{"id":1718898,"date":"2022-10-06T12:54:27","date_gmt":"2022-10-06T16:54:27","guid":{"rendered":"https:\/\/wordpress-1016567-4521551.cloudwaysapps.com\/?post_type=station&p=1718898"},"modified":"2022-10-06T18:35:21","modified_gmt":"2022-10-06T22:35:21","slug":"monumental-math-proof-solves-triple-bubble-problem-and-more","status":"publish","type":"station","link":"https:\/\/platodata.io\/plato-data\/monumental-math-proof-solves-triple-bubble-problem-and-more\/","title":{"rendered":"\u2018Monumental\u2019 Math Proof Solves Triple Bubble Problem and More"},"content":{"rendered":"
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 \u2014 or just predicting what large bubble clusters should look like \u2014 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.<\/p>\n
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\u2019t even have the luxury of limiting our attention to clusters of five bubbles \u2014 perhaps the best way to minimize surface area involves splitting one of the volumes across multiple bubbles.<\/p>\n
Even in the simpler setting of the two-dimensional plane (where you\u2019re 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, \u201cquickly, you can\u2019t really even get any plausible conjecture,\u201d said Emanuel Milman<\/a> of the Technion in Haifa, Israel.<\/p>\n But more than a quarter century ago, John Sullivan<\/a>, now of the Technical University of Berlin, realized that in certain cases, there is a guiding conjecture<\/a> to be had. Bubble problems make sense in any dimension, and Sullivan found that as long as the number of volumes you\u2019re trying to enclose is at most one greater than the dimension, there\u2019s a particular way to enclose the volumes that is, in a certain sense, more beautiful than any other \u2014 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.<\/p>\n Over the decade that followed, mathematicians wrote a series of groundbreaking papers proving Sullivan\u2019s conjecture when you\u2019re 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).<\/p>\n But proving Sullivan\u2019s conjecture for three volumes, the mathematician Frank Morgan<\/a> of Williams College speculated<\/a> in 2007, \u201ccould well take another hundred years.\u201d<\/p>\n<\/div>\n<\/div>\n Now, mathematicians have been spared that long wait \u2014 and have gotten far more than just a solution to the triple bubble problem. In a paper<\/a> posted online in May, Milman and Joe Neeman<\/a>, of the University of Texas, Austin, have proved Sullivan\u2019s 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.<\/p>\n 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\u2019s candidate, potentially starting mathematicians on the road to proving the conjecture for these cases too. \u201cMy impression is that they have grasped the essential structure behind the Sullivan conjecture,\u201d said Francesco Maggi<\/a> of the University of Texas, Austin.<\/p>\n Milman and Neeman\u2019s central theorem is \u201cmonumental,\u201d Morgan wrote in an email. \u201cIt\u2019s a brilliant accomplishment with lots of new ideas.\u201d<\/p>\n Our experiences with real soap bubbles offer tempting intuitions about what optimal bubble clusters should look like, at least when it comes to small clusters. The triple or quadruple bubbles we blow through soapy wands seem to have spherical walls (and occasionally flat ones) and tend to form tight clumps rather than, say, a long chain of bubbles.<\/p>\n But it\u2019s not so easy to prove that these really are the features of optimal bubble clusters. For example, mathematicians don\u2019t know whether the walls in a minimizing bubble cluster are always spherical or flat \u2014 they only know that the walls have \u201cconstant mean curvature,\u201d which means the average curvature stays the same from one point to another. Spheres and flat surfaces have this property, but so do many other surfaces, such as cylinders and wavy shapes called unduloids. Surfaces with constant mean curvature are \u201ca complete zoo,\u201d Milman said.<\/p>\n But in the 1990s, Sullivan recognized that when the number of volumes you want to enclose is at most one greater than the dimension, there\u2019s a candidate cluster that seems to outshine the rest \u2014 one (and only one) cluster that has the features we tend to see in small clusters of real soap bubbles.<\/p>\n To get a feel for how such a candidate is built, let\u2019s use Sullivan\u2019s approach to create a three-bubble cluster in the flat plane (so our \u201cbubbles\u201d will be regions in the plane rather than three-dimensional objects). We start by choosing four points on a sphere that are all the same distance from each other. Now imagine that each of these four points is the center of a tiny bubble, living only on the surface of the sphere (so that each bubble is a small disk). Inflate the four bubbles on the sphere until they start bumping into each other, and then keep inflating until they collectively fill out the entire surface. We end up with a symmetric cluster of four bubbles that makes the sphere look like a puffed-out tetrahedron.