Spirograph
A disk rolls inside (or outside) a fixed ring. A pen fixed to the rolling disk traces a curve called a hypotrochoid (inside) or epitrochoid (outside). These are the mathematical curves behind the classic Spirograph toy — parametric equations that produce patterns of startling beauty from simple circular motion.
x(t) = (R−r)·cos(t) + d·cos((R−r)t/r) · y(t) = (R−r)·sin(t) − d·sin((R−r)t/r)
y(t) = sin(t) − sin( t)
Parametric curves from rolling circles
A hypotrochoid is traced when a smaller circle of radius r rolls inside a larger circle of radius R, with a pen point at distance d from the center of the smaller circle. If the small circle rolls outside instead, the curve is an epitrochoid. Special cases include the ellipse (R = 2r, d ≠ r), the astroid (R = 4r, d = r), the cardioid (an epitrochoid with R = r), and the rhodonea (rose curves, when d = r).
When does the curve close?
The curve closes — returns exactly to its starting point — if and only if the ratio R/r is rational. If R/r = p/q in lowest terms, the curve closes after q full revolutions of the parameter t (that is, after the inner circle has rolled around p times). The number of petals or lobes equals p/gcd(p,q) for a hypotrochoid. When R/r is irrational, the curve never closes and eventually fills an annular region densely — an irrational winding on a torus.
Connection to Fourier series
Every spirograph curve is the sum of two circular motions at different angular velocities. This is exactly a finite Fourier series with two terms. The deep connection runs both ways: any closed curve in the plane can be approximated arbitrarily well by a sum of circular motions — which is to say, by a chain of rolling circles. This is the principle behind Fourier epicycle animations: a series of circles riding on circles, each at its own frequency and amplitude, tracing out any shape you can draw.
From toy to mathematics
The Spirograph toy was patented by Denys Fisher in 1965, but the mathematics goes back centuries. Albrecht Dürer studied these curves in 1525. The term “roulette” for curves traced by rolling circles was established in the 17th century. Today, hypotrochoids appear in engine design (the Wankel rotary engine’s rotor traces an epitrochoid), coin geometry (the British 50p is a Reuleaux heptagon, closely related), and the design of gears, cams, and optical instruments. The beauty of these curves emerges from the simplest premise: one circle rolling against another.