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<\/p>\nThe advantage of Rogers\u2019 method was that you didn\u2019t have to start with a particularly efficient lattice to get an efficient sphere packing. You just had to choose the right ellipsoid. But this introduced a new complication. Unlike a sphere, which is completely defined by a single number \u2014 its radius \u2014 an ellipsoid is defined by several axes of different lengths. The higher the dimension, the greater the number of directions you can stretch your ellipsoid in, and the more options you have for what your starting ellipsoid will look like.<\/p>\n
\u201cIn higher dimensions, you have no idea how to grow it. You have too much freedom,\u201d Klartag said.<\/p>\n
Mathematicians ultimately returned to Minkowski\u2019s approach, choosing to focus on finding the right lattices. They became more specialized in lattice theory and moved away from Rogers\u2019 focus on geometry.<\/p>\n
This strategy led to improvements in high-dimensional sphere packing. But for the most part, they only improved on Rogers\u2019 packing by a relatively small margin. Mathematicians still hoped for a bigger leap.<\/p>\n
For decades, they didn\u2019t get it. It would take an outsider to end the stagnation.<\/p>\n
An Outside Perspective<\/p>\n
Klartag, a mathematician at the Weizmann Institute of Science, was always intrigued by lattices and sphere packing. He just never had the time to learn much about them. He works in geometry, not lattice theory, and he usually studies convex shapes \u2014 shapes that don\u2019t jut inward. These shapes involve all sorts of symmetries, particularly in high dimensions. Klartag is convinced that this makes them extremely powerful: Convex shapes, he argues, are underappreciated mathematical tools.<\/p>\n
<\/p>\n
Boaz Klartag long suspected that methods from the field of convex geometry could be useful for sphere-packing problems. He just never had the time to test out his hunch \u2014 until now.<\/p>\n
Then last November, after completing a major project in his usual area of study, he noticed his calendar was uncharacteristically clear. \u201cI thought, I\u2019m 47 years old, all my life I wanted to study lattices, if I don\u2019t do it now then it\u2019s never going to happen,\u201d he said. He asked a friend, Barak Weiss<\/a> of Tel Aviv University, to mentor him in this new endeavor.<\/p>\n Weiss started a small seminar with Klartag and a handful of others to study the literature. Klartag\u2019s homework included a close reading of Minkowski\u2019s and Rogers\u2019 sphere-packing recipes.<\/p>\n When he read Rogers\u2019 trick for turning an ellipsoid into a sphere packing, he wondered why mathematicians had given up on the method. Ellipsoids are convex shapes, so Klartag knew lots of sophisticated ways to manipulate them. He also realized that the starting ellipsoids that Rogers had used were intuitive but inefficient. All he needed to do was construct a better ellipsoid \u2014 one that encompassed more space before its boundary hit other points in the lattice \u2014 and he could set a new packing record.<\/p>\n I thought, I\u2019m 47 years old, all my life I wanted to study lattices, if I don\u2019t do it now then it\u2019s never going to happen.<\/p>\n Boaz Klartag<\/p>\n He started with a method he knew well for growing and shrinking the boundary of an ellipsoid along each of its axes according to a random process. Whenever the boundary expanded enough to touch a new point in the lattice, he froze the ellipsoid\u2019s growth in that direction. This ensured that the point would never fall inside the ellipsoid. But the shape continued to inflate in every other direction, until it ran into another point. In this way, the ellipsoid would change shape in jerky, hesitating motions, gradually exploring the space around it. Eventually, the boundary would hit enough points to prevent the ellipsoid from growing further.<\/p>\n Over time, on average, the technique led the ellipsoid to increase in volume. But did it increase enough to surpass Rogers\u2019 intuitive ellipsoid?<\/p>\n Because Klartag\u2019s process was random, it produced a different ellipsoid every time he implemented it. He evaluated the range of possible volumes these ellipsoids might have. If he could find an ellipsoid that was larger in volume than the one Rogers had used decades earlier, he could then use Rogers\u2019 original method to turn it into a tighter sphere packing.<\/p>\n But Klartag couldn\u2019t find a single ellipsoid that was big enough. So he tweaked the details of his random growth process. After just a week or two, he was able to prove that, at least some of the time, this process would yield ellipsoids that were large enough to set a new record.