Teacher Information: Mandelbrot and Julia Sets
While there are a number of web resources that make reference to or actually
construct the incredibly intricate and beautiful Mandelbrot set, this virtual
manipulative is fast and versatile and makes connections and relationships accessible
to student users.
In exploring Mandelbrot and Julia sets, the most common approach is simply
to observe, experiment, and marvel at the beauty and complexity hidden within.
For students who have some experience with complex numbers, the options are
considerably richer. The construction of all such sets depends on repeatedly
applying (iterating) a simple function, f(x) = x2
+ c, where x and c are complex numbers. For certain choices
of x and c, the iteration values approach 0; for others, the iteration
values oscillate wildly or go off to infinity, and some stay a fixed distance from
the origin. For a fixed complex number c, the set of x-values that
do not go to 0 or infinity form the Julia-set for that c.
The Mandelbrot set M, which is displayed in the left panel when this
virtual manipulative is opened, is closely related to iteration of the same
function that defines all of the Julia sets. The simplest definition is probably
that M consists of all the complex number c for which iterating
f(x) = x2 + c, starting at 0, does not
go to infinity. Since f(0) = c, the sequence we are interested in
begins c, c2 + c, (c2 + c)2
+ c, ... . Remarkably, the Mandelbrot set has another defining property
in terms of Julia sets: M consists of all those complex numbers c
for which the corresponding Julia set is connected, all in a single piece.
The fractal nature of the Mandelbrot and Julia sets comes from the fact that
the behavior depends very strongly on the starting point. That is, iterating
the defining function for two different numbers can give completely different
results even though the two numbers are very close together. This implies that
the structure of these sets retains its beauty and complexity no matter how
much we magnify the lens with which we view them. To see this we have a repeated
zoom capability that can magnify an original by a factor of many millions. Starting
with any viewing panel, place the cursor anywhere, and click and drag to draw
a square. When the mouse button is released, a new portion of the Mandelbrot
or Julia set is recalculated for the region within the square and appears in
the other panel. To keep track of the magnification, after each zoom operation,
we are told how large the original would have to be to allow us to see the picture
displayed.
Choosing the option allows the user to
move the cursor in the control panel just to the right of the buttons, thus choosing
complex number c. When the mouse button is released, the Julia set for that
value of c is drawn in the left panel, and the zoom feature becomes available.
One of the most striking properties of Julia sets is their "self-similarity."
By choosing zoom-squares carefully, it should be possible to see how the portion
in a small square, when blown up in the second panel, looks just like that in the
first panel.
For many users, the most interesting Julia sets are those for points near the boundary of the Mandelbrot set, so choosing the button allows us to choose a point for a Julia set, but we see where our point is located relative to the Mandelbrot set. Students may want to look at several Julia sets for points near the boundary of the Mandelbrot set, to observe how they change from being connected (inside M) to disconnected (outside M).