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Mandeldraw
(for Java1.1-compatible Browsers and Viewers)
from Hartmut Neubauer-Stankiewicz
back to the Mandelbrot-Set
Guidance
- Click in the dialog "Julia MandelControls" on the field "start".
- Wait, until the whole figure is displayed.
- Now you can enlarge parts of the original Mandelbrot-Set-Figure. For that move the mouse to the lower left end of the desired window and pull with pressed (left) mouse button to the upper right end. During pulling the window is represented as rectangle. Then release the mouse button.
Additionally you should increase the number of "max. Iteration". Rule of thumb: The smaller the window, the higher the necessary number of "max. iterations". Subsequently, press again "start". Now the selected window will be enlarged.
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You can repeat this process as often as you like. Instead you can also display certain points of the Julia-Set. For that:
- click in the field "Julia"
- click with the mouse button in any field in the Mandelbrot-Set-Figure or that presently displayed window (tip: the peripheral areas are particularly interesting!);
- possibly fit again the "max. Iteration" (this time downwards; any value between 50 and 100 would be enough);
- and click once more on "start".
- After a short time the new figure is displayed. Similarly as in step 3 you again can let parts display enlarged or also adapt the edge coordinates by hand (in the order: left, down, right, above).
- You again can subsequently display an Mandelbrot-Figure, by clicking the field "Mandelbrot". Note: The original figure is just displayed, if you enter twice the number 0 into the two fields behind "C-Real/Imag:".
- You can always abort the process for computing and painting by clicking on "stop". - Note: With the »Applet Viewer« you can also modify the size of the display range. Immediately after you did this, do not draw a new rectangle! But click first once more the button "start".
- You do not need to change the value for "Obere Grenze" ("upper boundary"); however you can obtain additional effects, if you adapt the value of "Innenmuster" ("interior pattern"). For better understanding, however some more mathematics are necessary.
- The functions Fn on the complex numbers are defined as follows:
F0(z) = 0
F1(z) = -z
F2(z) = z2 - z
.
.
.
Fn(z) = (Fn-1(z))2 - z für n = 1, 2, 3 etc.
(mathematicians also call such functions »polynomials«)
The Mandelbrot-Set does now consist exactly out of the points z, for which the sequence {Fn(z) | n = 1, 2, 3 ...} is limited, therefore doesn't infinitely expands. This applies in particular to the zeros points of the polynomials Fn (for n = 1, 2, ...). These are the numbers of z, to which applies:
Fn(z) = 0
Since Fn is a polynomial of the degree 2n-1, so it has 2n-1 zero points in complexes. (For n = 66 this is about as much as the square number of the population of world!!) If you did not modify the value "4" for "Innenmuster" ("interior pattern") (generally: if this value is larger than 0.25), you can see very well some of these points: that are the points, which are emphasized by concentric circles. I call these points the "eyes" of a figure.
If you now modify the value of n = "max. Iteration", the following is noticeable:
- With each n > 0 an "eye" is in the center of the main body of the figure (exactly in the zero point).
- The large protruding part on the right of the main body has only with even n an "eye".
- If n is divisible by 3, then you can additionally discover - among other things - "eyes" in the large protruding parts above and down, in addition in the main body of a "satellite" (in such a way I call the small figures, which are similar to the entire Mandelbrot-Set) on the "antenna" on the right.
- If n is divisible by 4, you find an "eye" in the smaller protruding part, which borders as next at the the large protruding part on the right towards the "antenna", in the same way - among other things - in the smaller protruding parts on the left at the top and at the bottom, as well as in the "satellites", which you can find completely above and down.
Thus you can realize that each protruding part and each "satellite" has a "order": the smallest iteration number, with which one can find a zero point in the center of this figure.
(Remark: With "Innenmuster" ("interior pattern") larger than 0.25 the following calculation for the colour representation takes place: From the last calculated value Fn(z) the absolute value and the logarithm of it are calculated; this will be multiplied with the given number; of that the next lower integer is formed; as soon as this number changes, a colour change takes place.)
- There is still another possibility of arranging the Mandelbrot-Set-Figure coloured. For that enter a very small, but still positive number (e.g. "0,001") under "Innenmuster" ("interior pattern"). The stronger you enlarge a window and the higher the iteration number is, the smaller must this number be. In this case each partial section of the figure has another colour, which depends on the "order":
- The main body of the figure (order 1) has the colour red.
- The large protruding part on the right side (order 2) is red with small tendency too magenta.
- The section figures of the order 3 are magenta with a trace of red.
- The figures of the order 4 are magenta (rose).
- The colours for the figures of the orders 5 to 8 are enough from dark-orange to bright magenta.
- The colours for the figures of the orders 9 to 12 are enough from Indian-yellow to very bright magenta; from 13 to 16 the colours from yellow to white are enough. White are also all section figures with higher orders as well as the edges of the remaining figures.
- Still another important tip: Enter for decimal separation always a point, thus e.g.. »0.1« instead of »0,1«!
to the applet
Do you habe comments, suggestions, ... for this applet?
Then you can contact Hartmut Neubauer-Stankiewicz via Email.
In case of questions or suggestions just write an Email to me.
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