The cycloid is famous in physics, since it is the solution to the Brachistochrone problem. Johann Bernoulli posed this problem: If a bead slides without friction from one point to another along a track propelled by gravity, what is the shape of the track that minimizes the time. Obviously, the shortest distance would be a straight track, but it is better to go more steeply downhill to gain speed. This increases the distance travelled, but shortens the time. The optimal track is a cycloid that falls further than the final point. In one day Newton solved this problem and showed that it is a cycloid.
That’s the good Spirograph, with the push pins and the configurable track. I got my kids the currently available Spirograph when they were young (1990s); they removed the push pins and it sucked - you just couldn’t anchor down the stationary pieces well enough without push pins.
The absolute value of sine has a similar shape to the cycloid, but it is much different. For example, at the cusp ABS(SIN) has a slope of one, while for the cycloid, the slope is infinite.
A bicycle wheel wouldn’t, but a train wheel could. There’s a flange on train wheels that is larger that the rolling surface. Put the pen there and the line would form a small loop and cross over itself. When you see a train rolling forward, some small portions of it are moving backwards.
I goofed with some Spirograph drawing about two years ago. I found it quite a lot more entertaining than I remembered or expected. I got pretty good results with magnets on a steel serving tray. The key was adding a few strategic snips & folds of regular latex balloon between the gear and paper for grip. The magnets were only a couple bucks at Hobby Lobby.
And it’s also the solution to the isochron problem: A bead sliding without friction along a cycloid-shaped path will oscillate back and forth, and the time required for the oscillation will be independent of how far it’s sliding.
(this is approximately true for circular paths, which is the basis for a pendulum, but only approximately, and the approximation is best for small oscillations)
