circle
.
(Also see Figure
3)
Theorem 5 (ItselfSimilarity)
The Mandelbrot set is itself similar in a non exact sense.
Figure 4:
A plot of a zoom into the set.You can see a part of the set
with the same shape as the whole set. The resolution and
iteration depth is the same as in Figure 3

Theorem 6 (Real axis)
The intersection of the Mandelbrot set with the real axis is
the real interval
Theorem 7 (connected set)
The mandelbrot set is a connected set.
This result seems to contradict every picture of the set, where you
think that the set consists of separated insuli.(See Figure
5)
Figure 5:
A plot of a zoom into the set. You can see some connected
insuli of the Mandelbrot set

Theorem 8
Fractal dimension of the border
The border of the Mandelbrot set is a fractal structure with not yet
known fractal dimension.
Next: The Dynamic of the
Up: The Mandelbrot set
Previous: The Mandelbrot set
Tino Kluge
20001205