As a child, you may have played with a popular toy called a Spirograph ® ....

As a child, you may have played with a popular toy called a Spirograph ^® . With it one could draw some appealing geometric figures. The Spirograph consists of a small toothed disk with several holes in it and a larger ring with teeth on both inside and outside as shown in Figure 1.120. You can draw pictures by meshing the small disk with either the inside or outside circles of the ring and then poking a pen through one of the holes of the disk while turning the disk. (The large ring is held fixed.)

An idealized version of the Spirograph can be obtained by taking a large circle (of radius a ) and letting a small circle (of radius b ) roll either inside or outside it without slipping. A “Spirograph” pattern is produced by tracking a particular point lying anywhere on (or inside) the small circle. Exercises 34–37 concern this set-up.

(a) A cusp (or corner) occurs on either the hypocycloid or epicycloid every time the point P on the small circle touches the large circle. Equivalently,

this happens whenever the smaller circle rolls through 2π. Assuming that a/b is rational, how many cusps does a hypocycloid or epicycloid have? (Your answer should involve a and b in some way.) (b) Describe in words and pictures what happens when a/b is not rational.