HS #94 2023.5.11
The Cycloid
Last month several friends alerted me to a New York Times essay, “The Wondrous Connections between Mathematics and Literature.” It began with Melville’s reference of a “cycloid” in “Moby Dick.” Do you know what a cycloid is?
You can draw one. Secure a ruler to a piece of paper. Practice rolling a disc (a quarter or the top of a jar will do) along the edge of the ruler. Then put a pencil against the disc and trace its path as you roll the disc along the edge one full revolution. You just made a cycloid.
If that’s too much work, you can visualize a cycloid by imagining the path taken on a dark night by a lightning bug that gets caught in the tread of a bicycle tire. As the tire rolls along the sidewalk, the hapless lit-up firefly traces out cycloids.
What’s so special about them? If you want to build a slide that allows one to slide from point A to point B in the minimum time, a cycloid is your answer. Intuitively, one might think that a straight line is the best shape. A cycloid path is longer, but you have the advantage of getting a faster start.
There is sordid history to the discovery.
The Bernoulli brothers of the 17th century, as with many brothers, were competitive. Johann Bernoulli discovered that the cycloid gave the optimal path, but instead of just publishing his finding, he posed it as a challenge to the entire mathematical community of his day in an attempt to best his brother, Jacob. One person submitted a solution anonymously. When Johann Bernoulli saw the exquisitely beautiful solution, he said “This is from Sir Isaac” (referring to Isaac Newton). He was correct. When asked how he knew, he said, “You can tell the lion by the size of the paw.”
Called the Brachistochrone (shortest-time) Problem, it inspired an entire new area of mathematics called Calculus of Variations. I gave a talk on the problem at Hope College in 1987 when I applied for the mathematics position. So it was profitable for me as well. Making use of the alphabetical order of the names involved, the announcement read, “Inspired some of the greatest mathematicians in history including Bernoulli, Euler, Leibnitz, Newton. Pennings will talk on Tuesday.”
In fact, more recently I got wondering what sort of optimal path one would make if limited to adjoining straight line segments rather than a smooth curve. Getting a couple mathematician friends to join me (including former Hope student Mark Panaggio – now a researcher at Johns Hopkins), we solved it – to be published this fall. All a testament to the ever-growing circle of knowledge inspired by past discoveries.
The cycloid has another interesting property as well. A pendulum swinging a big arc takes longer to go back and forth than one swinging a small arc. That’s not surprising. But what IS surprising is that a pendulum constrained to follow the curve of a cycloid will always take the same amount of time, no matter how high or low the starting point. The physicist Huygens actually constructed an accurate clock using this property.
One last notable fact about cycloids is also related to literature. Sir Arthur Conan Doyle, the author of the Sherlock Holmes stories, wrote “Silver Blaze” in which the mystery of a stolen horse was solved by Holmes noting the curious event of the dog that didn’t bark in the night. The non-barking led Holmes to realize the dog must have been familiar with the theif.
Similarly, the cycloid has a curious “negative” quality – one I have never seen noted as such in mathematics literature. Everything else involving the circle involves the number pi. The circumference is pi times the length of the diameter. The area of a circle is pi times the square of the radius (hence the area in the rim of a basket is almost 4 times the cross section of a basketball!). The area of a sphere is 4pi times the square of the radius, and the volume is 4/3 pi times the cube of the radius. Pi is ubiquitous! Except the cycloid. The length of a cycloid is exactly 8 times the radius of the circle. How cool is that! Hope the firefly appreciates it.
