Mandelbroot and Julia

You can find a program investigating the Mandelbroot and Julia sets at this address: https://sourceforge.net/projects/heptagram-problem/files/Mandelbroot_Julia/

On startup, the program should look like this:

It shows the “fate” of the point z0= 0.22709+i0.39920 represented by the rightmost end of the spiral. The iterates are represented by the corners of the spiral; in this case the spiral seems to converge to some point; so the iterates of this particular value of z0 will stay bounded forever; as is shown by the black colour. You may choose a region of the Mandelbroot set for further investigation; press and hold the left mouse button, and pull out a rectangle, to see something like this:

When you release the left mouse button, you will see a screen like this:

Here you have some options for the choice of the next picture. I have accepted the proposals of the program using the OK button. This is what came out:

I repeated the process, as shown in the following three pictures:

You see a mini Mandelbroot set; I have started a “spiral” there, it was almost closed, and pressing the F7 button I asked the program to close it more precisely. Using the menu option “New Picture”, I chose to come back to the main Mandelbroot set, and we see a nice closed “spiral” here:

Spiral is not an appropriate designation, perhaps one might say path or trajectory, or something like that.

Finally let me show, that one can also get Julia sets, as shown in the following two pictures. A trick: when using the mouse, just click down and up on the same place. If you manage to draw a “rectangle” with no area, you will automatically be prompted to choose Julia sets; if you don’t follow the advice you ask for trouble. Julia sets where the c-value are close to the boundary of the Mandelbroot set tend to be more fascinating.