Solution further to some kinds of 'coverage' applications?
‘Monumental’ Math Proof Solves Triple Bubble Problem and More
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.
By Erica Klarreich, Contributing Correspondent October 6, 2022
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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. ... much more ... (Math/Technical)
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