<\/p>\n He immediately informed Weiss of his result. \u201cLet\u2019s meet next week and I\u2019ll tell you what my mistake was,\u201d Klartag told his mentor. But by then, Klartag had only grown more confident in his proof.<\/p>\n Closing In on the Truth<\/p>\n The proof checked out. Klartag\u2019s new starting ellipsoid, when turned into a sphere packing, gave the most substantial improvement in packing efficiency since Rogers\u2019 1947 paper. For a given dimension d, Klartag can pack d times the number of spheres that most previous results could manage. That is, in 100-dimensional space, his method packs roughly 100 times as many spheres; in a million-dimensional space, it packs roughly 1 million times as many.<\/p>\n Klartag had broken open a central problem in the world of lattices and sphere packing after just a few months of study and a few weeks of proof writing. \u201cIt feels almost unfair,\u201d he said. But that\u2019s often how mathematics works: Sometimes all a sticky problem needs is a few fresh ideas, and venturing outside one\u2019s immediate field can be rewarding. Klartag\u2019s familiarity with convex geometry, usually a separate area of study, turned out to be just what the problem required. \u201cThis idea was at the top of my mind because of my work,\u201d he said. \u201cIt was obvious to me that this was something I could try.\u201d<\/p>\n His result has also revived a debate in the field about the nature of the optimal packing in arbitrarily high dimensions. For a while, mathematicians considered highly symmetric, lattice-based packings to be the best way to arrange spheres as densely as possible. But in 2023, a team found a packing that didn\u2019t rely neatly on a repeating lattice<\/a>; before Klartag\u2019s result, it was the record to beat. Some mathematicians saw it as evidence that more disorder was needed in the search for an optimal sphere packing.<\/p>\n Now Klartag\u2019s work supports the notion that order and symmetry might be the way to go after all.<\/p>\n Moreover, there\u2019s been debate about just how dense sphere packings can get. Some mathematicians think Klartag\u2019s packing is just a hair away from optimal \u2014 practically as close as possible. Others think there\u2019s still room for improvement. \u201cI really have no idea what to believe at this point,\u201d said Marcus Michelen<\/a>, a mathematician at the University of Illinois, Chicago. \u201cI think all realities are still on the table.\u201d<\/p>\n The answer matters for potential applications to cryptography and communications. And so Klartag\u2019s result, while not immediately useful for those applications, has generated some tentative enthusiasm. \u201cThe problem is huge for engineers, and there\u2019s been little progress,\u201d said Or Ordentlich<\/a>, an information theorist at the Hebrew University. \u201cSo this gets us excited.\u201d<\/p>\n Klartag, for his part, hopes that his work will set off a return to the practices of Rogers\u2019 time, when the fields of convex geometry and lattice theory were far more connected. \u201cI think what we now understand about convex bodies should be useful for lattices, even beyond packing,\u201d he said.<\/p>\n \u201cMy goal is to make these two fields less disconnected than they are now,\u201d he added. \u201cThis was my plan, and I still want to pursue it.\u201d<\/p>\n","protected":false},"excerpt":{"rendered":"The advantage of Rogers\u2019 method was that you didn\u2019t have to start with a particularly efficient lattice to…\n","protected":false},"author":2,"featured_media":4906,"comment_status":"","ping_status":"","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[49],"tags":[199,79],"class_list":{"0":"post-4905","1":"post","2":"type-post","3":"status-publish","4":"format-standard","5":"has-post-thumbnail","7":"category-physics","8":"tag-physics","9":"tag-science"},"_links":{"self":[{"href":"https:\/\/www.newsbeep.com\/us\/wp-json\/wp\/v2\/posts\/4905","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.newsbeep.com\/us\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.newsbeep.com\/us\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.newsbeep.com\/us\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/www.newsbeep.com\/us\/wp-json\/wp\/v2\/comments?post=4905"}],"version-history":[{"count":0,"href":"https:\/\/www.newsbeep.com\/us\/wp-json\/wp\/v2\/posts\/4905\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.newsbeep.com\/us\/wp-json\/wp\/v2\/media\/4906"}],"wp:attachment":[{"href":"https:\/\/www.newsbeep.com\/us\/wp-json\/wp\/v2\/media?parent=4905"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.newsbeep.com\/us\/wp-json\/wp\/v2\/categories?post=4905"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.newsbeep.com\/us\/wp-json\/wp\/v2\/tags?post=4905"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}