Creating Iterations

Iterated operations and constructions in Sketchpad are always created by example, and are always defined in terms of points and parameters. Use the tools and menus to construct a figure in which a set of independent points or parameters is used to produce (through whatever mathematical relationships you wish) an equal number of dependent objects (points or calculated values). The independent objects represent the pre-images or seeds of your iteration; and the corresponding dependent objects represent the first iterated images of those initial objects. Then use Iterate in the Transform menu to indicate the correspondence between pre-images and images. The Iterate dialog box allows you to specify the number of times you wish to iterate the construction. The result is a collection of iterated images of the pre-images and of every object that depends on the pre-images.

Any parameter used to define an iteration must have geometric children.

In the illustration on the left, triangle ABC and its midpoints D, E, and F have been constructed. In the illustration on the right, the independent vertices of the triangle have been mapped to the triangle’s midpoints using the Iterate dialog box: A has been mapped to D, B to E, and C to F. This relationship—of a triangle mapped to its own midpoint triangle—has been iterated four times. The result is a series of images of the segments and points that defined the initial construction as the triangle is iterated toward its midpoint triangle.

More generally, if a geometric point pre-image P is used to construct a dependent point P', then the iterated image of that point—or the orbit of that point’s iteration—is the sequence of points P', P'', P''', and so forth.