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Lesson 1 – Page 3:


    It's helpful to have visuals to picture how the iteration and bailout processes work. As each point is calculated, a sequence of complex numbers, called an orbit, is generated. In the diagrams below, the red dot represents 0/0 – the value to which z is initialized. The green dot is c – the point whose orbit is being calculated, and whose color will be determined by the result.

    When the result of the Mandelbrot formula iterations meets the bailout criterion, the point is considered an "outside" point – colored black in these illustrations. If the result never crosses the bailout threshold, the point is an "inside" point, colored turquoise. Points colored white are outside points that are right on the border between the inside and outside areas. In the following illustrations, mouseover each image to see where the green dot lies in relation to the Mandelbrot set

Figure 5: c = 0.7125/0

    In Figure 5, the red dot represents the complex value with which the formula was initialized (z = 0/0). The green dot represents the point that was evaluated in the first iteration (c = 0.7125/0). You can see by the yellow segment, that the next iteration produced a value that went "off-screen". So the pixel represented by the green dot is an outside point. Mouseover the image at left to confirm that the green dot is, indeed, an outside (black) point.

Figure 6: c = -0.4375/-1.2

    Figure 6 bails out on the third iteration so the pixel represented by the green dot is an outside point.

Figure 7: c = -0.11875/0.75

    In some cases, the results of a point's iterations converge into a constant, repeating pattern of values and will never satisfy or "escape" the bailout condition. In Figure 7, the iterations produced by the point evaluated (green dot) settle quickly into a repeating pattern – from the red dot, to the green dot, to a third point making up the triangle. Those three values will repeat infinitely in this unchanging pattern without ever satisfying the bailout condition. In these cases, to prevent the computer from hanging infinitely in this iteration loop, we provide a maximum number of iterations and say, "If the bailout condition is not met by the maximum number (say, 100) of iterations, consider the point an 'inside' point." Remember that inside points of the Mandelbrot set are colored turquoise in these illustrations.

Figure 8: c = -0.763518/0.738808

    The point in Figure 8 bails out after 5 iterations, so it is an outside point.

Figure 9: c = -0.50625/0.56875

    The point in Figure 9 settles into a loop of five repeating values that will never cross the bailout threshold, so it is an inside point.

Figure 10: c = -0.7625/0.1625

    After many iterations, the point in Figure 10 bails out – outside point.

Figure 11: c = 0.2875/-0.35

    After many iterations, the point in Figure 11 settles into a repeating pattern that will never bail out – inside point.

Figure 12:
c =-0.39054087021840005066976260071379/

    This point, which is inside the Mandelbrot set, never bails out and never settles into a periodic orbit pattern.

    You can play around with these points and orbits too. Highlight, copy, and paste the following parameter set (all the text in purple below) into UF.

orbitExample {

    [Note: If UF reports missing formulas, or other errors, you are mostly likely missing the formulas needed for this image. Stop what you're doing. Go to Options|Update Public Formulas, download the Full formula collection, and then try again to paste the parameter set.]

    On the "segments" layer, right-click on one of the Point to track parameter fields (on the Outside tab of the Layer Properties tool window), select the Eyedropper tool , and then mouseover the image. The Fractal Mode tool window will display a preview of the iterations for any given point. Click on the image to draw the iterations. (To have the green dot reflect the location of first iteration, you will need to right-click in one of the Point to track parameter fields, choose Copy Complex Value, and then paste them (right-click again and choose Paste Complex Value) into one of the Point to track parameter fields on the layer named "point to track".)

    You will want to save this parameter set for future exploration, so create a parameter file called "Working with UF.upr" and save the image there. If you need a reminder how to save a parameter set, refer to the "Saving your fractal" section of the Quick Start Tutorial in UF's Help file.

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