Valuable Sites to Visit:

Let's Iterate

Recursion

Graphical Iteration

Iterated Systems

Logistic Equation and Bifurcation Diagram

Chua's Circuit

*Waterloo Fractal Compression

Differential Equations- Papers

Repetitions in Mathematics

The simplest way to understand iterations is to look at the definition of iteration :making repetitions. Iterations are functions that are repeated. For instance, start with the function (taken from Making Connections Through Iterations, Bannard, 1991):

g(x) = (x+3)/2

The second iteration produces:

g (g(x)) = ((x+3)/2 +3)/2 = (x +3+6)/4 = (x+9)/4

A third iteration produces:

g (g(g(x))) =((x+9)/4)+3)/2 = (x+9+12)/8= (x+21)/8

One could continue on indefinitely. The trick is to look for patterns. This is most easily done by graphing the function.

Graphing the function for population growth lead to the discovery by biologist Robert May that these systems bifurcate, that is they create images that break off and repeat themselves in two's:

Iterations have proven to be a useful tool in an understanding of chaos. Sierpinski's Triangle pictured above is a pictorial example of the use of iteration. The following activity provides a hands-on method for understanding these concepts more fully.

Activities:

Although the author of these lessons suggests using them for grades 4-8, I found that both of these are powerful lessons, even for older students, particularly if they have not been exposed to the principles of iteration.

• Iteration in Jurassic Park: a demonstration of iterations using the iterations by Michael Crichton in the chapters of Jarassic Park.
• Fractal Formation Through Iteration: an activity to demonstrate the principles of iteration and how they lead to the formation of fractals.