Above are hypocycloids, curves produced by fixing a point on the circumference of a small circle of radius b, rolling around the inside of a large circle of radius a > b. If a/b is rational, i.e. it can be expressed as a fraction, the path will return to it's starting point. If the ratio is irrational, the path will never touch the same spot on the circumference of the larger circle more than once, and images such as the ones above result.