The Fibonacci Numbers Hiding in Strange Spaces

The two men were trying to figure out when they could fit the ellipsoid inside the ball. This type of problem is easy to solve in geometry where shapes don't bend at all. Shapes can bend as much as you want as long as their volume doesn't change in other subfields of geometry.

Symmetric geometry is more difficult to understand. The answer depends on how long the ellipsoid is. A long, thin shape with a high eccentricity can be folded into a smaller shape. Things are simpler when the eccentricity is high.

The radius of the smallest ball was calculated by McDuff and Schlenk. The solution was based on a sequence of numbers where the next number is always the same as the previous two.

The mathematicians were left wondering what if you tried to make a cube out of your ellipsoid. Is it possible that more staircases would pop up?

There is a surprise.

Researchers found a few staircases here and a few more there. The Association for Women in Mathematics held a weeklong workshop in the summer of 2019. McDuff and Morgan Weiler were part of a working group that was put together at the event. They wanted to create a shape that would allow them to make infinitely many staircases.

symplectic shapes are a system of moving objects and can be visualized. Symplectic shapes are always described by an even number of variables. They're even-dimensional. The shapes that are four-dimensional or more are the most intriguing to mathematicians since they represent just one object moving along a fixed path.

Four-dimensional shapes are hard to see. Two-dimensional pictures that capture at least some information about the shape can be drawn by researchers. A four-dimensional ball becomes a right triangle.

Hirzebruch surfaces are the shapes that the group analyzed. The top corner of the right triangle is the location of each Hirzebruch surface. A number is used to measure how much you have cut off. When b is 0, you don't cut anything; when it's 1, you wipe out the entire triangle.

The group's efforts were not likely to bear fruit initially. Weiler said that they didn't find anything when they worked on it for a week. They didn't make much progress by early 2020. The paper they would write was called "No Luck in Finding Staircases."