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Fractal Geometry of the Mandelbrot Set (Cover Page)
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One of the principal goals of the chaos club was to acquaint students with different representations of the same idea. Iteration provides a wonderful opportunity to do this. During the first weeks of the club, students became familiar withthe notion of iteration by listing orbits, plotting histograms, and drawing time series. Working only with x2 + c for real values of x and c, students were exposed to such contemporary mathematical phenomena as chaos and bifurcations.
One image that unifies all of these ideas is the orbit or bifurcation diagram. While we did not have time in the chaos club to investigate this image fully, the relationship between this picture and the Mandelbrot set is described in [6].
A large portion of time in the club was devoted to explaining the Mandelbrot set. Students used the software package [12] to understand the meaning of the Mandelbrot set as the set of parameter values for which the orbit of 0 does not escape. Nowadays, teh web-based package [9] is much more suitable for this purpose. Of course, the idea of a parameter space is one that is quite unfamiliar to students. Nonetheless, with software that displays the fate of the orbit of 0 as a mouse is dragged over the Mandelbrot set, students quickly catch on to the idea.
One of the main experiments that the students performed was to discover the relationship between the geometry and the periods of the bulbs. Working in groups, students first magnified then printed various primary bulbs. Then they assembled them in a large mural depicting the Mandelbrot set together with the associated periods. This large view enabled students to see the fact that the period was equal to the number of spokes on the principal antenna without difficulty. This large image still hangs in the school's computer lab and is a source of many questions for current students.
Once the students understood the mathematics behind the periods of the bulbs, they naturally began looking for the pattern that governed the arrangement of the bulbs. This requires further work with the quadratic filled Julia sets and will be described in the next article in this series.
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