<\/p>\n Next, we place this sphere on top of an infinite flat plane, as if the sphere is a ball resting on an endless floor. Imagine that the ball is transparent and there\u2019s a lantern at the north pole. The walls of the four bubbles will project shadows on the floor, forming the walls of a bubble cluster there. Of the four bubbles on the sphere, three will project down to shadow bubbles on the floor; the fourth bubble (the one containing the north pole) will project down to the infinite expanse of floor outside the cluster of three shadow bubbles.<\/p>\n The particular three-bubble cluster we get depends on how we happened to position the sphere when we put it on the floor. If we spin the sphere so a different point moves to the lantern at the north pole, we\u2019ll typically get a different shadow, and the three bubbles on the floor will have different areas. Mathematicians have proved<\/a> that for any three numbers you choose for the areas, there is essentially a single way to position the sphere so the three shadow bubbles will have precisely those areas.<\/p>\n<\/div>\n<\/div>\n We\u2019re free to carry out this process in any dimension (though higher-dimensional shadows are harder to visualize). But there\u2019s a limit to how many bubbles we can have in our shadow cluster. In the example above, we couldn\u2019t have made a four-bubble cluster in the plane. That would have required starting with five points on the sphere that are all the same distance from each other \u2014 but it\u2019s impossible to place that many equidistant points on a sphere (though you can do it with higher-dimensional spheres). Sullivan\u2019s procedure only works to create clusters of up to three bubbles in two-dimensional space, four bubbles in three-dimensional space, five bubbles in four-dimensional space, and so on. Outside those parameter ranges, Sullivan-style bubble clusters just don\u2019t exist.<\/p>\n But within those parameters, Sullivan\u2019s procedure gives us bubble clusters in settings far beyond what our physical intuition can comprehend. \u201cIt\u2019s impossible to visualize what is a 15-bubble in [23-dimensional space],\u201d Maggi said. \u201cHow do you even dream of describing such an object?\u201d<\/p>\n Yet Sullivan\u2019s bubble candidates inherit from their spherical progenitors a unique collection of properties reminiscent of the bubbles we see in nature. Their walls are all spherical or flat, and wherever three walls meet, they form 120-degree angles, as in a symmetric Y shape. Each of the volumes you\u2019re trying to enclose lies in a single region, instead of being split across multiple regions. And every bubble touches every other (and the exterior), forming a tight cluster. Mathematicians have shown that Sullivan\u2019s bubbles are the only clusters that satisfy all these properties.<\/p>\n When Sullivan hypothesized that these should be the clusters that minimize surface area, he was essentially saying, \u201cLet\u2019s assume beauty,\u201d Maggi said.<\/p>\n But bubble researchers have good reason to be wary of assuming that just because a proposed solution is beautiful, it is correct. \u201cThere are very famous problems \u2026 where you would expect symmetry for the minimizers, and symmetry spectacularly fails,\u201d Maggi said.<\/p>\n For example, there\u2019s the closely related problem of filling infinite space with equal-volume bubbles in a way that minimizes surface area. In 1887, the British mathematician and physicist Lord Kelvin suggested that the solution might be an elegant honeycomb-like structure. For more than a century, many mathematicians believed this was the likely answer \u2014 until 1993, when a pair of physicists identified a better<\/a>, though less symmetric, option. \u201cMathematics is full \u2026 of examples where this kind of weird thing happens,\u201d Maggi said.<\/p>\n When Sullivan announced his conjecture in 1995, the double-bubble portion of it had already been floating around for a century. Mathematicians had solved the 2D double-bubble problem<\/a> two years earlier, and in the decade that followed, they solved it in three-dimensional space<\/a> and then in higher<\/a> dimensions<\/a>. But when it came to the next case of Sullivan\u2019s conjecture \u2014 triple bubbles \u2014 they could prove the conjecture<\/a> only in the two-dimensional plane, where the interfaces between bubbles are particularly simple.<\/p>\n Then in 2018, Milman and Neeman proved an analogous version of Sullivan\u2019s conjecture in a setting known as the Gaussian bubble problem. In this setting, you can think of every point in space as having a monetary value: The origin is the most expensive spot, and the farther you get from the origin, the cheaper land becomes, forming a bell curve. The goal is to create enclosures with preselected prices (instead of preselected volumes), in a way that minimizes the cost of the boundaries of the enclosures (instead of the boundaries\u2019 surface area). This Gaussian bubble problem has applications in computer science to rounding schemes and questions of noise sensitivity.<\/p>\nShadow Bubbles<\/strong><\/h2>\n
A Dark Art<\/strong><\/h2>\n