Lesson 1     Lesson 2     Lesson 3     Lesson 4     Lesson 5     Lesson 6     Additional Information
    Glossary of Terms     Links & Resources

Lesson 1 – Page 5:

Exploring the Mandelbrot coastline

Highlight, copy, and paste this parameter set into UF.

DefaultMandelbrot {
  title="Default Mandelbrot" width=500 height=400 layers=1
  method=multipass caption="Background" opacity=100
  center=-0.5/0 magn=1
  maxiter=100 percheck=off filename="Standard.ufm" entry="Mandelbrot"
  p_start=0/0 p_power=2/0 p_bailout=1e20
  transfer=none solid=4278215290
  transfer=linear solid=4286722382 filename="Standard.ucl"
  entry="Basic" p_type=Iteration
  linked=yes smooth=yes index=0 color=0 index=200 color=16777215
  smooth=yes index=0 opacity=255 index=200 opacity=255

    We will use this image as our starting place, so save the parameters in your "Working with UF.upr" file. [Note: Depending on your screen resolution, you may want to make the image bigger before you save it. You can do this by increasing either the Width or the Height parameters on the Image tab of the Fractal Properties tool window. Make sure "Maintain aspect ratio" is checked first.]

    This parameter set is my personal default fractal – customized with the settings I prefer to use when starting a new fractal. On the Formula tab, I have selected my preferences: Multi-pass Linear for the Drawing Method, Periodicity Checking is Off, Maximum Iterations is set at 100, and the Bailout value at 1e20 (which is shorthand for a 1 with twenty 0's after it).

    I like to begin every image with a grayscale gradient so that I don't become sidetracked or confused by colors while I'm exploring the fractal structure. Open the Gradient Editor and note that the simple gradient has two control points – one black, one white. If you right-click on the Gradient Editor, you'll see that Color and Opacity are linked and "Smooth Curves" is enabled. My default image also has a magenta color selected as the Solid Color on the Outside tab and a turquoise color selected as the Solid Color on the Inside tab. Because I'm using a grayscale gradient for Outside points, I want a completely different color for the Inside. You could select any color, but for the purposes of these tutorials, we will know that turquoise areas are always Inside areas.

    To ensure that I begin every image with these parameter and gradient settings, I have saved the parameters and selected it as the Default Parameter set on the Options|Options|Defaults tab. For the purposes of this course, I suggest using this Default Mandelbrot image. Later you are free to develop and save a customized default parameter set according to your own preferences.

    We know that all the solid black areas are outside points and the turquoise area contains inside points. If you zoom into either of these areas, you'll get more of the same – more solid black or more solid turquoise. The interesting fractal structure, however, is located on the border between the inside and outside areas – the white areas on our default fractal.

    Click and hold your left mouse button on any white area, then drag it slightly to invoke the selection box. Keep the selection box quite small and then right-click and select Zoom In (or press Enter). Though your location may vary, your fractal window should now be partially solid turquoise, and the rest filled with a "coastline" of intricate fractal structure in grayscale colors. In most cases, you should see the beginnings of some sort of spiral structure.

    One of my favorite areas to explore is the disk-side of Seahorse Valley, which is located in the area designated by the red box. (The structures on the cardioid side of this cleft actually look more like seahorses, but I'm fond of this particular area.)

    In case you zoomed in elsewhere before, Undo until you get back to the Default Mandelbrot. Then locate the Seahorse Valley area, which is the pinched point separating the largest disk from the main cardioid.

Figure 13: Seahorse Valley

    If we zoom into the boxed area in Figure 13, we get something like this:

Figure 14: Spiral shape on the west coastline of Seahorse Valley

    Zoom again into one of the cranium-like shapes:

     Figure 15 looks like it has lots of potential for exploration, but note all the turquoise indicating Inside points. This is because our Maximum Iterations is set too low for the depth to which we've zoomed.

Figure 15: Cranium-like spiral

    Change the Maximum Iterations setting on the Formula tab to 1000 so that the formerly turquoise areas fill with intricate fractal structure.

    This screen-shot is too small to see clearly, but look at your own fractal near the area boxed in red above. Can you see a very small turquoise shape? Zoom in on it:

Figure 16: Midget

... and we discover a mini-Mandelbrot shape, called a midget. Midgets found by deep-zooming into the Mandelbrot structure are often surrounded, as is the one in Figure 17, by very beautiful and intricate spirals. This midget was fairly easy to find. Others are more elusive and take some practice to find. The more exotic midgets can only be discovered by zooming deeply into the fractal structure.

Figure 17: Midget (zoom)

Back – Lesson 1Page 6

Copyright © 2005-2011 Janet Parke