{"680804":{"#nid":"680804","#data":{"type":"news","title":"What\u2019s the Shape of the Universe? Mathematicians Use Topology to Study the Shape of the World and Everything in\u00a0it","body":[{"value":"\u003Cdiv class=\u0022theconversation-article-body\u0022\u003E\u003Cp\u003EWhen you look at your surrounding environment, it might seem like you\u2019re living on a flat plane. After all, this is why you can navigate a new city using a map: a flat piece of paper that represents all the places around you. This is likely why some people in the past believed the earth to be flat. But most people now know that is far from the truth.\u003C\/p\u003E\u003Cp\u003EYou live on the surface of a giant sphere, like a beach ball the size of the Earth with a few bumps added. The surface of the sphere and the plane are two possible 2D spaces, meaning you can walk in two directions: north and south or east and west.\u003C\/p\u003E\u003Cp\u003EWhat other possible spaces might you be living on? That is, what other spaces around you are 2D? For example, the surface of a giant doughnut is another 2D space.\u003C\/p\u003E\u003Cp\u003EThrough a field called geometric topology, \u003Ca href=\u0022https:\/\/www.researchgate.net\/scientific-contributions\/John-B-Etnyre-10186406\u0022\u003Emathematicians like me\u003C\/a\u003E study all possible spaces in all dimensions. Whether trying to design \u003Ca href=\u0022https:\/\/www2.math.upenn.edu\/%7Eghrist\/preprints\/noticesdraft.pdf\u0022\u003Esecure sensor networks\u003C\/a\u003E, \u003Ca href=\u0022https:\/\/doi.org\/10.3389\/frai.2021.667963\u0022\u003Emine data\u003C\/a\u003E or use \u003Ca href=\u0022https:\/\/globalnews.ca\/news\/10037710\/origami-in-space\/\u0022\u003Eorigami to deploy satellites\u003C\/a\u003E, the underlying language and ideas are likely to be that of topology.\u003C\/p\u003E\u003Ch2\u003EThe Shape of the Universe\u003C\/h2\u003E\u003Cp\u003EWhen you look around the universe you live in, it looks like a 3D space, just like the surface of the Earth looks like a 2D space. However, just like the Earth, if you were to look at the universe as a whole, it could be a more complicated space, like a giant 3D version of the 2D beach ball surface or something even more exotic than that.\u003C\/p\u003E\u003Cfigure class=\u0022align-left zoomable\u0022\u003E\u003Cp\u003E\u003Ca href=\u0022https:\/\/images.theconversation.com\/files\/614228\/original\/file-20240819-17-hxuf1t.png?ixlib=rb-4.1.0\u0026amp;q=45\u0026amp;auto=format\u0026amp;w=1000\u0026amp;fit=clip\u0022\u003E\u003Cimg alt=\u0022A shape with a hole in the middle.\u0022 src=\u0022https:\/\/images.theconversation.com\/files\/614228\/original\/file-20240819-17-hxuf1t.png?ixlib=rb-4.1.0\u0026amp;q=45\u0026amp;auto=format\u0026amp;w=237\u0026amp;fit=clip\u0022 srcset=\u0022https:\/\/images.theconversation.com\/files\/614228\/original\/file-20240819-17-hxuf1t.png?ixlib=rb-4.1.0\u0026amp;q=45\u0026amp;auto=format\u0026amp;w=600\u0026amp;h=503\u0026amp;fit=crop\u0026amp;dpr=1 600w, https:\/\/images.theconversation.com\/files\/614228\/original\/file-20240819-17-hxuf1t.png?ixlib=rb-4.1.0\u0026amp;q=30\u0026amp;auto=format\u0026amp;w=600\u0026amp;h=503\u0026amp;fit=crop\u0026amp;dpr=2 1200w, https:\/\/images.theconversation.com\/files\/614228\/original\/file-20240819-17-hxuf1t.png?ixlib=rb-4.1.0\u0026amp;q=15\u0026amp;auto=format\u0026amp;w=600\u0026amp;h=503\u0026amp;fit=crop\u0026amp;dpr=3 1800w, https:\/\/images.theconversation.com\/files\/614228\/original\/file-20240819-17-hxuf1t.png?ixlib=rb-4.1.0\u0026amp;q=45\u0026amp;auto=format\u0026amp;w=754\u0026amp;h=632\u0026amp;fit=crop\u0026amp;dpr=1 