The Fibonacci Numbers Hiding in Unusual Areas

McDuff and Schlenk had been making an attempt to determine once they might match a symplectic ellipsoid—an elongated blob—inside a ball. Such a drawback, often called an embedding drawback, is fairly simple in Euclidean geometry, the place shapes don’t bend in any respect. It’s additionally simple in different subfields of geometry, the place shapes can bend as a lot as you want so long as their quantity doesn’t change.

Symplectic geometry is extra difficult. Right here, the reply depends upon the ellipsoid’s “eccentricity,” a quantity that represents how elongated it’s. An extended, skinny form with a excessive eccentricity will be simply folded right into a extra compact form, like a snake coiling up. When the eccentricity is low, issues are much less easy.

McDuff and Schlenk’s 2012 paper calculated the radius of the smallest ball that would match numerous ellipsoids. Their resolution resembled an infinite staircase primarily based on Fibonacci numbers—a sequence of numbers the place the following quantity is at all times the sum of the earlier two.

After McDuff and Schlenk unveiled their outcomes, mathematicians have been left questioning: What should you tried embedding your ellipsoid into one thing aside from a ball, like a four-dimensional dice? Would extra infinite staircases pop up?

A Fractal Shock

Outcomes trickled in as researchers uncovered a number of infinite staircases right here, a number of extra there. Then in 2019, the Affiliation for Ladies in Arithmetic organized a weeklong workshop in symplectic geometry. On the occasion, Holm and her collaborator Ana Rita Pires put collectively a working group that included McDuff and Morgan Weiler, a freshly graduated PhD from the College of California, Berkeley. They got down to embed ellipsoids into a sort of form that has infinitely many incarnations—ultimately permitting them to supply infinitely many staircases.

To visualise the shapes that the group studied, do not forget that symplectic shapes signify a system of transferring objects. As a result of the bodily state of an object makes use of two portions—place and velocity—symplectic shapes are at all times described by a good variety of variables. In different phrases, they’re even-dimensional. Since a two-dimensional form represents only one object transferring alongside a set path, shapes which are four-dimensional or extra are probably the most intriguing to mathematicians.

However four-dimensional shapes are unattainable to visualise, severely limiting mathematicians’ toolkit. As a partial treatment, researchers can typically draw two-dimensional footage that seize a minimum of some details about the form. Beneath the principles for creating these 2D footage, a four-dimensional ball turns into a proper triangle.

The shapes that Holm and Pires’ group analyzed are known as Hirzebruch surfaces. Every Hirzebruch floor is obtained by chopping off the highest nook of this proper triangle. A quantity, b, measures how a lot you’ve chopped off. When b is 0, you haven’t minimize something; when it’s 1, you’ve erased almost the entire triangle.

Initially, the group’s efforts appeared unlikely to bear fruit. “We spent every week engaged on it, and we didn’t discover something,” stated Weiler, who’s now a postdoc at Cornell. By early 2020, they nonetheless hadn’t made a lot headway. McDuff recalled considered one of Holm’s options for the title of the paper they might write: “No Luck in Discovering Staircases.”