A Geometric Iteration

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A Geometric Iteration

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When you create an iteration, it can be a geometric iteration (based on independent points), a numeric iteration (based on parameters), or a combination of the two. Here’s an example of using a  geometric iteration to create an iterated midpoint triangle.

To define a Sketchpad iteration, create the pre-image and then specify how its defining points or parameters should be mapped to produce the first iteration image. Sketchpad handles the iteration from there.

For instance, here are the pre-image and the first three images for the midpoint triangle iteration.


First Image

Second Image

Third Image

The first image repeats the pre-image construction, but starting from midpoints D, E, and F. The map that defines this iteration is shown on the right.

Follow the steps below to construct this iterated midpoint triangle.




1.Construct the pre-image triangle exactly as shown above.


Don’t construct more objects than necessary in the pre-image. In this example, the pre-image construction includes three sides of the triangle and the three midpoints D, E, and F. Segments DE, EF, and DF will be constructed as part of the first image. If you also constructed these segments as part of the pre-image, they  would be constructed again, resulting in unnecessary duplicated segments.

2.Select the independent seed points A, B, and C that define the pre-image.

3.Choose Transform | Iterate. The Iterate dialog box appears with the left column filled in.


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

Use the Structure and Display pop-up menus to control how the iteration is constructed and displayed.


4.Click point D in the sketch to fill in the top box of the right column. This specifies that point D should play the same role in the first image that point A played in the pre-image,

5.Similarly click points E and F in the sketch to complete the iteration map. As you complete each row of the map, Sketchpad displays the first three partial images based on the part of the map you’ve already completed.

6.After you complete the entire map, click the Iterate button to confirm the iteration rule and construct the iteration images.

Sketchpad produces an iterated image object for each object affected by your mapping. In this example there are six iterated image objects, one for each vertex of your original triangle and one for each side. You can select and manipulate each iterated image object separately. For example, you could hide or clear the three iterated image objects of your original triangle’s vertices, or you could color each of the three iterated image objects of your original triangle’s edges with a different color.

Sketchpad normally displays iterations to a depth of three, showing the first, second, and third iteration image. Use Iteration Properties to change the depth of the iteration in order to show more or fewer iteration images, or select any of the iterated image objects and press + or on the keyboard.


You can also use a numeric value (a parameter or a calculation) to control the depth of an iteration. After selecting the seed points and/or values, select one additional parameter or calculation, and then hold Shift and choose Transform | Iterate to Depth. (Notice the change in the command name.) The whole number part of the value you selected last will define the number of iterations. (When you  choose Iterate, the minimum depth is one; when you choose Iterate to Depth, the minimum depth is zero.)

Use Iterate to create fractals such as the Sierpiński Gasket by specifying more than one mapping of your pre-image. See Multiple Iteration Maps for more information.