754w, https:\/\/images.theconversation.com\/files\/614228\/original\/file-20240819-17-hxuf1t.png?ixlib=rb-4.1.0\u0026amp;q=30\u0026amp;auto=format\u0026amp;w=754\u0026amp;h=632\u0026amp;fit=crop\u0026amp;dpr=2 1508w, https:\/\/images.theconversation.com\/files\/614228\/original\/file-20240819-17-hxuf1t.png?ixlib=rb-4.1.0\u0026amp;q=15\u0026amp;auto=format\u0026amp;w=754\u0026amp;h=632\u0026amp;fit=crop\u0026amp;dpr=3 2262w\u0022 sizes=\u0022(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px\u0022\u003E\u003C\/a\u003E\u003C\/p\u003E\u003Cfigcaption\u003E\u003Cspan class=\u0022caption\u0022\u003EA doughnut, also called a torus, is a shape that you can move across in two directions, just like the surface of the Earth.\u003C\/span\u003E \u003Ca class=\u0022source\u0022 href=\u0022https:\/\/commons.wikimedia.org\/wiki\/File:Simple_Torus.svg\u0022\u003E\u003Cspan class=\u0022attribution\u0022\u003EYassineMrabet via Wikimedia Commons\u003C\/span\u003E\u003C\/a\u003E\u003Cspan class=\u0022attribution\u0022\u003E, \u003C\/span\u003E\u003Ca class=\u0022license\u0022 href=\u0022http:\/\/creativecommons.org\/licenses\/by-nc-sa\/4.0\/\u0022\u003E\u003Cspan class=\u0022attribution\u0022\u003ECC BY-NC-SA\u003C\/span\u003E\u003C\/a\u003E\u003C\/figcaption\u003E\u003C\/figure\u003E\u003Cp\u003EWhile you don\u2019t need topology to determine that you are living on something like a giant beach ball, knowing all the possible 2D spaces can be useful. Over a century ago, mathematicians figured out \u003Ca href=\u0022https:\/\/doi.org\/10.1007\/978-3-642-34364-3\u0022\u003Eall the possible 2D spaces\u003C\/a\u003E and many of their properties.\u003C\/p\u003E\u003Cp\u003EIn the past several decades, mathematicians have learned a lot about all of the possible 3D spaces. While we do not have a complete understanding like we do for 2D spaces, we do \u003Ca href=\u0022https:\/\/bookstore.ams.org\/gsm-151\u0022\u003Eknow a lot\u003C\/a\u003E. With this knowledge, physicists and astronomers can try to determine what \u003Ca href=\u0022https:\/\/doi.org\/10.3390\/universe2010001\u0022\u003E3D space people actually live in\u003C\/a\u003E.\u003C\/p\u003E\u003Cp\u003EWhile the answer is not completely known, there are many \u003Ca href=\u0022https:\/\/www.quantamagazine.org\/what-shape-is-the-universe-closed-or-flat-20191104\/\u0022\u003Eintriguing and surprising possibilities\u003C\/a\u003E. The options become even more complicated if you consider time as a dimension.\u003C\/p\u003E\u003Cp\u003ETo see how this might work, note that to describe the location of something in space \u2013 say a comet \u2013 you need four numbers: three to describe its position and one to describe the time it is in that position. These four numbers are what make up a 4D space.\u003C\/p\u003E\u003Cp\u003ENow, you can consider what 4D spaces are possible and in which of those spaces do you live.\u003C\/p\u003E\u003Ch2\u003ETopology in Higher Dimensions\u003C\/h2\u003E\u003Cp\u003EAt this point, it may seem like there is no reason to consider spaces that have dimensions larger than four, since that is the highest imaginable dimension that might describe our universe. But a branch of physics called \u003Ca href=\u0022https:\/\/www.space.com\/17594-string-theory.html\u0022\u003Estring theory\u003C\/a\u003E suggests that the universe has many more dimensions than four.