Math is riddled with questions that seem silly or contrived, yet end up being surprisingly deep. Take a simple geometry question that dates back to 1917. Lay an infinitely thin needle on a table, then turn it so that it points in every possible direction. Depending on how you do this, you can sweep out all sorts of shapes: a circle, a triangle, the spiky contours of a sea urchin. What's the area of the smallest shape you can make this way?
Mathematicians have proved that
there's no limit to how small it can be. Furthermore, they've
formulated different versions of the problem, asking about the size of such shapes for notions of "size" other than area. These variations seem even more arbitrary than the original question — who cares?
As it turns out, such questions are intimately connected to some of the most central research programs in mathematics. Last year,
I wrote several stories about how a variation of this problem underpins a tower of important conjectures about the behavior of functions, including one about the physics of waves. "Somehow, this geometry of lines pointing in many different directions is ubiquitous in a large portion of mathematics," one mathematician told me.
Some "toy" problems stay just that — intriguing brainteasers that don't become broadly relevant. But many others become laboratories of sorts: places for mathematicians to test out new ideas, explore novel connections and revisit math's very foundations.
This sort of thing happens too often to be a coincidence. It illustrates the hidden complexity of simple problems and shows just how intertwined different fields of mathematics can be.
In a conversation with
Quanta published in 2018, Tadashi Tokieda, a mathematician at Stanford University, spoke about how he
plays with actual toys and everyday objects — strips of paper, balls in a bowl, rolls of quarters — to uncover mathematical insights. "If you come a little fresher, and a little more naïve, you can look all over the place … and find your own surprises," he said.
Earlier this year, Kevin Hartnett
wrote about a series of results in this vein, detailing mathematicians' quest to discover the fattest possible Möbius strip and other "optimal" shapes. Richard Schwartz, one of the leaders of the project,
spoke to Quanta a few years ago about what attracts him to such questions. By virtue of being so easy to state, toy problems often captivate mathematicians from different research areas and with varying levels of expertise — promoting novel approaches and diverse ways of thinking.
Moreover, "I feel if it's a simple problem that hasn't been solved, it probably has some kind of hidden depth to it," Schwartz said.
The centuries-old "sphere packing" problem is a perfect example of this. How can you arrange spheres so that they fill as much space as possible? In 1611, Johannes Kepler conjectured that to get the optimal answer in 3D, you should stack spheres in a pyramid shape, akin to how oranges get piled in a grocery store. But it wasn't until 1998 that a mathematician named Thomas Hales finally proved this.
In higher dimensions, the problem, though still straightforward, is even harder. In 2016,
Quanta reported on
a groundbreaking result that showed how to optimally pack eight- and 24-dimensional spheres into the densest possible configurations. Central to the work (for which Maryna Viazovska was
awarded the Fields Medal in 2022) were complicated mathematical functions called modular forms.
Modular forms cropping up in this context came as a surprise to many mathematicians. But perhaps it no longer should: As
I reported last year, these functions have also turned up in unexpected ways in number theory, combinatorics, topology, cryptography and even string theory. The sphere-packing problem, because it's so simple and natural, has helped to cement just how fundamental this highly technical piece of mathematical technology can be.
