## The Geometer's Sketchpad Spirograph Lab, Part 4

### Mike Riedy

TOC || Page 2 || Page 3 || Page 4 || Intro Lab

### Part 4: Discoveries and the Math Involved

Cusps and Phases:

Let's explain what's going on when you double-click on the Spirograph button. GSP is tracing the path of Son of Bert as he rotates around his circle. However, Son of Bert's circle is also rotating around Roger's circle. That is why you get such a bizarre tracing.

There are two things in the tracing that you need to look at: the cusps and the period of rotation. A cusp is an indentation in the tracing (see the picture below). The period is the number of rotations it takes for the tracing to get back to where it started. In the picture below you will see that the tracing has only one cusp and a period of one. This tracing is said to be "in phase" because the curve comes back to itself after a certain period. Most tracings will eventually be in phase if they are left running long enough.

More than likely your spirograph machine will look like the one below. The machine has gone through three rotations and the trace is still out of phase.

How many cusps are in the picture?

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Let the animation run for a while.

Approximately how many periods does the point go through before the tracing is in phase?

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Try changing the radii ('radii' is the plural of 'radius') of Roger's circle and Bert's circle. Double-click Spirograph to trace a new curve.

How does the change in radii affect the tracing?
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Now the really hard part. Your job is to study the relation between the radii of the circles and the phase of the tracing. Play with the radii until you have a tracing that is in phase with one cusp. Try to come as close as possible. Once you get one cusp in phase save it on your disk as "One Cusp".

What is the relation between Roger's radius and Bert's radius?

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Now try to create, as close as possible, a tracing that is in phase with two cusps. Save it as "Two Cusps".

Now what is the relation between Roger's radius and Bert's radius?

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By now you have probably noticed a pattern. If you generalize the pattern you can create a spirograph machine that will create a tracing with any number of cusps and still be in phase.

Let's say, for example, that we wanted an in-phase tracing with five cusps. What would be the relation between the radii of the two circles?

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