{"id":137234,"date":"2026-06-23T12:11:35","date_gmt":"2026-06-23T16:11:35","guid":{"rendered":"https:\/\/www.simonsfoundation.org\/?p=137234"},"modified":"2026-06-23T12:11:43","modified_gmt":"2026-06-23T16:11:43","slug":"the-twisted-mathematics-of-richard-schwartz-and-mobius-bands","status":"publish","type":"post","link":"https:\/\/www.simonsfoundation.org\/2026\/06\/23\/the-twisted-mathematics-of-richard-schwartz-and-mobius-bands\/","title":{"rendered":"The Twisted Mathematics of Richard Schwartz and M\u00f6bius Bands"},"content":{"rendered":"","protected":false},"excerpt":{"rendered":"","protected":false},"author":432,"featured_media":137554,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"_relevanssi_hide_post":"","_relevanssi_hide_content":"","_relevanssi_pin_for_all":"","_relevanssi_pin_keywords":"","_relevanssi_unpin_keywords":"","_relevanssi_related_keywords":"","_relevanssi_related_include_ids":"","_relevanssi_related_exclude_ids":"","_relevanssi_related_no_append":"","_relevanssi_related_not_related":"","_relevanssi_related_posts":"","_relevanssi_noindex_reason":"","footnotes":""},"categories":[],"tags":[],"class_list":["post-137234","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","news_type-article"],"acf":{"block_editor":[{"acf_fc_layout":"video","embed_code":"https:\/\/vimeo.com\/1202522225?share=copy&fl=sv&fe=ci","title":"An animation of hill climbing, a mathematical optimization technique used by mathematician Richard Schwartz when he was researching how to make a torus shape with the smallest number of vertices. The algorithm starts with an arbitrary solution to a problem and then makes incremental changes to the solution to find a more optimized outcome.","caption":"Courtesy of Richard Schwartz","video_autoplay":true,"video_loop":true,"video_width":"","video_height":"","video_container_size":"video_large"},{"acf_fc_layout":"text","text":"

Take a rectangular strip of paper, give it a half twist and connect its short ends to each other. You have made a M\u00f6bius band, also known as a M\u00f6bius strip or loop. It is a familiar staple of recreational mathematics and a simple illustration of the mathematical concept of nonorientability. (That is, it has no universal \u2018up\u2019 or \u2018down\u2019 direction.) Plus, it\u2019s just fun to play with.<\/span><\/p>\r\n

If you start with a long, skinny strip of paper, it\u2019s easy to make a M\u00f6bius band. With shorter strips, it gets harder to twist and connect the ends. You\u2019ll eventually hit the point where you physically can\u2019t make a M\u00f6bius band because the ratio between the length and width is too low. There is a clever folding pattern that produces a M\u00f6bius band from a strip of paper with an aspect ratio of the square root of 3 (around 1.73) to 1. Even though this construction has been described in papers since as early as 1930, no one was able to prove that it was impossible to twist a strip with a smaller aspect ratio into a M\u00f6bius band.<\/span><\/p>"},{"acf_fc_layout":"image","type":"image_medium","image":137576,"title":"A M\u00f6bius strip is a surface that connects to itself with a half twist. Mathematician Richard Schwartz recently proved the minimum aspect ratio possible for such a band. ","caption":"Krishnavedala\/Wikimedia","caption_full":"","less_margin":false},{"acf_fc_layout":"text","text":"

When Richard Schwartz heard about the conundrum, he asked himself, \u201cWhat\u2019s so hard about this problem? I could do this.\u201d Schwartz is a math professor at Brown University and four-time Simons Fellow in Mathematics<\/a> with broad interests, from geometry and topology to writing colorful, mind-bending children\u2019s books about math. <\/span><\/p>\r\n

His first instinct when faced with the M\u00f6bius strip question was to model it on the computer. That exploration gave him some immediate gratification, an improvement on what were then the best-known bounds of the problem based on an argument that involved cutting the M\u00f6bius band along a particular set of lines. When he first programmed it into the computer, he erroneously assumed those cuts left him with a parallelogram. \u201cThere followed three years where I was trying to push this argument further using the parallelogram idea,\u201d he says. His chain of reasoning kept getting more complicated, but the bound never budged. Finally, he started playing with a paper model. \u201cI cut one open, and oh my God, you get a trapezoid.\u201d The entire parallelogram argument evaporated, taking with it his original improvement.<\/span><\/p>\r\n

While he was trying to correct his earlier erroneous paper, he realized, \u201cWhen I did the calculation right, it solved the whole conjecture.\u201d In just a few days, he was able to prove<\/a> that the square root of 3 to 1 is indeed the lower limit of aspect ratios for a M\u00f6bius band. He circulated his proof among some other interested mathematicians and, with their feedback and improvements, \u201cit got to be a razor-sharp argument,\u201d he says. \u201cI\u2019m thrilled with it.\u201d<\/span><\/p>\r\n

The optimal M\u00f6bius band is the answer to a problem of geometric optimization. Such problems ask, given certain geometric constraints, what the limits are of the shapes that fit those constraints \u2014 for instance, the smallest or the largest. The questions are often natural and intuitive (such as identifying the widest M\u00f6bius strip), but actually proving that a particular shape is optimal can be fiendishly difficult. Schwartz\u2019s misadventure among the specious parallelograms illustrates the combination of playfulness and tenacity that allows him to latch on to these curiosity-driven questions and see them through to the end.<\/span><\/p>\r\n

During a trip to the Institut des Hautes \u00c9tudes Scientifiques (IHES) outside Paris during his 2024\u20132025 Simons Fellowship, Schwartz turned his attention to another geometric optimization problem involving a well-known shape: an origami-like folding of a flat surface into a torus. A torus is the mathematical name for the shape of an inner tube or the layer of glaze covering a doughnut. The term refers to the shape topologically: That is, the shape can be stretched, twisted and deformed and, so long as it isn\u2019t torn or glued, still be called a torus.<\/span><\/p>\r\n

A flat torus is a representation of the torus as a square, or any parallelogram, with opposite sides \u2018identified.\u2019 That is, the sides behave like the sides in the classic arcade game Asteroids. When the user-controlled spaceship travels off the right side of the screen, it reappears on the left side, continuing in the same direction. Just as in the game, the sides of the flat torus are linked. <\/span><\/p>\r\n

If you actually sat at your desk, took a square of paper and taped the edges together to realize that flat torus in three-dimensional space, it wouldn\u2019t be smooth and pretty like an inner tube \u2014 it would be folded or crumpled in some places. The smooth surface of an inner tube requires the stretching of rubber. A flat torus can\u2019t fit in three-dimensional space in a smooth way without distortion, but if you allow some folding, you can do it. <\/span><\/p>"},{"acf_fc_layout":"code","code":"

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