  Bertrand's theorem proves that inverse square and Hooke's law-type central
forces are the only ones for which all bounded orbits are closed. Similar
analysis was used to show that for other central force laws there exist closed
orbits for a discrete set of angular momentum and energy values. These orbits
can in general be characterized as ``spirograph''-like, although specific
orbits look more ``star''-like or ``triangular.'' We use the results of a
perturbative version of Bertrand's theorem to predict which values of angular
momentum and energy result in closed orbits, and what their shapes will be.
