Although not a mathematician myself, I find math endlessly fascinating about for the way one can take a seemingly impossible problem and translate it or express it in another context, and all of a sudden that hard problem can become easy. Of course, you have to make the right choice of a new context, a task which is itself not always easy or predictable. And “easy” may be relative. Still, when it works, the result is intriguing and inspiring. For example, see how a proof involving rectangles and curves was worked out by asking instead about Möbius strips and Klein bottles (already fun objects in their own right).
Briefly, the question is this: given a smooth, continuous closed curve, can you find four points along the curve that define a rectangle of any aspect ratio? Closed curves end up where they started; continuous curves have no gaps; smooth curves have no corners. Frankly, it’s a little bit amazing that we can say much of anything concrete about such curves, given how many and varied possible shapes there are that satisfy those minimal criteria. Yet we can, including the rectangle thing. All you have to do is think fourth dimensionally. You can read about the details in the linked article, or the full paper. (I don’t know enough math to follow the whole thing, but I’d like to hear the story behind this acknowledgment: “We thank Peter Feller and Patrick Orson for stimulating discussions on asubtropical island at the outset of this work.”) But the bottom line is that the final proof involves thinking about the curve question using a different geometry than the one we are most familiar with from high school math, and to build on results involving Klein bottles.
Another aspect of the story I liked was the way in which it grew out of our current quarantine context. Now, of course I don’t think that any circumstantial good which comes out of quarantine can in any way balance the tremendous cost of the pandemic. I wouldn’t even claim that it makes sense to try to compare or relate the two. The pandemic has a real cost, period. Separately, human creativity can still operate even under constraints which were not designed for the purpose of fostering creativity. And so the constraints of quarantine still allowed for some elegant and intriguing mathematics.
Will that math help us solve practical problems? I can’t say. It’s not always obvious which mathematical results will find concrete applications or lay the groundwork for other results that have concrete applications. Maybe it will only ever serve to give mathematicians more opportunities to express their creativity and find more abstract beauty, which is not too shabby. For now, I’m content to enjoy it as a good thing unto itself.
Check out the details on an upcoming online mini-conference on July 25th, organized by ESN and the American Scientific Affiliation.