What is the Mandelbrot Set?
The frayed edges of the Mandelbrot set are special. It is full of patterns. If you pick out a small box of the edges and calculate colours with the formula of the Mandelbrot set, you get different colourful patterns depending on the place.
I choose the way with real numbers, because not many people are familiar with complex ones. You use the following formulas for the Mandelbrot set. x I explain the calculation with point P I start with point N(0|0).
The first term a The second term is derived from the coordinates of the
given point P The third term is derived from the coordinates of the
previous point P x In this way you get the sequence 0.81, 0.74, 0.51,
1.0, 0.74, 1.1, 1.8, 2.4, ... belonging to
the given point P
The first sequence is convergent with limit equal to 0.31. (The points with convergent sequences form the Mandelbrot set.) The other sequences are divergent. The terms get larger
and larger without limits, but not in the same way.
I make a note of these numbers in tabular form:
You give the colour black to the first point with the
convergent sequence. So the Mandelbrot set becomes black.
Now the points can be drawn.
The computer cannot avoid one mistake: It cannot calculate all terms of a sequence. If it investigates only 50 terms for instance, maybe the sequence doesn't exceed 2 in number, nevertheless it is divergent. You make the mistakes less crass, if you determine more terms (e.g. 500). Nearly everyone who writes computer programs and is interested in computer graphics has had a try with the Mandelbrot set. It is an unforgettable experience if a simple program produces the complicated Mandelbrot set for the first time. In former times it took hours and hours (Commodore 64 nostalgia!). Every own attempt of programming fades beside the standard
program
After starting the program the Mandelbrot set appears.
You form a small box with the mouse and move it with pressed mouse button
to an edge. Then you press the enter key. A new picture appears on the
screen. If you want you can look for a new place and press the enter key.
The patterns repeat. You recognize self similarities.
-0.567709792 < x < -0.557685031 and 0.638956191 < y < 0.646482313 You set a palette in grey with Colors/Load Color Map.../altern.map
.
If you save a picture, the coordinates of the boxes are
also recorded. You find them with You see the coordinates with You can feed the coordinates into the computer via You must choose
The program makes possible a trip through the fractal
geometry beyond generating the Mandelbrot set. You must choose new formulas
with
You see three Julia sets below. The pictures belong to the inside, the edges and the outside of the Mandelbrot set going from the left to the right. The picture in the middle shows fully detailed patterns with big depths.
You also come to Julia sets inside Winfract, if
you choose Fractals/Formula... Julia. Now it is possible to give
precisely two coordinates. You can't do it with the mouse button.
German Albert Kluge
Alexander F.Walz
Christian Gloor
Christian Symmank
Hanno Rein
Manfred Thole
Thomas Hövel
Wikipedia
English Eric W. Weisstein
Jules Ruis
M. Eric Carr
Michael Frame, Benoit Mandelbrot, and Nial Neger
Robert Munafo
Wikipedia
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