The Fibonacci numbers are hiding in strange spaces

McDuff and Schlenk were trying to figure out when to fit a symplectic ellipsoid — an elongated blob — into a ball. This type of problem, known as an embedding problem, is quite simple in Euclidean geometry, where shapes don’t bend at all. It’s also easy in other sub-areas of geometry, where shapes can bend as much as you want as long as their volume doesn’t change.

Symplectic geometry is more complicated. Here the answer depends on the “eccentricity” of the ellipsoid, a number that indicates how long it is. A long, thin shape with a high eccentricity can easily be folded into a more compact shape, such as a coiled hose. If the eccentricity is low, it is less easy.

McDuff and Schlenk’s 2012 paper calculated the radius of the smallest ball that would fit on several 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 revealed their results, mathematicians wondered: What if you tried to embed your ellipsoid in something other than a ball, such as a four-dimensional cube? Would there be more infinite stairs popping up?

A fractal surprise

The results trickled in when researchers discovered a few infinite stairs here, and a few more there. Subsequently, the Association for Women in Mathematics organized a week in 2019 workplace in symplectic geometry. During the event, Holm and her collaborator Ana Rita Pires composed a working group including McDuff and Morgan Weiler, a recently graduated PhD from the University of California, Berkeley. They wanted to embed ellipsoids in a kind of shape that has infinite incarnations — eventually allowing them to produce infinitely many stairs.

Dusa McDuff and colleagues have mapped an ever-expanding zoo of infinite stairs.Thanks to Barnard College

To visualize the shapes 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 are even-dimensional. Because a two-dimensional shape represents only 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 images that capture at least some information about the shape. According to the rules for creating these 2D images, a four-dimensional ball becomes a right triangle.

The shapes that Holm and Pires’s group analyzed are called Hirzebruch surfaces. Each Hirzebruch surface is obtained by chopping off the top corner of this right triangle. A number, b, measure how much you have chopped off. When b is 0, you didn’t cut anything; if it is 1, you have almost erased the entire triangle.

At first, it seemed unlikely that the group’s efforts would bear fruit. “We’ve been working on it for a week and we haven’t found anything,” said Weiler, who is now a postdoc at Cornell. At the beginning of 2020 they had not made much progress. McDuff remembered one of Holm’s suggestions for the title of the article they would write: “No Luck in Finding Staircases.”

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