# The Mathematical Tourist

## By Ivars Peterson

September 25, 2008

## Waltz of the Cycloids

Mathematical vocabulary isn't something that I normally associate with classical music, so I was caught by surprise recently when an announcer on a local radio station gamely tried to explain the term "cycloid."

Why? The announcer was introducing a piece of music called the Cycloids Waltz (Cycloiden Walzer), written by Johann Strauss Jr. Strauss had dedicated his composition to the "Gentlemen Technical Students at Vienna University" and conducted its debut at their ball in the Sofienbad-Saal on Feb. 10, 1858. The apt title not only came directly out of mathematical vocabulary that would be familiar to the students but also hinted at the whirling movements of a waltz.

A cycloid is the path traced by a point on the rim of a circle that rolls (without slipping) along a straight line.

Galileo Galilei (1564–1642) studied the cycloid around 1599 and gave the curve its name, using the Greek word for "circle" as its main element.

A segment of a cycloid also represents the shape of the curve along which a bead sliding from rest and accelerated by gravity will slip from one point to another in the least time, a problem originally posed in 1696 by Johann Bernoulli (see the brachistochrone problem). Moreover, a bead sliding on a cycloid will exhibit simple harmonic motion, with a period independent of the starting point.

Galileo suggested that the cycloid would be the strongest possible arch for a bridge, and many concrete viaducts do have cycloidal arches. Cogwheels often have cycloidal sides to reduce friction as gears mesh.

In a chapter on the cycloid in Martin Gardner's 6th Book of Mathematical Diversions from Scientific American, Gardner notes that the cycloid has been called the "Helen of geometry," not only because of its beautiful properties but also because it has been the object of so many historic quarrels between eminent mathematicians.

The version of Cycloiden, Walzer, Op. 207, that I heard on the radio was recorded for the Marco Polo label by the Slovak State Philharmonic Orchestra, Kosice. The author of the accompanying album notes remarks that the title page design for the first piano edition of the piece featured a circle, encompassing the name of the work and its composer and the dedication to the students. The designer then surrounded this feature with representations of various tools of the technician (set square, compasses, theodolite, and so on) and depictions of several engineering achievements, such as the steamship, steam engine, blast furnace, and plough. But no cycloid.

Strauss' composition isn't the only one in the musical literature that refers to the cycloid. The Library of Congress collection of American sheet music includes Cycloid Polka, written by Charles Kinkel in 1873. Unfortunately, I can't play the piano or find a recording, so I have no idea what this piece sounds like. But I do know there's usually a lot of lively circling when you dance a polka.

Comments are welcome. You can reach Ivars Peterson at ipeterson@maa.org or visit the blog version of this article at http://mathtourist.blogspot.com/.

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