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.

Hi John

I hope this note finds you well. Always enjoy your posts. As an aside: The brachistochrone, as your readers might enjoy learning, was the solution to a problem pose by Johann Bernoulli in late 1696. Responses came from the best mathematical minds of the time. Among them: Leibniz, Jakob Bernoulli, l’Hopital and one by Newton. Newton’s solution (took him 12 hours to do it through an all nighter, offered w/o proof) was presented anonymously. As Bernoulli examined the solution, the authorship was clear to him, and he coined the idiom “ex ungue leonem” , to know “the lion from its claws.”

More on building a brachistochrone can be found here: http://www.math.usma.edu/people/rickey/papers/BuildBrachio-penult.pdf

Have a great afternoon

Daniel

Simply wonderful!

Does it really work for rolling as well as (frictionless) sliding?

Interesting; I didn’t know anything about tautochrones before. However, what you state about their relationship to brachistochrones, “its solution is the same curve as the tautochrone. So different problems, same solution.”, is a bit misleading, I think.

The solution to both are segments of curves in the cycloid family – but, for a GIVEN pair of points, there are actually an infinite number of tautochrones through them, but only one brachistochrone – therefore, most tautochrones are not brachistochrones, at least not for the same points. This can be demonstrated by considering the following:

Take two given points, the higher one called A and the lower called B, and assume we have found a tautochrone (call it ^AB^ ) through them. Let us also assume this tautochrone happens to be the brachistochrone for the points A and B as well. Now, any point C on the tautochrone ^AB^ between them, also defines a tautochrone, ^CB^, by the definition of a tautochrone. If ^CB^ also was the brachistochrone for points C and B, this leads to a physically demonstrable absurdity – the

minimumdescent time between two points is a constant regardless of the difference in positions of the points! I imagine such a fundamental physical constant would be of great interest. :-)Anyway, thanks for introducing me to tautochrones – I do wish the interesting problems that are tackled by the calculus of variations were cataloged and explained somewhere else than graduate level math texts… Any recommendations?