https://www.nature.com/articles/d41586-023-01801-8 Skip to main content Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript. Advertisement Advertisement Nature * View all journals * Search * Log in * Explore content * About the journal * Publish with us * Subscribe * Sign up for alerts * RSS feed 1. nature 2. news 3. article * NEWS * 01 June 2023 This infinite tiling pattern could end a 60-year mathematical quest First truly single aperiodic tile discovered that can cover an infinite surface without repeating itself. * Davide Castelvecchi 1. Davide Castelvecchi View author publications You can also search for this author in PubMed Google Scholar * Twitter * Facebook * Email You have full access to this article via your institution. Download PDF a pattern of polygon tiles with black outlines and some coloured green transitioning from straight edges to curved A shape (highlighted in green) that can endlessly tile an area without making a repeating pattern has been discovered.Credit: D. Smith et al./arXiv After 60 years of searching, mathematicians might have finally found a true single 'aperiodic' tile -- a shape that can cover an infinite plane, but never make a repeating pattern. Periodic tilings have translational symmetry: a honeycomb pattern, for example, can be repeated forever and looks identical after being shifted in any of six directions by any number of cells. But in aperiodic tilings any such shift is impossible. A shape breakthrough In March, a team announced an important breakthrough in the search for an aperiodic tile. David Smith, a hobbyist mathematician based in Bridlington, UK, discovered a shape that he suspected could be an aperiodic tile and, together with three professional mathematicians, Smith wrote up a proof that his tile -- together with its mirror, or flipped, image -- could be used to build infinite aperiodic tilings of the plane^1. (The proof has not yet been peer reviewed, although mathematicians have reportedly said that it seems to be rigorous.) a transformed polygon with wavy edges and blue, red and black dots on the outline The pattern is made up of single tiles such as this wavy one.Credit: D. Smith et al./arXiv Smith's shape was not a single aperiodic tile, because it and its mirror image are effectively two separate tiles -- and both versions were required for tiling the entire plane. But now the same group of mathematicians has reported a modified version of their original tile that can build aperiodic tilings without being flipped^2.This proof was posted on the preprint server arXiv and has not yet been peer reviewed. The first aperiodic tilings were discovered in the 1960s, and they involved 20,426 tile types. After various improvements, Roger Penrose, a mathematician at the University of Oxford, UK -- who won a Nobel Prize in Physics in 2020 for his foundational work on the theory of black holes -- discovered the first aperiodic tiling made of only two tile types that were not merely mirror images of each other. Penrose's tilings now adorn the patio of Oxford's mathematics department. doi: https://doi.org/10.1038/d41586-023-01801-8 References 1. Smith, D., Myers, J. S., Kaplan, C. S. & Goodman-Strauss, C. Preprint at https://arxiv.org/abs/2303.10798 (2023). 2. Smith, D., Myers, J. S., Kaplan, C. S. & Goodman-Strauss, C. Preprint at https://arxiv.org/abs/2305.17743 (2023). Download references Subjects * Mathematics and computing Latest on: Mathematics and computing Towards quantum machine learning Towards quantum machine learning Spotlight 24 MAY 23 Rewriting the quantum-computer blueprint Rewriting the quantum-computer blueprint Outlook 24 MAY 23 Commercializing quantum computers step by step Commercializing quantum computers step by step Outlook 24 MAY 23 Nature Careers Jobs * Faculty Positions at SUSTech Department of Biomedical Engineering The Department of Biomedical Engineering (BME) , Southern University of Science and Technology (SUSTech), seeks outstanding applicants for full-tim... Shenzhen Department of Biomedical Engineering, SUSTech [] * Postdoctoral Fellows/Research scientists We are seeking for highly motivated postdoctoral fellows to join in Dr. Thanh Hoang?s lab in the University of Michigan. Our research encompasses a... United States, Ann Arbor, University of Michigan University of Michigan * Postdoctoral Research Fellow POSTDOCTORAL POSITION -- DEPARTMENT OF UROLOGY - BOSTON CHILDREN'S HOSPITAL AND HARVARD MEDICAL SCHOOL A postdoctoral position is open in the Urolo... Boston, Massachusetts (US) Boston Children's Hospital (BCH) [] * Specialist in Genetic Trials and Precision Medicine The position We are looking for a highly qualified and innovative Specialist in Genetics Trials and Precision Medicine, to implement cutting-edge h... Oxford, Oxfordshire Novo Nordisk [] * Post doctor (2 years) within carbon burial in Arctic lakes The Department of Ecology and Environmental Science is offering a postdoctoral scholarship within a project on carbon burial in Arctic lakes. Umea, Sweden Umea University [] You have full access to this article via your institution. Download PDF Subjects * Mathematics and computing Advertisement Sign up to Nature Briefing An essential round-up of science news, opinion and analysis, delivered to your inbox every weekday. Email address [ ] [ ] Yes! Sign me up to receive the daily Nature Briefing email. I agree my information will be processed in accordance with the Nature and Springer Nature Limited Privacy Policy. Sign up * Close Nature Briefing Sign up for the Nature Briefing newsletter -- what matters in science, free to your inbox daily. Email address [ ] Sign up [ ] I agree my information will be processed in accordance with the Nature and Springer Nature Limited Privacy Policy. Close Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing Explore content * Research articles * News * Opinion * Research Analysis * Careers * Books & Culture * Podcasts * Videos * Current issue * Browse issues * Collections * Subjects * Follow us on Facebook * Follow us on Twitter * Subscribe * Sign up for alerts * RSS feed About the journal * Journal Staff * About the Editors * Journal Information * Our publishing models * Editorial Values Statement * Journal Metrics * Awards * Contact * Editorial policies * History of Nature * Send a news tip Publish with us * For Authors * For Referees * Language editing services * Submit manuscript Search Search articles by subject, keyword or author [ ] Show results from [All journals] Search Advanced search Quick links * Explore articles by subject * Find a job * Guide to authors * Editorial policies Nature (Nature) ISSN 1476-4687 (online) ISSN 0028-0836 (print) nature.com sitemap About Nature Portfolio * About us * Press releases * Press office * Contact us Discover content * Journals A-Z * Articles by subject * Nano * Protocol Exchange * Nature Index Publishing policies * Nature portfolio policies * Open access Author & Researcher services * Reprints & permissions * Research data * Language editing * Scientific editing * Nature Masterclasses * Live Expert Trainer-led workshops * Research Solutions Libraries & institutions * Librarian service & tools * Librarian portal * Open research * Recommend to library Advertising & partnerships * Advertising * Partnerships & Services * Media kits * Branded content Career development * Nature Careers * Nature Conferences * Nature events Regional websites * Nature Africa * Nature China * Nature India * Nature Italy * Nature Japan * Nature Korea * Nature Middle East * Privacy Policy * Use of cookies * Your privacy choices/Manage cookies * Legal notice * Accessibility statement * Terms & Conditions * Your US state privacy rights Springer Nature (c) 2023 Springer Nature Limited