Chaos Theory and Fractal Geometry

Chaos Theory is relatively new and sources of information abound on the Internet, especially for those who want to know more about fractals. But there is so much more than fractals to share with your students. I have included some of those sites I found most informative, but some are very technical. These may augment your knowledge without necessitating a course in Chaos at the local university.

This site has been designed to offer educators a course in Chaos Theory and Fractal Geometry. The course encompasses the following topics:

• The mathematics and history of Chaos Theory
• Iterations and recursions
• Dynamical systems and how they relate to real world situations
• Graphing non-linear equations and the creation of strange attractors
• The mathematics of fractal, including fractal generators for students to create their own fractals
• Mandelbrot and Julia sets
• Measurement and Scale

This course offers text, relevant web sites, interactive Internet activities to reinforce concepts, and proposals for research topics and presentation modes for students. It is hoped that student projects may be presented for future students to use as models. The course is designed for high school students with a knowledge of Algebra. However, there are links to a number of sites that may be used for younger students.

Teacher Support:

• links to web sites that cover advanced topics in Chaos Theory and Fractal Geometry . These sites are marked by an asterisk. There is very little offered in traditional texts on these subjects, so please feel free to use this course or portions to augment your own offerings to students.
• message board at Making Order Out of Chaos Website

Online Text : Exploring Chaos and Fractals.

NCTM strands covered:

• discrete math
• measurement
• patterns and functions
• logic and language

Goals:

to familiarize students and teachers with the following concepts and skills:

• graphing non-linear equations,
• iterations and feedback loops,
• mathematics as an experimental science,
• infinite perimeters in a finite space,
• application of the principles of Chaos in real-life situations.

Assessment:

This course is designed for students to explore interactive sites in the topics listed above. This type of learning is best demonstrated through projects. One possibility is to have students explore a topic in greater depth than presented in the course and present the information through a web site or HyperStudio program that can be loaded onto a web site.

Acknowledgements and Recommended Reading:

The strange attractor featured in the site map (the spiral image) is the work of Tom Stilson at Stanford University. He has some wonderful images to view on his website.

Some wonderful texts for further reading include:

Chaos: Making a New Science by James Gleick (1987)

Leadership and the New Science by Margaret Wheatley (1992)

And to augment with some activities not available on the Internet:

Discrete Mathematics Across the Curriculum K-12, available through NCTM (1991)