A fractal is a geometric pattern that repeats itself at smaller and smaller scales to produce irregular shapes and surfaces that cannot be represented by classical geometry......think about the spirals in some seashells. That irregular pattern can be considered to be a fractal.

Fractals often exhibit the property of self-similarity and represent the many irregularly shaped objects in nature. A self-similar object is one whose component parts resemble the whole. Any of its smallest parts are similar in shape to its larger structure---like a broccoli flower or a fern leaf.

Sample some of the following activities to learn more about fractals.


Places To Go

The following are places to go (some real and some virtual) to find out about fractals.

Dance of Chance
Visit the Museum of Science in Boston and check out their exhibit on fractals and patterns in nature.
Fractal World
Fractals: eternal chaos and eternal order. This page contains colorful, decorative and very unusual images.
Golden Ratio
Travel to the Parthenon and look at its proportions. The proportions represent the Golden Ratio. Two quantities are in the Golden Ratio “if the ratio of the sum of the quantities to the larger quantity is equal to the ratio of the larger quantity to the smaller one.”
Hunting the Hidden Dimension
Spend time at PBS with mathematicians who claim that fractals can deepen our understanding of nature.
Visit the World Wide Web to understand fractals because this informative website claims that the internet is a fractal.

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People To See

Benoit Mandelbrot
Meet Benoit Mandelbrot. He is a French mathematician who helped develop chaos theory. He also developed fractal geometry which is used to find order in apparently erratic shapes and processes. Mandelbrot invented the term fractal. He based the term on fractus, a Latin word meaning a broken stone with an irregular surface. 
How to use "The Fractal Microscope"
Get to know the creators of a fractal microscope.
Who was Fibonacci?
Meet Mr. Fibonacci. The mathematician who first wrote about Fibonacci numbers was a man named Leonardo who lived in Pisa, Italy in the early 1200s. He signed his name as Fibonacci which means “son of Bonaccio”. So he is now known as Mr. Fibonacci.

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Things To Do

A Fractals Unit for Elementary and Middle School Students
This is a great site. It will lead you to fractal information and activities that you can actually understand.
The Fibonacci Sequence
Apparently the Fibonacci sequence can be found in the Mandelbrot set. Check it out.
The Fract-ED Information Pages
Explore this introductory fractal tutorial.
Check out the fractal of the day and view past fractals in the Fractal Gallery.
Fractals Theme Page
Oh boy! Join the Chaos Club!
Julia and Mandelbrot Set Explorer
After you knock yourself out making Mandelbrot sets, make some Julia sets.
The Leap Fractal Game
Choose your own difficulty level in this challenging game of Leap Fractal.
Mandelbrot Explorer
Create your own Mandelbrot sets. In case you've forgotten, a Mandelbrot set is the set of all complex c such that iterating z -> z^2+c does not go to infinity (starting with z=0). See the creations that other Mandelbrot aficionados have made.
NCTM Interactive Fractal Tool
Use this online tool to play with and create fractals.
Sprott's Fractal Gallery
Fractal music! Listen to fractal music while you view the Fractal of the Day. The inventor of fractals, Benoit Mandelbrot, also is a proponent of chaos theory. From this site, you can view a chaos demonstration and experience "strange attractors".
What Is a Fractal? And Who Is This Guy Mandelbrot?
See how this site calls fractals "worlds within worlds".

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Teacher Resources

Google Earth fractals
Contribute your own images to this collaborative site where individuals use Google Earth to locate fractal pictures.

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  • Crownover, Richard M. Introduction to Fractals and Chaos. Boston: Jones and Bartlett, c1995.  
  • Dixon, Robert. Mathographics. New York: Dover Publications, 1991.
  • Laplante, Phillip A. Fractal Mania. Blue Ridge Summit, PA: Tab Books, 1993.
  • Peitgen, Heinz-Otto. Chaos and Fractals:New Frontiers of Science. New York: Springer-Verlag, c1992.
  • Pickover, Clifford A.Chaos in Wonderland: Visual Adventures in a Fractal World. New York: St. Martin's Press, c1994.
  • Turcotte, Donald Lawson. Fractals and Chaos in Geology and Geophysics. Cambridge, U.K.; New York: Cambridge University Press, c1997.