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MAMT 590 Special Topics: Fractals and Topology

Fall 2011

**Instructor: **Tom Hull

**Time:** Mondays and Wednesdays from 5:30pm - 7:20pm, starting September 26, 2011.

**Location:** CSP 187 (Center for Sciences and Pharmacy building)

Fractals have been said to describe the "Geometry of Nature." Whether or not this is actually true is somewhat controversial. But it is true that fractals have given scientists, engineers, and mathematicians new tools with which to study nature. Instead of trying to model natural objects with "Euclidean" tools like circles and lines, fractals allow one to model nature with objects that look bumpy and rough, in the same way that a tree leaf or a rock looks bumpy and rough.

"What are fractals," you ask? Take this course and find out! They are a very new topic in mathematics; the work "fractal" was only invented in the 1970s, which makes fractal geometry a very recent development in math (especially compared to, say, calculus or Euclidean geometry). In this class you will learn:

- How to identify fractals.
- How to make pictures of them on a computer, using the software package
*Mathematica*. - How to characterize fractals and use them to model things in the real world.
- About certain "classic" fractals, like self-similar fractals, iterative function systems, and the Mandelbröt Set and Julia Sets.

Also, we will study how fractals fit into a branch of math called Topology. (Point-Set Topology, to be exact.) This will allow us to explore in-depth one of the most suprising (and lucrative) applications of fractals: Fractal image compression on computers.

**Textbooks:** There are two books for this course:

*Superfractals*, by Michael F. Barnsley, Cambridge University Press, 2006, ISBN 978-0521844932*Chaos: Making a New Science*, by James Gleick, Penguin (Anv Rep edition), ISBN 978-0143113454

NOTE: Previously on this web page I posted a different, out-of-print book as our textbook, hoping that I would be able to find enough copies for us to use it as our book. This has proved to be far too difficult and expensive, and so I am using the above (both in-print) books instead. If you already managed to find a copy of the other book, please contact me and we can try to work something out. (Another positive, though, is that the above two books are much cheaper than the book I was planning to use! :-)

Please contact me if you have any questions! thull@wne.edu

Last updated: 8/12/2011