This took me a while to wrap my head around. Basically, the shapes are generated every time the faster one “passes” the slower one. In the first example, we had 13 and 8, so in one full cycle the faster one passes the slower one 5 times. Each time, it generates a cardioid. Cardioid are epicycloids, as seen here.
Since each pass will occur in an identical amount of time, we end up with rotational symmetry. Forgive my poor drawing skills:
It’s easy to miss the fact that they are
Land of mathematics