Oct 7, 2022 07:12 PM
https://www.quantamagazine.org/monumenta...-20221006/
The decades-old Sullivan’s conjecture, about the best way to minimize the surface area of a bubble cluster, was thought to be out of reach for three bubbles and up — until a new breakthrough result.
INTRO: When it comes to understanding the shape of bubble clusters, mathematicians have been playing catch-up to our physical intuitions for millennia. Soap bubble clusters in nature often seem to immediately snap into the lowest-energy state, the one that minimizes the total surface area of their walls (including the walls between bubbles). But checking whether soap bubbles are getting this task right — or just predicting what large bubble clusters should look like — is one of the hardest problems in geometry. It took mathematicians until the late 19th century to prove that the sphere is the best single bubble, even though the Greek mathematician Zenodorus had asserted this more than 2,000 years earlier.
The bubble problem is simple enough to state: You start with a list of numbers for the volumes, and then ask how to separately enclose those volumes of air using the least surface area. But to solve this problem, mathematicians must consider a wide range of different possible shapes for the bubble walls. And if the assignment is to enclose, say, five volumes, we don’t even have the luxury of limiting our attention to clusters of five bubbles — perhaps the best way to minimize surface area involves splitting one of the volumes across multiple bubbles.
Even in the simpler setting of the two-dimensional plane (where you’re trying to enclose a collection of areas while minimizing the perimeter), no one knows the best way to enclose, say, nine or 10 areas. As the number of bubbles grows, “quickly, you can’t really even get any plausible conjecture,” said Emanuel Milman of the Technion in Haifa, Israel.
But more than a quarter century ago, John Sullivan, now of the Technical University of Berlin, realized that in certain cases, there is a guiding conjecture to be had. Bubble problems make sense in any dimension, and Sullivan found that as long as the number of volumes you’re trying to enclose is at most one greater than the dimension, there’s a particular way to enclose the volumes that is, in a certain sense, more beautiful than any other — a sort of shadow of a perfectly symmetric bubble cluster on a sphere. This shadow cluster, he conjectured, should be the one that minimizes surface area.
Over the decade that followed, mathematicians wrote a series of groundbreaking papers proving Sullivan’s conjecture when you’re trying to enclose only two volumes. Here, the solution is the familiar double bubble you may have blown in the park on a sunny day, made of two spherical pieces with a flat or spherical wall between them (depending on whether the two bubbles have the same or different volumes).
But proving Sullivan’s conjecture for three volumes, the mathematician Frank Morgan of Williams College speculated in 2007, “could well take another hundred years.”
Now, mathematicians have been spared that long wait — and have gotten far more than just a solution to the triple bubble problem. In a paper posted online in May, Milman and Joe Neeman, of the University of Texas, Austin, have proved Sullivan’s conjecture for triple bubbles in dimensions three and up and quadruple bubbles in dimensions four and up, with a follow-up paper on quintuple bubbles in dimensions five and up in the works.
And when it comes to six or more bubbles, Milman and Neeman have shown that the best cluster must have many of the key attributes of Sullivan’s candidate, potentially starting mathematicians on the road to proving the conjecture for these cases too. “My impression is that they have grasped the essential structure behind the Sullivan conjecture,” said Francesco Maggi of the University of Texas, Austin.
Milman and Neeman’s central theorem is “monumental,” Morgan wrote in an email. “It’s a brilliant accomplishment with lots of new ideas...” (MORE - details)
The decades-old Sullivan’s conjecture, about the best way to minimize the surface area of a bubble cluster, was thought to be out of reach for three bubbles and up — until a new breakthrough result.
INTRO: When it comes to understanding the shape of bubble clusters, mathematicians have been playing catch-up to our physical intuitions for millennia. Soap bubble clusters in nature often seem to immediately snap into the lowest-energy state, the one that minimizes the total surface area of their walls (including the walls between bubbles). But checking whether soap bubbles are getting this task right — or just predicting what large bubble clusters should look like — is one of the hardest problems in geometry. It took mathematicians until the late 19th century to prove that the sphere is the best single bubble, even though the Greek mathematician Zenodorus had asserted this more than 2,000 years earlier.
The bubble problem is simple enough to state: You start with a list of numbers for the volumes, and then ask how to separately enclose those volumes of air using the least surface area. But to solve this problem, mathematicians must consider a wide range of different possible shapes for the bubble walls. And if the assignment is to enclose, say, five volumes, we don’t even have the luxury of limiting our attention to clusters of five bubbles — perhaps the best way to minimize surface area involves splitting one of the volumes across multiple bubbles.
Even in the simpler setting of the two-dimensional plane (where you’re trying to enclose a collection of areas while minimizing the perimeter), no one knows the best way to enclose, say, nine or 10 areas. As the number of bubbles grows, “quickly, you can’t really even get any plausible conjecture,” said Emanuel Milman of the Technion in Haifa, Israel.
But more than a quarter century ago, John Sullivan, now of the Technical University of Berlin, realized that in certain cases, there is a guiding conjecture to be had. Bubble problems make sense in any dimension, and Sullivan found that as long as the number of volumes you’re trying to enclose is at most one greater than the dimension, there’s a particular way to enclose the volumes that is, in a certain sense, more beautiful than any other — a sort of shadow of a perfectly symmetric bubble cluster on a sphere. This shadow cluster, he conjectured, should be the one that minimizes surface area.
Over the decade that followed, mathematicians wrote a series of groundbreaking papers proving Sullivan’s conjecture when you’re trying to enclose only two volumes. Here, the solution is the familiar double bubble you may have blown in the park on a sunny day, made of two spherical pieces with a flat or spherical wall between them (depending on whether the two bubbles have the same or different volumes).
But proving Sullivan’s conjecture for three volumes, the mathematician Frank Morgan of Williams College speculated in 2007, “could well take another hundred years.”
Now, mathematicians have been spared that long wait — and have gotten far more than just a solution to the triple bubble problem. In a paper posted online in May, Milman and Joe Neeman, of the University of Texas, Austin, have proved Sullivan’s conjecture for triple bubbles in dimensions three and up and quadruple bubbles in dimensions four and up, with a follow-up paper on quintuple bubbles in dimensions five and up in the works.
And when it comes to six or more bubbles, Milman and Neeman have shown that the best cluster must have many of the key attributes of Sullivan’s candidate, potentially starting mathematicians on the road to proving the conjecture for these cases too. “My impression is that they have grasped the essential structure behind the Sullivan conjecture,” said Francesco Maggi of the University of Texas, Austin.
Milman and Neeman’s central theorem is “monumental,” Morgan wrote in an email. “It’s a brilliant accomplishment with lots of new ideas...” (MORE - details)