|A. Complex Iteration. A review of
complex arithmetic: the background needed to use the formulas that generate pictures of Julia
sets and of the Mandelbrot set.
|B. Julia Sets. For a complex number c, the filled-in
Julia set of c is the set of all z for which the iteration z → z2 + c does
not diverge to infinity. The Julia set is the boundary of the filled-in Julia set. For almost all c, these
sets are fractals.
|C. The Mandelbrot set. The Mandelbrot set is the
set of all c for which the iteration z → z2 + c, starting from z = 0, does
not diverge to infinity. Julia sets are either connected (one piece) or a dust of infinitely many points.
The Mandelbrot set is those c for which the Julia set is connected.
|D. Combinatorics of the
Mandelbrot Set. Associated with each disc and cardioid of the Mandelbrot set is a cycle.
There are simple rules relating the cycle of a feature to those of nearby features. From this
we can build a map of the Mandelbrot set.
|E. Some features of the Mandelbrot set boundary.
The boundary of the Mandelbrot set contains infinitely many copies of the Mandelbrot set. In fact, as
close as you look to any boundary point, you will find infinitely many little Mandelbrots. The boundary
is so "fuzzy" that it is 2-dimensional. Also, the boundary is filled with points where a little bit
of the Mandelbrot set looks like a little bit of the Julia set at that point.
|F. Scalings in the Mandelbrot Set.
The Mandelbrot set includes infinitely many smaller copies of itself. These can be organized
into hierarchical sequences for which the ratio of the sizes of successive copies approaches
a limiting value. Some of these give the Feigenbaum constant associated with the logistic map,
others give new constants. Some give integer limits.
|G. Complex Newton's Method.
Julia sets related to finding the roots of equations. SImilar features arise in magnetic pendula and in
light reflected within a pyramid of shiny spheres.
|H. Universality of the Mandelbrot Set.
Newton's method for a family of cubic polynomials revealed more copies of the Mandelbrot set.
Yet Newton's method is nothing like z → z2 + c. Further investigation
shows we're surrounded by Mandelbrot sets.
|I. Here is Ray Girvan's page on the
Mandelbrot Monk. Was the Mandelbrot set discovered in the 13th
century? Read the page carefully.
|J. Fractals in Literature.
Not only are fractals present in the structure of literature, sometimes they are the
subject of literature.
|K. Fractals in Art.
Because it exhibits a balance of familiarity and novelty, the Mandelbrot set is more
interesting than the Sierpinski gasket. This aesthetic maxim is familiar in art.
|Here is a recent class project on the Mandelbrot set.