Iterations and Iterated Images

To iterate an action or an operation is to repeat it some number of times. Mathematically, iteration refers to the process of repeatedly applying some mathematical construction, calculation, or other operation to the previous result of that same operation. The operation must define an output in terms of some input, and the iteration uses the output of one step as the input for the next step.

Sketchpad allows you to iterate any of the mathematical relationships you use to construct relationships in a sketch. You can use iterations to create repeated transformations (such as tessellations), to produce fractals and other self-similar objects, or to generate other sequences and series.

In algebra, an iteration is the repetition of a calculation that uses an input value to calculate an output value. The iteration repeatedly applies the calculation to the value that resulted from the previous calculation—the output from one step is the input for the next. To begin the process, there must be a starting value; this value is called the seed. Consider the calculation “add 2” applied to the seed 5. When you apply this operation once to 5, the result is 7 (because 5 + 2 = 7). When you then apply the rule to the first result (7), the second result is 9 (because 7 + 2 = 9). Iterating this operation produces the sequence of values 7, 9, 11, 13, 15, 17, … .

In geometry, an iteration uses an operation performed on a set of geometric objects that produces a new set of objects. The original set of objects is the input, and the new set is the output. To begin the process, there must be a starting set of objects; these starting objects are the pre-image. Consider the transformation “translate to the right by 1 cm.” If you apply this transformation to an initial pre-image ∆ABC, the first image result is ∆A'B'C', translated 1 cm to the right. Iterating this transformation produces a sequence of triangles congruent to the initial pre-image ∆ABC, each shifted 1 cm to the right of the previous triangle in the sequence.

In these examples, it’s helpful to think of the operation—“add 2” or “translate to the right by 1 cm”—as distinct from any individual value or triangle in the sequence; rather, think of it as mapping each value or image in the sequence to the next value or image in the sequence. Thus, one might say that 47 maps to 49 under the operation “add 2.” The entire iteration is defined by the pre-image (the seed) and the mapping operation. When you apply the operation to the pre-image once, the result is the first image of your pre-image according to your mapping operation. As you iterate the operation, you generate the second, third, and fourth images, and so on.

Subtopics:
Creating Iterations
Working with Iterations