Points off a rolling wheel make traces. What makes those traces have symmetry?
In the diagram the point P can move to different places around the dotted circle. Each position P takes will fix a corresponding position for P'. As P moves around on that circle what will P' do?
A cheap and simple toy with lots of mathematics. Can you interpret the images that are produced? Can you predict the pattern that will be produced using different wheels?
Choose positions for P' and plot the corresponding positions of P.