Roulette Curves

Choose a wheel and a pen and create your own beautiful patterns.

How many cusps (end points) does your pattern have? Try a different wheel, will that have more or fewer cusps?

Find the wheel that makes the most cusps!

This device works like a Spirograph. The patterns are made by tracing a point on circle as it rolls inside a fix circle. These patterns are called roulette curves.

When the larger circle is 4 times bigger than the smaller circle, you will trace a curve with four cusps (end points). Similarly, when the larger circle is k times bigger than the smaller circle, you will trace a curve with k cusps. When the ratio between the larger circle and the smaller circle is a rational number, like k/n, it will trace a curve with k cusps. You can find the ratio by counting the teeth. But, if the ratio between the two circles is an irrational number, then the curve will continue forever and never close. In our exhibit, the outside wheel has 148 teeth, while the five inner-wheels have 40, 50, 60, 65, 80 teeth.