The Fibonacci Numbers Hiding in Strange Spaces

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.