trochoid

one circle rolling on another, the trace of a third thing

The trochoid is the curve traced by a point carried on a circle that rolls on another circle. Three numbers: the fixed circle (R), the rolling one (r), and how far the pen is from the rolling centre (d). The ratio R/r is everything — rational closes, irrational fills the ring forever. The curve belongs to neither circle alone. It's the trace of a third thing.