Title: A Negative Answer to Ulam’s Problem 19 From the Scottish Book
Author and Year: Dmitry Ryabogin, 2022
Journal: Annals of Mathematics
If a solid object floats in water in every position, is it necessarily a sphere? In a paper published this year in the Annals of Mathematics, Dmitry Ryabogin proves the answer is “no”.
This was a question that mathematicians discussed while sipping coffee and tea at the Scottish Café in 20th century Poland. Initially, they would scribble their ideas on the marble tabletops. This practice was not ideal because the record of their discussions was lost every night when the waiters cleaned the tables. The story goes that it was Lucja Braus, Stefan Banach’s wife, who had the idea to start a notebook to be kept at the Scottish Café. The book has since been dubbed the “Scottish Book”, after the name of the café. Its contents gradually grew as the thinkers added interesting problems.
Problem 19 in the Scottish Book, written by Stanislaw Ulam, concerned an object floating in water. A solid sphere will float in water regardless of its orientation, as long as the solid is less dense than the liquid. Another crucial feature of a sphere in order to have this property is that it has uniform density: It is the same material throughout. If half of the sphere is foam and the other half is rubber, for example, then the object will not float at every orientation.
The mathematicians at the Scottish Café wanted to know if there was any other solid shape of uniform density that would float at every position. For instance, a rubber duck easily floats if it is oriented upwards or on its head, but it is a frustrating task to try to get it to float lying sideways.
The answer was not immediately obvious. So Ulam recorded the question in the Scottish Book, hopeful that future mathematicians might have better luck at finding a solution.
In a recent paper, Dmitry Ryabogin solves Problem 19 from the Scottish Book. He shows there exists a convex object (a solid is convex if it curves outward at every point) which has the same density throughout, floats in water in every orientation, and is not a sphere. Furthermore, Ryabogin proves an analogous result for dimensions greater than three. A hypersphere is a generalization of a sphere in greater than three dimensions; In any dimension, Ryabogin shows there exists a theoretical object which is not a hypersphere but floats in water in every position.
What is the object that floats but is not a sphere? It is almost a sphere. Ryabogin constructs a surface which is a small perturbation away from a sphere. He accomplishes this by reducing the problem to finding a solution to a system of two integral equations. Instead of a usual equation involving an unknown variable, an integral equation involves an unknown function. This is similar to a differential equation, except instead of derivatives, the equation involves integrals of the unknown function.
Integral equations in general can be challenging to solve, since there is no strategy that works for every equation. Through clever analysis, Ryabogin finds a solution which is defined as a body of revolution: A continuous two-dimensional function rotated about a fixed axis which creates a three-dimensional solid.
Figure 1: An example of a body of revolution. The function f(x) is rotated around the x-axis to define a three-dimensional object. Note that this object is not convex, because it curves inward in the middle. Source: https://www.nagwa.com/en/explainers/103158132069/
In dimensions greater than three it is impossible to visualize the object, but the strategy is the same. Ryabogin’s proof is a negative answer to the question Ulam posed more than 70 years ago.
Future Open Questions
Stanislaw Ulam was a Polish mathematician who made far-reaching contributions to mathematics and physics. He is famous for working on the Manhattan project after emigrating to the United States. During his time in Lwów, Poland (now Lviv, Ukraine), he was a regular customer of the Scottish Café. Other frequenters of the cafe included Stefan Banach, Marc Kac, and even the Hungarian mathematician John von Neumann, who was one of the founders of game theory.
Ulam was the most prolific contributor to the Scottish Book. He wrote forty problems on his own and collaborated with others for fifteen more problems. The book was closed in 1941 with the advent of World War II, when Ulam and others fled to the United States. In total, 193 problems were recorded. Though Problem 19 can finally be marked as solved, many problems remain open for future mathematicians to continue to tackle.