By Edward Scheinerman

Many years ago, as a freshman in college, I had a great deal of interest in mathematics, but was uncertain about pursuing that as a career. At the same time, the field of medicine was attractive and familiar to me and seemed like a safer bet. I was pleased to learn that being a mathematics major was completely compatible with fulfilling premedical requirements, so I followed both paths at the beginning of my undergraduate career. But I also knew that a decision loomed. Would I be applying to mathematics graduate programs or to medical school? How was I to decide?

To that end, I invented a pair of challenges for myself as a guide to making this choice. On the one hand, I would see how I performed on the MCAT exam. This made sense to me because that test is designed to assess one’s preparation for success in medicine. On the other hand, I thought up a mathematical problem unlike any I had ever encountered. I figured that if I could solve that problem, then perhaps it was ok to pursue a career as a mathematician.

Here’s the problem: Find a function f with the property that f(f(x)) = sin x for all real x. In other words, find a compositional square root of the sine function.

Dear reader, since you are finding this account in Math Values you might reasonably conclude that I solved this problem and did poorly on the MCATs. And that would be a correct inference if I followed my own, self-imposed rules. But, in fact, the opposite was the case. Though I achieved a few heuristic results on the mathematics question, I was not able to create a true solution. And the MCAT result was fine. Nevertheless, and quite happily, I ignored my own criteria and chose a career as a mathematician.

Through the years I occasionally thought about the compositional square root of the sine question. I returned to it in earnest recently when an eager and talented undergraduate student, Tongtong Chen, asked if she could work on research with me. It took only a few minutes to explain the question, but then many months of looking at it in earnest. The happy news is that we not only found a function f with the required property, we found infinitely many such functions; most of the solutions are “ugly” but we also identified some nice answers.

So rather than solving this problem near the beginning of my mathematical journey, the solution came decades later. Solution or not, I made the right choice for me.

For the interested reader, Tongtong and I wrote up our results as an article for the American Mathematical Monthly and it recently appeared [Tongtong Chen & Edward Scheinerman (2022) Finding a Compositional Square Root of Sine, The American Mathematical Monthly, 129:9, 816-830, DOI: 10.1080/00029890.2022.2105054].

Edward Scheinerman is the Vice Dean for Faculty and a Professor of Applied Mathematics and Statistics at the Whiting School of Engineering, Johns Hopkins University. He is the author of several books including The Mathematics Lover’s Companion.