\u003C\/p\u003E\u003Cp\u003EThere are also practical applications of thinking about higher dimensional spaces, such as \u003Ca href=\u0022https:\/\/doi.org\/10.1007\/1-4020-4266-3_05\u0022\u003Erobot motion planning\u003C\/a\u003E. Suppose you are trying to understand the motion of three robots moving around a factory floor in a warehouse. You can put a grid on the floor and describe the position of each robot by their x and y coordinates on the grid. Since each of the three robots requires two coordinates, you will need six numbers to describe all of the possible positions of the robots. You can interpret the possible positions of the robots as a 6D space.\u003C\/p\u003E\u003Cp\u003EAs the number of robots increases, the dimension of the space increases. Factoring in other useful information, such as the locations of obstacles, makes the space even more complicated. In order to study this problem, you need to study high-dimensional spaces.\u003C\/p\u003E\u003Cp\u003EThere are countless other scientific problems where high-dimensional spaces appear, from modeling the \u003Ca href=\u0022https:\/\/doi.org\/10.1017\/CBO9781316410486\u0022\u003Emotion of planets\u003C\/a\u003E \u003Ca href=\u0022https:\/\/www.science.org\/content\/article\/physicists-discover-whopping-13-new-solutions-three-body-problem\u0022\u003Eand spacecraft\u003C\/a\u003E to trying to understand the \u003Ca href=\u0022https:\/\/www.ias.edu\/ideas\/2013\/lesnick-topological-data-analysis\u0022\u003E\u201cshape\u201d of large datasets\u003C\/a\u003E.\u003C\/p\u003E\u003Ch2\u003ETied Up In Knots\u003C\/h2\u003E\u003Cp\u003EAnother type of problem topologists study is how one space can sit inside another.\u003C\/p\u003E\u003Cp\u003EFor example, if you hold a knotted loop of string, then we have a 1D space (the loop of string) inside a 3D space (your room). Such loops are called mathematical knots.\u003C\/p\u003E\u003Cp\u003EThe \u003Ca href=\u0022https:\/\/www.britannica.com\/science\/knot-theory\u0022\u003Estudy of knots\u003C\/a\u003E first grew out of physics but has become a central area of topology. They are essential to how scientists understand \u003Ca href=\u0022https:\/\/bookstore.ams.org\/gsm-20\u0022\u003E3D and 4D spaces\u003C\/a\u003E and have a delightful and subtle structure that researchers are \u003Ca href=\u0022https:\/\/doi.org\/10.1016\/B978-0-444-51452-3.X5000-X\u0022\u003Estill trying to understand\u003C\/a\u003E.\u003C\/p\u003E\u003Cfigure class=\u0022align-center zoomable\u0022\u003E\u003Cp\u003E\u003Ca href=\u0022https:\/\/images.theconversation.com\/files\/614230\/original\/file-20240819-17-qmwj95.png?ixlib=rb-4.1.0\u0026amp;q=45\u0026amp;auto=format\u0026amp;w=1000\u0026amp;fit=clip\u0022\u003E\u003Cimg alt=\u0022Illustrations of 15 connected loops of string with different crossings\u0022 src=\u0022https:\/\/images.theconversation.com\/files\/614230\/original\/file-20240819-17-qmwj95.png?ixlib=rb-4.1.0\u0026amp;q=45\u0026amp;auto=format\u0026amp;w=754\u0026amp;fit=clip\u0022 srcset=\u0022https:\/\/images.theconversation.com\/files\/614230\/original\/file-20240819-17-qmwj95.png?ixlib=rb-4.1.0\u0026amp;q=45\u0026amp;auto=format\u0026amp;w=600\u0026amp;h=447\u0026amp;fit=crop\u0026amp;dpr=1 600w, https:\/\/images.theconversation.com\/files\/614230\/original\/file-20240819-17-qmwj95.png?ixlib=rb-4.1.0\u0026amp;q=30\u0026amp;auto=format\u0026amp;w=600\u0026amp;h=447\u0026amp;fit=crop\u0026amp;dpr=2 1200w, https:\/\/images.theconversation.com\/files\/614230\/original\/file-20240819-17-qmwj95.png?ixlib=rb-4.1.0\u0026amp;q=15\u0026amp;auto=format\u0026amp;w=600\u0026amp;h=447\u0026amp;fit=crop\u0026amp;dpr=3 1800w, https:\/\/images.theconversation.com\/files\/614230\/original\/file-20240819-17-qmwj95.png?ixlib=rb-4.1.0\u0026amp;q=45\u0026amp;auto=format\u0026amp;w=754\u0026amp;h=562\u0026amp;fit=crop\u0026amp;dpr=1 