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    Mandlebrot Fractal Geometry set

    Mandelbrot Fractal Geometry set - 0
    Mandelbrot Fractal Geometry set - 0
    Start.

    Mandelbrot set with continuously coloured environment.
    Mandelbrot Fractal Geometry set - 1
    Mandelbrot Fractal Geometry set - 1
    Step 1 - Head and Shoulder

    Gap between the "head" and the "body", also called the "seahorse valley".
    Mandelbrot Fractal Geometry set - 2
    Mandelbrot Fractal Geometry set - 2
    Step 2 - Seahorse Valley

    On the left double-spirals; on the right "seahorses".
    Mandelbrot Fractal Geometry set - 3
    Mandelbrot Fractal Geometry set - 3
    Step 3 - Seahorse

    "Seahorse" upside down
    Mandelbrot Fractal Geometry set - 4
    Mandelbrot Fractal Geometry set - 4
    Step 4 - Seahorse Tail

    The central endpoint of the "seahorse tail" is also a Misiurewicz point.
    Mandelbrot Fractal Geometry set - 5
    Mandelbrot Fractal Geometry set - 5
    Step 5 - Seahorse Tail (Part)

    Part of the "tail" — there is only one path consisting of the thin structures that lead through the whole "tail". This zigzag path passes the "hubs" of the large objects with 25 "spokes" at the inner and outer border of the "tail"; thus the Mandelbrot set is a simply connected set, which means there are no islands and no loop roads around a hole.
    Mandelbrot Fractal Geometry set - 6
    Mandelbrot Fractal Geometry set - 6
    Step 6 - Double Hook

    Satellite. The two "seahorse tails" are the beginning of a series of concentric crowns with the satellite in the center.
    Mandelbrot Fractal Geometry set - 7
    Mandelbrot Fractal Geometry set - 7
    Step 7 - Satellite

    Each of these crowns consists of similar "seahorse tails"; their number increases with powers of 2, a typical phenomenon in the environment of satellites. The unique path to the spiral center passes the satellite from the groove of the cardioid to the top of the "antenna" on the "head".
    Mandelbrot Fractal Geometry set - 8
    Mandelbrot Fractal Geometry set - 8
    Step 8 - Satellite Antenna

    Several satellites of second order may be recognized.
    Mandelbrot Fractal Geometry set - 9
    Mandelbrot Fractal Geometry set - 9
    Step 9 - Satellite Head and Shoulder

    The "seahorse valley" of the satellite. All the structures from the start of the zoom reappear.
    Mandelbrot Fractal Geometry set - 10
    Mandelbrot Fractal Geometry set - 10
    Step 10 - Satellite Seahorse Valley

    Double-spirals and "seahorses" - unlike the 2nd image from the start they have appendices consisting of structures like "seahorse tails"; this demonstrates the typical linking of n+1 different structures in the environment of satellites of the order n, here for the simplest case n=1.
    Mandelbrot Fractal Geometry set - 11
    Mandelbrot Fractal Geometry set - 11
    Step 11 - Satellite Double Spiral

    Double-spirals with satellites of second order - analog to the "seahorses" the double-spirals may be interpreted as a metamorphosis of the "antenna".
    Mandelbrot Fractal Geometry set - 12
    Mandelbrot Fractal Geometry set - 12
    Step 12 - Satellite Spirally Wheel with Julia Islands

    In the outer part of the appendices islands of structures may be recognized; they have a shape like Julia sets; the largest of them may be found in the center of the "double-hook" on the right side.
    Mandelbrot Fractal Geometry set - 13
    Mandelbrot Fractal Geometry set - 13
    Step 13 - Satellite Seahorse Tail with Julia Islands

    Part of the "double-hook"
    Mandelbrot Fractal Geometry set - 14
    Mandelbrot Fractal Geometry set - 14
    Step 14 - Julia Islands

    The islands above seem to consist of infinitely many parts like Cantor sets, as is actually the case for the corresponding Julia set Jc. However they are connected by tiny structures so that the whole represents a simply connected set. The tiny structures meet each other at a satellite in the center that is too small to be recognized at this magnification.
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