The tautochrone is a curve such that a ball rolling down the curve takes the same amount of time to reach the bottom, no matter where along the curve it starts. (The name comes from the Greek tauto for same and chrono for time.) It doesn’t sound like such a curve should be possible because balls starting further up the curve have longer to travel. However, balls starting higher also have more potential energy, and so they travel further but faster. See the video below for a demonstration.
[The video is entitled “brachistochrone race” rather than “tautochrone race.” The brachistochrone problem is to find the curve of fastest descent. But its solution is the same curve as the tautochrone. So different problems, same solution.]
I first heard of the tautochrone as a differential equation problem to find its equation. But someone could run into it in an American literature class.
Clifford Pickover’s new book The Physics Book has a chapter on the tautochrone. (In this book, “chapters” are only two pages: one page of prose and one full-page illustration.) Pickover points out a passage in Moby Dick that discusses a bowl called a try-pot that is shaped like a tautochrone in the radial direction.
[The try-pot] is a place also for profound mathematical meditation. It was in the left hand try-pot of the Pequod, with the soapstone diligently circling round me, that I was first indirectly struck by the remarkable fact, that in geometry all bodies gliding along the cycloid, my soapstone for example, will descend from any point in precisely the same time.