754w, https:\/\/images.theconversation.com\/files\/614230\/original\/file-20240819-17-qmwj95.png?ixlib=rb-4.1.0\u0026amp;q=30\u0026amp;auto=format\u0026amp;w=754\u0026amp;h=562\u0026amp;fit=crop\u0026amp;dpr=2 1508w, https:\/\/images.theconversation.com\/files\/614230\/original\/file-20240819-17-qmwj95.png?ixlib=rb-4.1.0\u0026amp;q=15\u0026amp;auto=format\u0026amp;w=754\u0026amp;h=562\u0026amp;fit=crop\u0026amp;dpr=3 2262w\u0022 sizes=\u0022(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px\u0022\u003E\u003C\/a\u003E\u003C\/p\u003E\u003Cfigcaption\u003E\u003Cspan class=\u0022caption\u0022\u003EKnots are examples of spaces that sit inside other spaces.\u003C\/span\u003E \u003Ca class=\u0022source\u0022 href=\u0022https:\/\/commons.wikimedia.org\/wiki\/File:Knot_table.svg\u0022\u003E\u003Cspan class=\u0022attribution\u0022\u003EJkasd\/Wikimedia Commons\u003C\/span\u003E\u003C\/a\u003E\u003C\/figcaption\u003E\u003C\/figure\u003E\u003Cp\u003EIn addition, knots have many applications, ranging from \u003Ca href=\u0022https:\/\/www.ias.edu\/ideas\/2011\/witten-knots-quantum-theory\u0022\u003Estring theory\u003C\/a\u003E in physics to \u003Ca href=\u0022https:\/\/doi.org\/10.1002\/bmb.20244\u0022\u003EDNA recombination\u003C\/a\u003E in biology to \u003Ca href=\u0022https:\/\/doi.org\/10.1017\/CBO9780511626272\u0022\u003Echirality\u003C\/a\u003E in chemistry.\u003C\/p\u003E\u003Ch2\u003EWhat Shape Do You Live On?\u003C\/h2\u003E\u003Cp\u003EGeometric topology is a beautiful and complex subject, and there are still countless exciting questions to answer about spaces.\u003C\/p\u003E\u003Cp\u003EFor example, the \u003Ca href=\u0022https:\/\/bookstore.ams.org\/gsm-20\u0022\u003Esmooth 4D Poincar\u00e9 conjecture\u003C\/a\u003E asks what the \u201csimplest\u201d closed 4D space is, and the \u003Ca href=\u0022https:\/\/www.quantamagazine.org\/mathematicians-prove-this-knot-cannot-solve-major-problem-20230202\/\u0022\u003Eslice-ribbon conjecture\u003C\/a\u003E aims to understand how knots in 3D spaces relate to surfaces in 4D spaces.\u003C\/p\u003E\u003Cp\u003ETopology is currently useful in science and engineering. Unraveling more mysteries of spaces in all dimensions will be invaluable to understanding the world in which we live and solving real-world problems.\u003C!-- Below is The Conversation\u0027s page counter tag. Please DO NOT REMOVE. --\u003E\u003Cimg style=\u0022border-color:!important;border-style:none;box-shadow:none !important;margin:0 !important;max-height:1px !important;max-width:1px !important;min-height:1px !important;min-width:1px !important;opacity:0 !important;outline:none !important;padding:0 !important;\u0022 src=\u0022https:\/\/counter.theconversation.com\/content\/235635\/count.gif?distributor=republish-lightbox-basic\u0022 alt=\u0022The Conversation\u0022 width=\u00221\u0022 height=\u00221\u0022 referrerpolicy=\u0022no-referrer-when-downgrade\u0022\u003E\u003C!-- End of code. If you don\u0027t see any code above, please get new code from the Advanced tab after you click the republish button. The page counter does not collect any personal data. More info: https:\/\/theconversation.com\/republishing-guidelines --\u003E\u003C\/p\u003E\u003Cp\u003E\u0026nbsp;\u003C\/p\u003E\u003Cp\u003E\u003Cem\u003EThis article is republished from \u003C\/em\u003E\u003Ca href=\u0022https:\/\/theconversation.com\u0022\u003E\u003Cem\u003EThe Conversation\u003C\/em\u003E\u003C\/a\u003E\u003Cem\u003E under a Creative Commons license. Read the \u003C\/em\u003E\u003Ca href=\u0022https:\/\/theconversation.com\/whats-the-shape-of-the-universe-mathematicians-use-topology-to-study-the-shape-of-the-world-and-everything-in-it-235635\u0022\u003E\u003Cem\u003Eoriginal article\u003C\/em\u003E\u003C\/a\u003E\u003Cem\u003E.