Mandlebrot Fractal Geometry set

Mandelbrot Fractal Geometry set - 0
Mandelbrot Fractal Geometry set - 0

Start. Mandelbrot set with continuously coloured environment.

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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".

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Mandelbrot Fractal Geometry set - 2
Mandelbrot Fractal Geometry set - 2

Step 2 - Seahorse Valley On the left double-spirals; on the right "seahorses".

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Mandelbrot Fractal Geometry set - 3
Mandelbrot Fractal Geometry set - 3

Step 3 - Seahorse "Seahorse" upside down

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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.

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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.

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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.

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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".

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Mandelbrot Fractal Geometry set - 8
Mandelbrot Fractal Geometry set - 8

Step 8 - Satellite Antenna Several satellites of second order may be recognized.

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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.

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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.

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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".

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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.

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Mandelbrot Fractal Geometry set - 13
Mandelbrot Fractal Geometry set - 13

Step 13 - Satellite Seahorse Tail with Julia Islands Part of the "double-hook"

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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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