Goal:
Your goal in this experiment is to view a remarkable
relationship between the structure of the Mandelbrot set near special
c-values and the shape of the corresponding filled Julia sets.
Goal:
Your goal in this experiment is to view a remarkable
relationship between the structure of the Mandelbrot set near special
c-values and the shape of the corresponding filled Julia sets.
Procedure:
Using the
Quadratic map applet
for at least 5 different decorations attached directly to
the main cardioid, magnify the central portion (the junction point)
of the antenna from
which all of the spokes emanate. Zoom in on this point several
times using the magnify feature of the program. Now, using the
mouse, click on this point to draw
the filled Julia set for this c-value. Magnify the corresponding junction
points in these filled Julia sets.
Results:
Compare the shape of the magnified Mandelbrot
set near the
junction point with the corresponding magnifications of the filled Julia set.
Are there any
similarities between these two images? Explain in an essay.
In your essay include the c-values you worked with as well as
several pictures explaining your observations.
Notes and Questions:
The similarity between magnified portions
of the Mandelbrot set and the corresponding filled Julia sets holds only near
certain c-values such as the central junctions of the antenna. These are
c-values for which 0 is eventually periodic. Such c-values are called
Misiurewicz points.