{"id":136661,"date":"2024-12-13T19:23:49","date_gmt":"2024-12-13T12:23:49","guid":{"rendered":"https:\/\/hotvideos24.online\/?p=136661"},"modified":"2024-12-13T19:23:49","modified_gmt":"2024-12-13T12:23:49","slug":"after-more-than-50-years-maths-sofa-problem-may-finally-be-solved","status":"publish","type":"post","link":"https:\/\/hotvideos24.online\/?p=136661","title":{"rendered":"After More Than 50 Years, Math&#8217;s &#8220;Sofa Problem&#8221; May Finally Be Solved"},"content":{"rendered":"<p> <script async src=\"https:\/\/pagead2.googlesyndication.com\/pagead\/js\/adsbygoogle.js?client=ca-pub-3711241968723425\"\r\n     crossorigin=\"anonymous\"><\/script>\r\n<ins class=\"adsbygoogle\"\r\n     style=\"display:block\"\r\n     data-ad-format=\"fluid\"\r\n     data-ad-layout-key=\"-fb+5w+4e-db+86\"\r\n     data-ad-client=\"ca-pub-3711241968723425\"\r\n     data-ad-slot=\"7910942971\"><\/ins>\r\n<script>\r\n     (adsbygoogle = window.adsbygoogle || []).push({});\r\n<\/script><br \/>\n<\/p>\n<div>\n<p id=\"isPasted\">In the world of mathematics, two things always hold true: firstly, some of the most stubborn and complex problems often have surprisingly real-world applications; and secondly, for people who spend all their time actually <em>in<\/em> that real world, those problems can seem\u2026 well, <a href=\"https:\/\/www.iflscience.com\/maths-bunkbed-conjecture-has-been-proven-false-after-40-years-76998\" target=\"_blank\" rel=\"noopener noreferrer\">pretty silly<\/a>.<\/p>\n<p>Take, for example, the \u201csofa problem\u201d: a conundrum that has both stumped mathematicians for decades, and also been \u201csolved\u201d by just about anybody who\u2019s ever moved house in their life. It\u2019s a question of how to move a curved sofa around a 90-degree corner \u2013 yes, just like in that one episode of <em>Friends<\/em> you\u2019re all now quoting.<\/p>\n<h2><strong>The math of moving<\/strong><\/h2>\n<p>Technically, the sofa problem <a href=\"http:\/\/www.nealrwagner.com\/pubs\/corner\/corner_final.pdf\" target=\"_blank\" rel=\"noopener noreferrer\">is this<\/a>: What is the region of largest area which can be moved around a right-angled corner in a corridor of width one? It was first formally stated <a href=\"https:\/\/www.proquest.com\/openview\/6ba832c6011ae61b969e221ee51a0b09\/1?pq-origsite=gscholar&amp;cbl=30748\" target=\"_blank\" rel=\"noopener\">in 1966<\/a> by the Austrian-Canadian mathematician Leo Moser \u2013 though it had been a topic of conversation around the mathematical water coolers for many years before that \u2013 and until now, never conclusively solved.<\/p>\n<p>Now, you\u2019ll notice that there\u2019s no mention of the eponymous settee in this formulation, and indeed the first piece of \u201cfurniture\u201d suggested as a solution was actually a \u201cpiano\u201d. Nevertheless, the \u201csofa\u201d terminology soon took off, mostly because \u2013 well, look at it:<\/p>\n<div class=\"fr-image-container no-background\" data-asset-id=\"80875\" data-reactroot=\"\">\n<div class=\"fr-image\"><picture title=\"\"><source media=\"(min-width: 1000px)\" srcset=\"https:\/\/assets.iflscience.com\/assets\/articleNo\/77210\/iImg\/80875\/the-gerver-sofa-showing-each-of-the-18-segments-making-up-its-boundary-it-looks-like-kind-of-an-old-style-phone-receiver-seen-from-above-l.webp\" type=\"image\/webp\"><source media=\"(min-width: 1000px)\" srcset=\"https:\/\/assets.iflscience.com\/assets\/articleNo\/77210\/iImg\/80875\/the-gerver-sofa-showing-each-of-the-18-segments-making-up-its-boundary-it-looks-like-kind-of-an-old-style-phone-receiver-seen-from-above-l.png\" type=\"image\/png\"><source media=\"(min-width: 568px)\" srcset=\"https:\/\/assets.iflscience.com\/assets\/articleNo\/77210\/iImg\/80875\/the-gerver-sofa-showing-each-of-the-18-segments-making-up-its-boundary-it-looks-like-kind-of-an-old-style-phone-receiver-seen-from-above-m.webp\" type=\"image\/webp\"><source media=\"(min-width: 568px)\" srcset=\"https:\/\/assets.iflscience.com\/assets\/articleNo\/77210\/iImg\/80875\/the-gerver-sofa-showing-each-of-the-18-segments-making-up-its-boundary-it-looks-like-kind-of-an-old-style-phone-receiver-seen-from-above-m.png\" type=\"image\/png\"><source media=\"(max-width: 567px)\" srcset=\"https:\/\/assets.iflscience.com\/assets\/articleNo\/77210\/iImg\/80875\/the-gerver-sofa-showing-each-of-the-18-segments-making-up-its-boundary-it-looks-like-kind-of-an-old-style-phone-receiver-seen-from-above-s.webp\" type=\"image\/webp\"><source media=\"(max-width: 