Fourteen years ago, the mathematicians Dusa McDuff and Felix Schlenk stumbled upon a hidden geometric garden that is only now beginning to flower. The pair were interested in a certain kind of oblong shape, one that could be squeezed and folded up in very particular ways and stuffed inside a ball. They wondered: For a certain shape, how big does the ball need to be?
As their results began to crystallize, at first they didn’t notice the striking patterns emerging. But a colleague who reviewed their work spotted the famed Fibonacci numbers — a list whose entries have popped up again and again in nature and throughout centuries of mathematics. They’re closely related, for example, to the exalted golden ratio, which has been studied in art, architecture and nature since the ancient Greeks.
Fibonacci numbers “always make mathematicians happy,” said Tara Holm, a mathematician at Cornell University. Their appearance in McDuff and Schlenk’s work, she added, was “some indication that there’s something there there.”
Their landmark result was published in 2012 in the Annals of Mathematics, widely considered the top journal in the field. It revealed the existence of staircase-like structures with infinitely many steps. The size of each step in these “infinite staircases” was a ratio of Fibonacci numbers.
As the staircase ascended, the steps became smaller and smaller, the top of the staircase crushing up against the golden ratio. Neither the golden ratio nor the Fibonacci numbers has any apparent relationship to the problem of fitting a shape inside a ball. It was bizarre to find these numbers lurking within McDuff and Schlenk’s work.
Then earlier this year, McDuff uncovered another clue to this mystery. She and several others revealed not just infinitely more staircases, but intricate fractal structures. Their results are “not something that I remotely expected to see arising naturally in this kind of problem,” said Michael Usher, a professor at the University of Georgia.
The work has revealed hidden patterns in seemingly unrelated areas of math — a reliable sign that something important is afoot.
The Shape of Motion
These problems don’t take place in the familiar world of Euclidean geometry, where objects hold their shape. Instead, they operate by the strange rules of symplectic geometry, where shapes represent physical systems. For example, consider a simple pendulum. At any given moment, the pendulum’s physical state is defined by where it is and how fast it’s going. If you plot all the possibilities for those two values — the pendulum’s location and velocity — you’ll get a symplectic shape that looks like the surface of an infinitely long cylinder.
You can modify symplectic shapes, but only in very particular ways. The end result must reflect the same system. The only thing that can change is how you measure it. These rules ensure that you don’t mess with the underlying physics.
McDuff and Schlenk had been trying to figure out when they could fit a symplectic ellipsoid — an elongated blob — inside a ball. This type of problem, known as an embedding problem, is pretty easy in Euclidean geometry, where shapes don’t bend at all. It’s also straightforward in other subfields of geometry, where shapes can bend as much as you like as long as their volume doesn’t change.
Symplectic geometry is more complicated. Here, the answer depends on the ellipsoid’s “eccentricity,” a number that represents how elongated it is. A long, thin shape with a high eccentricity can be easily folded into a more compact shape, like a snake coiling up. When the eccentricity is low, things are less simple.
McDuff and Schlenk’s 2012 paper calculated the radius of the smallest ball that could fit various ellipsoids. Their solution resembled an infinite staircase based on Fibonacci numbers — a sequence of numbers where the next number is always the sum of the previous two.
After McDuff and Schlenk unveiled their results, mathematicians were left wondering: What if you tried embedding your ellipsoid into something other than a ball, like a four-dimensional cube? Would more infinite staircases pop up?
A Fractal Surprise
Results trickled in as researchers uncovered a few infinite staircases here, a few more there. Then in 2019, the Association for Women in Mathematics organized a weeklong workshop in symplectic geometry. At the event, Holm and her collaborator Ana Rita Pires put together a working group that included McDuff and Morgan Weiler, a freshly graduated Ph.D. from the University of California, Berkeley. They set out to embed ellipsoids into a type of shape that has infinitely many incarnations — eventually allowing them to produce infinitely many staircases.
To visualize the shapes that the group studied, remember that symplectic shapes represent a system of moving objects. Because the physical state of an object uses two quantities — position and velocity — symplectic shapes are always described by an even number of variables. In other words, they’re even-dimensional. Since a two-dimensional shape represents just one object moving along a fixed path, shapes that are four-dimensional or more are the most intriguing to mathematicians.
But four-dimensional shapes are impossible to visualize, severely limiting mathematicians’ toolkit. As a partial remedy, researchers can sometimes draw two-dimensional pictures that capture at least some information about the shape. Under the rules for creating these 2D pictures, a four-dimensional ball becomes a right triangle.
The shapes that Holm and Pires’ group analyzed are called Hirzebruch surfaces. Each Hirzebruch surface is obtained by chopping off the top corner of this right triangle. A number, b, measures how much you’ve chopped off. When b is 0, you haven’t cut anything; when it’s 1, you’ve erased nearly the whole triangle.
Initially, the group’s efforts seemed unlikely to bear fruit. “We spent a week working on it, and we didn’t find anything,” said Weiler, who is now a postdoc at Cornell. By early 2020, they still hadn’t made much headway. McDuff recalled one of Holm’s suggestions for the title of the paper they would write: “No Luck in Finding Staircases.”
But the group eventually found their footing, and in October 2020 they posted a paper that excavated infinite staircases for certain values of b.
This past March, McDuff, Weiler and Nicki Magill — a student of Holm’s who began working with McDuff during the coronavirus pandemic — posted a preprint in which they nearly completed the project of analyzing the embeddings of ellipsoids in Hirzebruch surfaces. “It’s amazing,” said Holm. “It’s so beautiful.”
When they did so, another surprise arose. If you look at all the values of b for which an infinite staircase appears, you get yet another fractal structure — an arrangement of points with features that defy common sense. Called a Cantor set, it has more points than the rational numbers — yet somehow the Cantor set’s points are more spread out.
“They really developed this beautiful picture with the staircase symmetries that I’m still trying to fully absorb,” said Daniel Cristofaro-Gardiner, a mathematician at the University of Maryland.
Although the new work has produced more infinite staircases than any previous results, symplectic embeddings and their accompanying staircases remain mostly a mystery, as Hirzebruch surfaces comprise only a tiny fraction of the possible symplectic shapes. “I still feel like we’re a little bit in the woods and we haven’t quite gotten up to the cloud level where we can sort of see the whole picture,” said Holm. “It’s an exciting moment, because I think we will get there.”