\n<\/p>\n
Imagine that you want to know the most efficient way to make a torus\u2014a doughnut-shaped mathematical object\u2014from origami paper. But this torus, which is a surface, looks drastically different than the outside of a glazed bakery doughnut. Instead of seeming almost perfectly smooth, the torus that you envision is jagged with many faces, each of which is a polygon. In other words, you want to construct a polyhedral torus with faces that are shapes such as triangles or rectangles.<\/p>\n
Your peculiar-looking shape will be trickier to construct than one with a smooth surface. The complexity of the problem only grows if you decide that you want to envision constructing something similar but in four or more dimensions.<\/p>\n
Mathematician Richard Evan Schwartz of Brown University tackled the problem in a recent study by working backward from an existing polyhedral torus to answer questions about what would be needed to construct it from scratch. He posted his findings to a preprint server in August 2025.<\/p>\n
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Schwartz was able to find a solution to a long-standing question: What\u2019s the minimum number of vertices (corners) needed to make polyhedral tori with a property called intrinsic flatness? The answer, Schwartz found, is eight vertices. He first demonstrated that seven vertices aren\u2019t enough. He then discovered an example of an intrinsically flat polyhedral torus with eight vertices.<\/p>\n
\u201cIt\u2019s very striking that Rich Schwartz was able to entirely solve this well-known problem,\u201d says Jean-Marc Schlenker, a mathematician at the University of Luxembourg. \u201cThe problem looks elementary but had been open for many years.\u201d<\/p>\n
Schwartz\u2019s finding essentially provides the minimum number of vertices that a polyhedral torus needs so that it can be flattened. But one detail\u2014what it means to be \u201cintrinsically flat\u201d rather than simply \u201cflat\u201d\u2014is a bit complicated to parse. The notion is also central to connecting Schwartz\u2019s results to the question of building polyhedral tori from scratch.<\/p>\n
Since the 1960s mathematicians have known that intrinsically flat versions of mathematical objects exist. Actually finding those objects is a different beast, Schwartz notes. Describing polyhedral tori as intrinsically flat isn\u2019t quite equivalent to simply saying that they\u2019re flat like a piece of paper. Instead it means that these surfaces have the same dimensions as (or, as mathematicians say, \u201care isometric to\u201d) tori that are smooshed flat. \u201cAnother way to say it is that if you compute the angle sums around each vertex, it adds up to 2\u03c0 everywhere,\u201d Schwartz says.<\/p>\n
According to Schlenker, Schwartz\u2019s finding is very on-brand for his expertise. Yet for many years, Schwartz was so stumped by the problem that he set it aside.<\/p>\n
But Schwartz wasn\u2019t initially convinced. \u201cBasically, they shoved it in my face, and at some point, years passed. I actually thought it was too hard of a problem,\u201d he says. The difficulty stemmed from the large dimensions that seemed to be involved. \u201cEven for just seven or eight [vertices], it seems that you would have to look at 20-some-odd-dimensional space,\u201d he says.<\/p>\n
But when the three mathematicians reunited in 2025, Schwartz learned that Leli\u00e8vre\u2019s roommate, Vincent Tugay\u00e9,had found an example that worked with nine vertices. \u201cIt was a really pretty thing\u201d that Tugay\u00e9, a high school teacher with a Ph.D. in physics, exhibited at math outreach fairs in Paris, Schwartz says. \u201cI thought, \u2018Well, this one\u2019s got to be the best,\u2019\u201d adds Schwartz, who then set out to settle whether his intuition was correct.<\/p>\n
To approach the question of whether the cases with seven or eight vertices would work, Schwartz focused on answering \u201cHow do I cut down the dimension?\u201d He generated a lot of ideas about how to do so for the seven vertices case. Yet he ultimately stumbled upon a mathematical gift of sorts: a little known 1991 paper that \u201cgoes about 80 percent of the way to proving that you can\u2019t do it with seven vertices,\u201d he says. \u201cThen I just finished it off.\u201d<\/p>\n
Still thinking that the eight vertices case also wouldn\u2019t work, he then tried to use a similar approach to prove that claim. When he found he couldn\u2019t rule out some cases, he decided to figure out what properties an eight-vertex torus would need to have to be intrinsically flat. Using an approach that he describes as \u201cheavily supervised machine learning,\u201d Schwartz then found an eight-vertex example that did work.<\/p>\n
\u201cWhat’s most striking, I think, is that it\u2019s another example of the specific skills that Rich Schwartz has developed, blending traditional mathematical investigation with computational methods,\u201d Schlenker says. \u201cHe finds beautiful geometric ideas to prove some results but also writes elaborate programs to search for and find examples. Very few mathematicians are capable of bringing those two strands together so harmoniously.\u201d<\/p>\n
If you enjoyed this article, I\u2019d like to ask for your support. Scientific American<\/span> has served as an advocate for science and industry for 180 years, and right now may be the most critical moment in that two-century history.<\/p>\n I\u2019ve been a Scientific American<\/span> subscriber since I was 12 years old, and it helped shape the way I look at the world. SciAm <\/span>always educates and delights me, and inspires a sense of awe for our vast, beautiful universe. I hope it does that for you, too.<\/p>\n If you subscribe to Scientific American<\/span>, you help ensure that our coverage is centered on meaningful research and discovery; that we have the resources to report on the decisions that threaten labs across the U.S.; and that we support both budding and working scientists at a time when the value of science itself too often goes unrecognized.<\/p>\n In return, you get essential news, captivating podcasts, brilliant infographics, can’t-miss newsletters, must-watch videos, challenging games, and the science world’s best writing and reporting. You can even gift someone a subscription.<\/p>\n There has never been a more important time for us to stand up and show why science matters. I hope you\u2019ll support us in that mission.<\/p>\n","protected":false},"excerpt":{"rendered":" Imagine that you want to know the most efficient way to make a torus\u2014a doughnut-shaped mathematical object\u2014from origami paper. But this torus, which is a surface, looks drastically different than the outside of a glazed bakery doughnut. Instead of seeming almost perfectly smooth, the torus that you envision is jagged with many faces, each of<\/p>\n","protected":false},"author":1,"featured_media":44076,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[58],"tags":[11436,8203,22329,4005],"class_list":{"0":"post-44075","1":"post","2":"type-post","3":"status-publish","4":"format-standard","5":"has-post-thumbnail","7":"category-science","8":"tag-discover","9":"tag-mathematicians","10":"tag-shapes","11":"tag-ways"},"_links":{"self":[{"href":"https:\/\/naijaglobalnews.org\/index.php?rest_route=\/wp\/v2\/posts\/44075","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/naijaglobalnews.org\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/naijaglobalnews.org\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/naijaglobalnews.org\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/naijaglobalnews.org\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=44075"}],"version-history":[{"count":0,"href":"https:\/\/naijaglobalnews.org\/index.php?rest_route=\/wp\/v2\/posts\/44075\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/naijaglobalnews.org\/index.php?rest_route=\/wp\/v2\/media\/44076"}],"wp:attachment":[{"href":"https:\/\/naijaglobalnews.org\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=44075"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/naijaglobalnews.org\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=44075"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/naijaglobalnews.org\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=44075"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}