567px)\" srcset=\"https:\/\/assets.iflscience.com\/assets\/articleNo\/77210\/iImg\/80875\/the-gerver-sofa-showing-each-of-the-18-segments-making-up-its-boundary-it-looks-like-kind-of-an-old-style-phone-receiver-seen-from-above-s.png\" type=\"image\/png\"><img decoding=\"async\" class=\"inline-image fr-fic fr-dib\" data-asset-id=\"80875\" src=\"https:\/\/assets.iflscience.com\/assets\/articleNo\/77210\/iImg\/80875\/Gerver.svg.png\" alt=\"The Gerver sofa, showing each of the 18 segments making up its boundary\" title=\"The Gerver sofa, showing each of the 18 segments making up its boundary\" loading=\"lazy\"\/><\/source><\/source><\/source><\/source><\/source><\/source><\/picture><\/div><figcaption class=\"fr-figcaption\">\n<p>The Gerver sofa, showing each of the 18 segments making up its boundary.<\/p>\n<\/figcaption><\/div>\n<p>It was the determination of a lower bound that gave rise to the iconic sofa shape: in a 1968 treatise named, we kid you not, <em><a href=\"https:\/\/archive.org\/details\/hammersley1968\/page\/83\/mode\/2up\" target=\"_blank\" rel=\"noopener\">On the enfeeblement of mathematical skills by &#8216;Modern Mathematics&#8217; and by similar soft intellectual trash in schools and universities<\/a><\/em>, John Hammersley showed with some relatively simple calculus that this shape gave an area of (\u03c0\/2) + (2\/\u03c0) \u2013 roughly 2.2074.\u00a0<\/p>\n<p>In fact, he went further. In the same paper, he proved that an upper bound on the area was given by 2\u221a2 \u2013 roughly 2.8284. It had only been a couple of years, but the sofa problem was already on its way to a solution: the exact figure hadn\u2019t been nailed down yet, but mathematicians knew it had to be between these two values. Surely it wouldn\u2019t take much more work to find the true answer?<\/p>\n<p>Fast forward 25 years, though, and Hammersley\u2019s bounds were still the best we had. That was, until Rutgers mathematician Joseph Gerver stepped up to the plate, offering a sofa constructed from 18 <a href=\"https:\/\/www.iflscience.com\/what-is-the-mandelbrot-set-and-where-did-it-come-from-77169\" target=\"_blank\" rel=\"noopener\">analytically smooth<\/a> connected curve sections. The \u201cGerver sofa\u201d, as it became known, increased the lower bound to 2.2195.<\/p>\n<p>It would be another quarter-century again before the range of possible solutions would be pared down even further: <a href=\"https:\/\/arxiv.org\/abs\/1706.06630\" target=\"_blank\" rel=\"noopener\">in 2018<\/a>, mathematicians Yoav Kallus and Dan Romik used a computer-assisted proof to shave the upper bound down to 2.37.\u00a0<\/p>\n<p>It was a big improvement on Hammersley\u2019s original bounds \u2013 but that exact solution was still evading capture.<\/p>\n<h2><strong>Baek in the game<\/strong><\/h2>\n<p>It would have been around the same time as Kallus and Romik were working on their solution that Jineon Baek, a postdoctoral researcher at Yonsei University in Seoul, Korea, first started thinking about the sofa problem. Now, seven years later, he reckons he\u2019s cracked it in a proof that has yet to be peer-reviewed.<\/p>\n<p>\u201cI dedicated a lot of time to this, without any publication so far,\u201d he told <a href=\"https:\/\/www.newscientist.com\/article\/2459500-mathematicians-have-figured-out-the-best-sofa-shape-for-moving-around\/\" target=\"_blank\" rel=\"noopener\">New Scientist<\/a>. \u201cThe fact that now I can say to the world that I committed something valuable to this problem is validating.\u201d\u00a0<\/p>\n<p>For a question so easily stated and imagined, Baek\u2019s proof was no small undertaking. Spanning more than 100 pages, it does far more than simply brute force the problem or continuously shave off ever-smaller slices of area. Rather, it is, Romik told New Scientist, a \u201cwonderful development\u201d.<\/p>\n<p>\u201cI know I could never have done this,\u201d Romik said. \u201cI don\u2019t have a feeling of regret, or like, how could I miss this, because it\u2019s clear it\u2019s just not the sort of thinking that I think I would have been able to. [Baek] was just coming at it from a completely different direction.\u201d<\/p>\n<p><iframe loading=\"lazy\" title=\"Is Math The Greatest Subject In The World?