\u003C\/em\u003E\u003C\/p\u003E\u003C\/div\u003E","summary":"","format":"full_html"}],"field_subtitle":"","field_summary":[{"value":"\u003Cp\u003EWhether trying to design secure sensor networks, mine data or use origami to deploy satellites, the underlying language and ideas are likely to be that of topology.\u003C\/p\u003E","format":"limited_html"}],"field_summary_sentence":[{"value":"Whether trying to design secure sensor networks, mine data or use origami to deploy satellites, the underlying language and ideas are likely to be that of topology."}],"uid":"27469","created_gmt":"2025-02-28 14:22:35","changed_gmt":"2026-03-19 13:16:17","author":"Kristen Bailey","boilerplate_text":"","field_publication":"","field_article_url":"","location":"Atlanta, GA","dateline":{"date":"2025-02-28T00:00:00-05:00","iso_date":"2025-02-28T00:00:00-05:00","tz":"America\/New_York"},"extras":[],"hg_media":{"676431":{"id":"676431","type":"image","title":"You can describe the shape you live on in multiple dimensions. vkulieva\/iStock via Getty Images Plus","body":"\u003Cp\u003EYou can describe the shape you live on in multiple dimensions. \u003Ca href=\u0022https:\/\/www.gettyimages.com\/detail\/illustration\/green-neon-wireframe-shapes-collection-3d-royalty-free-illustration\/1509927575?phrase=math+torus\u0026amp;adppopup=true\u0022\u003Evkulieva\/iStock via Getty Images Plus\u003C\/a\u003E\u003C\/p\u003E","created":"1740770532","gmt_created":"2025-02-28 19:22:12","changed":"1740770532","gmt_changed":"2025-02-28 19:22:12","alt":"You can describe the shape you live on in multiple dimensions. vkulieva\/iStock via Getty Images Plus","file":{"fid":"260217","name":"file-20240816-23-nnp9id-copy.jpg","image_path":"\/sites\/default\/files\/2025\/02\/28\/file-20240816-23-nnp9id-copy.jpg","image_full_path":"http:\/\/hg.gatech.edu\/\/sites\/default\/files\/2025\/02\/28\/file-20240816-23-nnp9id-copy.jpg","mime":"image\/jpeg","size":512466,"path_740":"http:\/\/hg.gatech.edu\/sites\/default\/files\/styles\/740xx_scale\/public\/2025\/02\/28\/file-20240816-23-nnp9id-copy.jpg?itok=bQNfZoeS"}}},"media_ids":["676431"],"related_links":[{"url":"https:\/\/theconversation.com\/whats-the-shape-of-the-universe-mathematicians-use-topology-to-study-the-shape-of-the-world-and-everything-in-it-235635","title":"Read This Article on The Conversation"}],"groups":[{"id":"1278","name":"College of Sciences"},{"id":"658168","name":"Experts"},{"id":"1214","name":"News Room"},{"id":"1188","name":"Research Horizons"},{"id":"1279","name":"School of Mathematics"}],"categories":[],"keywords":[{"id":"173647","name":"_for_math_site_"},{"id":"187915","name":"go-researchnews"}],"core_research_areas":[],"news_room_topics":[{"id":"71911","name":"Earth and Environment"}],"event_categories":[],"invited_audience":[],"affiliations":[],"classification":[],"areas_of_expertise":[],"news_and_recent_appearances":[],"phone":[],"contact":[{"value":"\u003Ch5\u003EAuthor:\u003C\/h5\u003E\u003Cp\u003E\u003Ca href=\u0022https:\/\/theconversation.com\/profiles\/john-etnyre-1553642\u0022\u003EJohn Etnyre\u003C\/a\u003E, Professor of Mathematics, Georgia Institute of Technology\u003C\/p\u003E\u003Ch5\u003EMedia Contact:\u003C\/h5\u003E\u003Cp\u003EShelley Wunder-Smith\u003Cbr\u003E\u003Ca href=\u0022mailto:shelley.wunder-smith@research.gatech.edu\u0022\u003Eshelley.wunder-smith@research.gatech.edu\u003C\/a\u003E\u003C\/p\u003E","format":"limited_html"}],"email":[],"slides":[],"orientation":[],"userdata":""}}}