Math Teachers Head to Classroom For Fractal Geometry Program

Math teachers from around the Valley will be discussing such things as the iterated function system, fractal dimension, the Mandelbrot set and the Sierpinski gasket at a three-day workshop at California State University, Fresno starting Monday.

The Fractal Geometry Workshop for mathematics educators is a satellite version of a weeklong program offered at Yale University. Fresno State is one of only two locations in the country selected to offer the workshop.

 Mike Fisher, an assistant professor of mathematics at Fresno State, and Nial Neger, a retired high school math teacher from Connecticut and consultant to the Yale program, will be facilitating the workshop.  Last summer, Fisher spent two weeks at Yale learning how to implement such a program.

Fractal geometry is a geometry of nature. A good example of a natural fractal is a fern, in which each frond appears to be made out of smaller fronds.

Techniques from fractal geometry have been used to test for patterns in DNA sequences and financial data and to develop fractal cellular telephone antennas and fractal capacitors.  One of the most high profile uses of ideas from fractal geometry is in the area of computer graphics. Techniques from fractal geometry have been used to create graphics for movies such as “Star Trek II: The Wrath of Khan,” “The Return of the Jedi,” “Titantic,” “Apollo 13” and “The Day After Tomorrow.”

(Editors: Mike Fisher is available for interviews. To schedule a time, call Shirley Armbruster at 593-1815.)

Fractal Geometry

Fractal geometry is a geometry of nature.  As Benoit Mandelbrot wrote in his famous book, The Fractal Geometry of Nature, “Clouds are not spheres, mountains are not cones, and lightning does not travel in a straight line.”  Although these natural shapes aren’t easily described by the geometry we learned in high school, one important feature that these shapes do have in common is that they are (more-or-less) self-similar.

What this means is that the object looks like that it is made of smaller copies of itself.  A good example of a natural fractal is a fern. 

Notice how each frond appears to be made out of smaller fronds.  Of course, we can’t continue “zooming in” on the fern and continue to see smaller “ferns” forever, but there is definitely a repeating pattern at work here.  Self-similarity in nature has been observed for quite a while.  Jonathan Swift (1667-1745) wrote in On Poetry. A Rhapsody:

                                    So, naturalists observe, a flea

                                    Hath smaller fleas that on him prey;

                                    And these have smaller still to bite ‘em;

                                    And so proceed ad infinitum.

A common example of a mathematical fractal is the Sierpinski gasket.

Notice how the gasket is made up of three smaller copies of itself, and each of those copies are made up of three smaller copies, and so on…

One of the main differences between natural and mathematical fractals is that that we can “zoom in” on mathematical fractals forever (or at least until our computer crashes).

Although many of the mathematical ideas from fractal geometry were developed in the early part of the twentieth century by mathematicians such as Gaston Julia, Felix Hausdorff, Helge von Koch, and Georg Cantor, it wasn’t until Benoit Mandelbrot, a mathematician at IBM and at Yale University, wrote a series of papers in the 1950’s, 60’s and 70’s that the field of fractal geometry was unified.  (In fact, it was Mandelbrot who coined the term ‘fractal.’)

One of the most famous mathematical fractals, that has made it into mainstream culture, is a fractal named for Mandelbrot himself: the Mandelbrot set.

Mandelbrot discovered this set in 1979 while he was a researcher at IBM.  There are many interesting qualities and features that have been discovered about this set since Mandelbrot first laid his eyes on it.  One fascinating and bizarre feature of the Mandelbrot set is that its boundary (think of this as the outline of the black part) is two-dimensional!  Even though the Mandelbrot set has been extensively studied, there are still many mathematical questions about the Mandelbrot set that remain unanswered.

Below are some “zooms” into various parts of the Mandelbrot set.

As you can see from the above (and below) picture(s), we can often find “baby” Mandelbrot sets as we zoom in on the Mandelbrot set.

Being a field so able to describe certain aspects of nature, it is not surprising to hear that there are many applications of fractal geometry.  Many people have used techniques from fractal geometry to test for patterns in DNA sequences and financial data.  Others have developed fractal cellular telephone antennas and fractal capacitors.  Certainly one of the most high profile uses of ideas from fractal geometry is in the area of computer graphics. 

Techniques from fractal geometry have been used to create graphics for movies such as Star Trek II: The Wrath of Khan, The Return of the Jedi, The Titantic, Apollo 13, and, more recently, The Day After Tomorrow.

 EDITORS and NEWS/PUBLIC AFFAIRS DIRECTORS: Press releases can be downloaded at www.fresnostatenews.com. University Relations also provides releases for news media companies via e-mail. To be added to the distribution list, send your e-mail address to tomu@csufresno.edu.