\" allowtransparency=\"true\" height=\"150\" width=\"100%\" style=\"border: none; min-width: min(100%, 430px);height:150px;\" scrolling=\"no\" data-name=\"pb-iframe-player\" src=\"https:\/\/www.podbean.com\/player-v2\/?from=embed&amp;i=izr9m-12f34e8-pb&amp;share=1&amp;download=1&amp;fonts=Arial&amp;skin=666666&amp;font-color=auto&amp;rtl=0&amp;logo_link=episode_page&amp;btn-skin=8bbb4e&amp;size=150\"><span class=\"fr-mk\" style=\"display: none;\">\u00a0<\/span><\/iframe><\/p>\n<p>Without getting into the nitty-gritty, the proof goes like this: first, Baek said that the optimal sofa, whatever it turned out to be, had to have three specific properties \u2013 it had to be monotone, balanced, and have a rotation angle \u03c0\/2. Again, these are quite technical to define, but essentially it boils down to this: the \u201csofa\u201d we\u2019ve been using so far is pretty much the right shape already.<\/p>\n<p>Secondly, Baek set about proving a condition on how this sofa would move around the corner \u2013 a small thing, but crucial for completing the final step: defining the upper bound for the area of this sofa, and then showing that it was equal to Gerver\u2019s lower bound.<\/p>\n<p>That\u2019s right: after 32 years, it turns out Gerver was right all along.<\/p>\n<p>\u201cI am of course very happy about all of this,\u201d Gerver told New Scientist. \u201cI am 75 years old, and Baek can\u2019t be more than 30. He has a lot more energy, stamina and surviving brain cells than I do, and I am glad that he picked up the baton. I am also very happy that I lived long enough to see him finish what I started.\u201d<\/p>\n<h2><strong>Put your feet up<\/strong><\/h2>\n<p>So, is the sofa problem now complete? Well, technically, it remains to be seen. As with all mathematical proofs, it needs to be peer-reviewed for accuracy \u2013 a process that Baek is quietly hopeful for.\u00a0<\/p>\n<p>\u201cI can\u2019t say that I\u2019m confident 100 per cent, because we are humans, we make errors,\u201d he told New Scientist. \u201cBut still, I did my best to be as confident as I can.\u201d<\/p>\n<p>But if your hopes of solving the sofa problem yourself have been dashed by this news, take heart: since Baek defined his sofa so strictly, you can always choose a different shape for your own.\u00a0<\/p>\n<p>It might not make quite as good a sofa for your living room of course, but there\u2019s really nothing stopping you from going\u2026 Baek to the drawing board, you might say.<\/p>\n<p>The proof can be found <a href=\"https:\/\/arxiv.org\/abs\/2411.19826\" target=\"_blank\" rel=\"noopener\">on the ArXiv preprint server<\/a>.<\/p>\n<\/div>\n<p><script async src=\"https:\/\/pagead2.googlesyndication.com\/pagead\/js\/adsbygoogle.js?client=ca-pub-3711241968723425\"\r\n     crossorigin=\"anonymous\"><\/script>\r\n<ins class=\"adsbygoogle\"\r\n     style=\"display:block\"\r\n     data-ad-format=\"fluid\"\r\n     data-ad-layout-key=\"-fb+5w+4e-db+86\"\r\n     data-ad-client=\"ca-pub-3711241968723425\"\r\n     data-ad-slot=\"7910942971\"><\/ins>\r\n<script>\r\n     (adsbygoogle = window.adsbygoogle || []).push({});\r\n<\/script><br \/>\n<br \/><div data-type=\"_mgwidget\" data-widget-id=\"1660802\">\r\n<\/div>\r\n<script>(function(w,q){w[q]=w[q]||[];w[q].push([\"_mgc.load\"])})(window,\"_mgq\");\r\n<\/script>\r\n<br \/>\n<br \/><a href=\"https:\/\/www.iflscience.com\/after-more-than-50-years-maths-sofa-problem-may-finally-be-solved-77210\">Source link <\/a><\/p>\n","protected":false},"excerpt":{"rendered":"<p>In the world of mathematics, two things always hold true: firstly, some of the most stubborn and complex problems often have surprisingly real-world applications; and secondly, for people who spend &hellip; <a href=\"https:\/\/hotvideos24.online\/?p=136661\" class=\"more-link\">Read More<\/a><\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[8628],"tags":[],"class_list":["post-136661","post","type-post","status-publish","format-standard","hentry","category-science","entry"],"_links":{"self":[{"href":"https:\/\/hotvideos24.online\/index.php?rest_route=\/wp\/v2\/posts\/136661","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/hotvideos24.online\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/hotvideos24.online\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/hotvideos24.online\/index.php?rest_route=\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/hotvideos24.online\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=136661"}],"version-history":[{"count":0,"href":"https:\/\/hotvideos24.online\/index.php?rest_route=\/wp\/v2\/posts\/136661\/revisions"}],"wp:attachment":[{"href":"https:\/\/hotvideos24.online\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=136661"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/hotvideos24.online\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=136661"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/hotvideos24.online\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=136661"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}