The Fibonacci Numbers Hiding in Strange Spaces

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 PhD 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.”