Question 19.126

1. Generations of children have enjoyed the popular toy Spirograph, which allows the user to trace out symmetric patterns. A pencil or pen is placed in a hole in one of several plastic circular disks with teeth on the outside rim. The disk is then meshed in the teeth of another plastic circle and rotated around its inside or outside. Each plastic piece is labeled with the number of teeth that it has on its circumference.

Obtain a copy of a Spirograph or closely related toy, or else visit www.wordsmith.org/~anu/java/spirograph.html, which offers an interactive Java application (which you can download) that mimics what the Spirograph toy does.

  1. Experiment to determine, from the numbers of teeth on the rotating circular disk and the fixed circle, what symmetry pattern the result will have.
  2. Choose a rotating circular disk and fixed circle for which the ratio of the number of teeth reduces to a whole number. For each of several “offsets” (holes to choose for the pencil or pen), trace overlapping designs. What symmetry pattern do you get for the design taken as a whole? Repeat this experiment for other pairs of pieces and try to reach a general conclusion.
  3. Write up, in a page or so, a description of your experiments and what conclusions you reached.