The process has been halted before with complete . The related sequence of hypotenuse lengths is described by the recurrence relation The closed form is Other well-known visual representations to investigate in connection with the method of iteration with recurrence include the golden rectangle that contains Bernoulli's logarithmic spiral, as in figure 2 . The first four iteration stages that lead to the fascinating Sierpinski carpet, encountered in the study of the geometry of fractals, are shown in figure 3 . Additional iteration patterns that explore perimeter and area using this model can be considered. Students should be encouraged to develop their own creative constructions based on the method of iteration with recurrence.
Students readily see that the triangle itself is based on a recursive relationship.
That is, initially, they find that to name the members of row 8, they need to know row 7, and in turn row 6, and so on, working backward to the start. The student should also realize that in some situations the iteration process seems to yield a general recurrence relation, and yet the recurrence relation does not solve the original problem. For example, consider the problem of finding all the different lengths for a line segment on a 10 x 10 pin geoboard. In fact, the pattern breaks down with the 6 x 6 pin board. Yet another option for treating recursion is the spreadsheet.