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Topic: Interview with Bob Devaney
Replies: 4   Last Post: Apr 12, 1993 7:31 PM

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Evelyn Sander

Posts: 187
Registered: 12/3/04
Devaney Part 2: Mandelbrot Set
Posted: Apr 9, 1993 5:29 PM
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The following is the second in a two part series of articles
based on an interview with Boston University professor Bob
Devaney.

Educators often spend time considering what mathematics concepts
are appropriate for high school students. According to
Professor Devaney, the process of iteration should
be taught in high school. The concept comes up in savings and
interest problems and Newton's method. Chaos, fractals, talking
about the Mandelbrot set: these are fun ways to introduce the
iteration. Also, high school students often have seen a bit
of complex arithmetic. The Mandelbrot set is a nice illustration
of this concept.

Professor Devaney thinks that detailed discussion of dynamics
is most appropriate for after school math clubs, or as a last
topic at the end of the term when there is often some dead time
anyway. However, he says: "Students spend about two or
three weeks learning how to factor cubic polynomials; there is
no reason why they shouldn't also spend a couple weeks learning
about the Mandelbrot set."

As to the actual presentation of the Mandelbrot set appropriate for
high school students, here is a general outline. It is perhaps
even of interest to people who are not in high school but only
know the Mandelbrot set from seeing pictures of it. The explanation
should involve quite a bit of computer demonstration, but perhaps
this description will give the idea.

Consider the real one dimensional quadratic map f(x)=x*x+1. Iterate zero
under this map. The first few iterates are 1,2,5,26. It is fairly
clear that under successive iterations, one gets an arbitrarily
large number. Now consider the map g(x)=x*x+0. In this case,
g(0)=0, so there is a fixed point at the origin. For h(x)=x*x-1,
h(0)=-1, and h(-1)=0, so zero is a period two point.

The iterates of the origin under maps of the form of f, g, and
h are generally hard to calculate. Putting these maps into a
common form, add a parameter, and consider f(x,c)=x*x+c.
Consider c=-1.1. If you try this in your head, you see that it
is not so easy.

At this stage, one needs a program which graphs iterates of the
origin under f. Using this program, experiment with different
values of c to see what sort of behavior occurs. At c=-2, the
picture becomes chaotic.

Now switch to the complex plane. In other words, consider the
function f(z,c) defined above, except that now z and c can be
complex numbers. For example, for c=i, I calculate the first two
iterates of zero: f(0,i)=i and f(i,i)=-1+i.

The Mandelbrot set is defined to be all the c values for which
iterates of the origin stay bounded under iteration. At this
point, one should use a computer program to draw the Mandelbrot
set. Then, use a program with two windows representing copies
of the complex plane: in one window, the user picks a complex
c value with a mouse. After the user picks c, the other window
shows iterates of the origin. This program gives
the user an idea of what it means for iterates to get
arbitrarily large.

It is at this point possible to learn about the period of the
periodic points in the Mandelbrot set based on the point's
placement in one of the bulb shaped parts of the Mandelbrot set.
For this and further discussion, I refer to Professor
Devaney's books.

The above explanation is a fairly simple and fun way to teach
the concepts of iteration, complex arithmetic, and boundedness.
It answers the concern that all this popularization of
fractals and the Mandelbrot set is just a matter of looking at
pretty pictures without any knowledge of the math; although
this is not a complete explanation of the mathematics involved,
it does seem to give a partial understanding with little needed
background.







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