Iterating by Example

In Sketchpad, you define an iteration rule by example, creating both pre-image and image and specifying how they correspond to each other.

For instance, the first figure starts with triangle ABC. Triangle DEF is the midpoint triangle, constructed using as its vertices the midpoints of the sides of triangle ABC.

To repeat this construction, you’d construct a new midpoint triangle using the midpoints of the sides of triangle DEF, and then another using the midpoints of the sides of that triangle, and so forth.

To specify this repetition as an iteration rule in Sketchpad, you define a map that identifies the image for each pre-image:

Even though the repeated construction includes the sides of the triangles as well as the vertices, you specify the rule using only points A, B, C, D, E, and F. Sketchpad figures out what other objects depend on the pre-image points and includes them in the iteration. In this example, Sketchpad will include the sides of the triangle in the iteration.

For best results, construct the entire pre-image but construct only the points of the image. Let Sketchpad construct the other parts of the image.

For example, to iterate the midpoint triangle, construct the pre-image (triangle ABC) and the image points (midpoints D, E, and F), but not the remaining image objects (the segments connecting D, E, and F). Once you’ve constructed these objects, select the independent points that define the pre-image (points A, B, and C) and choose Iterate from the Transform menu.

Sketchpad displays the dialog box shown here, allowing you to define the corresponding image for each selected pre-image point. For each pre-image point in the triangle, click on the midpoint to which that pre-image should map. As you click each midpoint, Sketchpad displays the results of iterating the pre-image triangle towards your mapped images. Once you’ve mapped each of the three pre-image points to a unique midpoint, click Iterate to confirm the iteration rule and close the dialog box.

You may need to drag the dialog box out of the way so that you can see your original triangle and its midpoints.

Sketchpad produces a set of iterated images for each object affected by your mapping. In this example there are six iterated images, one for each vertex and side of your original triangle. You can select and manipulate each iterated image separately. For example, you could hide or delete the three iterated images of your original triangle’s vertices, or you could color each of the three iterated images of your original triangle’s edges with a different color. You can also use Iteration Properties to change the number of iterations that Sketchpad displays.

You can use Iterate to create fractals by specifying more than one mapping of your pre-images to first images. See Multiple Iteration Map for more information.