Fourteen years ago, two mathematicians stumbled upon a hidden garden that is now starting to flower. The pair were interested in a shape that could be folded up and stuffed inside a ball. How big does the ball have to be?
They didn't notice the striking patterns emerging until their results began to crystallise. A colleague who reviewed their work spotted a famous list of numbers that have appeared in nature and throughout centuries of mathematics. They are related to the golden ratio, which has been studied in art, architecture and nature.
Tara Holm is a mathematician at Cornell University. She said that their appearance in McDuff and Schlenk's work was indicative of something.
Their landmark result was published in 2012 in the Annals of Mathematics. It showed that there were staircase-like structures with a lot of steps. Each step in theinfinite staircases had a ratio of the numbers.
The steps became smaller and the top of the staircase became larger. There is no apparent correlation between the golden ratio and the problem of fitting a shape into a ball. It was odd to see these numbers in McDuff and Schlenk's work.
McDuff found another clue to the mystery. She and other people revealed intricate structures. A professor at the University of Georgia said that their results are not something that he expected.
Hidden patterns in seemingly unrelated areas of math have been revealed by the work.
These problems are not found in the world of Euclidean geometry. They operate by the rules of symplectic geometry. A simple pendulum could be considered. The pendulums physical state is determined by where it is and how fast it is moving. A symplectic shape that looks like the surface of an infinitely long cylinder can be obtained if you plot all the possibilities for those two values.
Only very specific ways can you modify the shape. The same system must be reflected in the final result. How you measure it is the only thing that can be changed. The underlying physics are ensured by these rules.
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 they came up with was based on the idea of a sequence of numbers where the next number is the sum of 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?
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 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 described by an even number of variables because of the physical state of an object. 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."
In October 2020 they posted a paper that excavated infinite staircases for certain values of b after finding their footing.
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