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This study, using microCT and custom visual data analyses of growing stingray skeletons, systematically examines tessellation patterns and morphologies of the many thousand interacting tesserae covering the hyomandibula (a skeletal element critical to feeding), over a two\u2010fold developmental change in hyomandibula length. The number of tesserae remains surprisingly constant, even as the hyomandibula expands isometrically, with all hyomandibulae displaying self\u2010similar distributions of tesserae shapes\/sizes. Although the distribution of tesserae geometries largely agrees with the rules for polyhedra tiling of complex surfaces\u2014dominated by hexagons and a minor fraction of pentagons and heptagons, but very few other polygons\u2014the agreement with Euler's classic mathematical laws is not perfect. Contrary to the assumed uniform growth rate (which is shown would create geometric incompatibilities), larger tesserae grow faster to accommodate skeletal expansion. It is hypothesized that this local regulation of global system complexity is driven by tension (from cartilaginous core expansion) in the fibers connecting tesserae, with strain\u2010responsive cells orchestrating local mineral apposition.<\/jats:p>","DOI":"10.1002\/advs.202407641","type":"journal-article","created":{"date-parts":[[2024,11,7]],"date-time":"2024-11-07T14:40:03Z","timestamp":1730990403000},"update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Growth of a Tessellation: Geometric rules for the Development of Stingray Skeletal Patterns"],"prefix":"10.1002","volume":"11","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7884-5017","authenticated-orcid":false,"given":"Binru","family":"Yang","sequence":"first","affiliation":[{"name":"Max Planck Institute of Colloids and Interfaces Potsdam Science Park  Am M\u00fchlenberg 1 OT Golm 14476 Potsdam Germany"},{"name":"Cluster of Excellence Matters of Activity Image Space Material Humboldt\u2010Universit\u00e4t zu Berlin  Unter den Linden 6 10099 Berlin Germany"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1955-309X","authenticated-orcid":false,"given":"David","family":"Kn\u00f6tel","sequence":"additional","affiliation":[{"name":"Zuse Institute Berlin (ZIB)  Takustr. 7 14195 Berlin Germany"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jana","family":"Ciecierska\u2010Holmes","sequence":"additional","affiliation":[{"name":"Max Planck Institute of Colloids and Interfaces Potsdam Science Park  Am M\u00fchlenberg 1 OT Golm 14476 Potsdam Germany"},{"name":"Cluster of Excellence Matters of Activity Image Space Material Humboldt\u2010Universit\u00e4t zu Berlin  Unter den Linden 6 10099 Berlin Germany"},{"name":"Institut f\u00fcr Biologie Humboldt\u2010Universit\u00e4t zu Berlin  Philippstra\u00dfe 13, Haus 2 10115 Berlin Germany"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8630-2461","authenticated-orcid":false,"given":"Jan","family":"W\u00f6lfer","sequence":"additional","affiliation":[{"name":"Institut f\u00fcr Biologie Humboldt\u2010Universit\u00e4t zu Berlin  Philippstra\u00dfe 13, Haus 2 10115 Berlin Germany"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3495-3996","authenticated-orcid":false,"given":"J\u00falia","family":"Chaumel","sequence":"additional","affiliation":[{"name":"Max Planck Institute of Colloids and Interfaces Potsdam Science Park  Am M\u00fchlenberg 1 OT Golm 14476 Potsdam Germany"},{"name":"Department of Organismic and Evolutionary Biology Harvard University  26 Oxford Street Cambridge MA 02138 USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8714-4992","authenticated-orcid":false,"given":"Paul","family":"Zaslansky","sequence":"additional","affiliation":[{"name":"Department for Operative Preventive and Pediatric Dentistry Charit\u00e9\u2010Universit\u00e4tsmedizin Berlin  A\u00dfmannshauser Str. 4\u20136 14197 Berlin Germany"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1550-7245","authenticated-orcid":false,"given":"Daniel","family":"Baum","sequence":"additional","affiliation":[{"name":"Zuse Institute Berlin (ZIB)  Takustr. 7 14195 Berlin Germany"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4437-7830","authenticated-orcid":false,"given":"Peter","family":"Fratzl","sequence":"additional","affiliation":[{"name":"Max Planck Institute of Colloids and Interfaces Potsdam Science Park  Am M\u00fchlenberg 1 OT Golm 14476 Potsdam Germany"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5026-6216","authenticated-orcid":false,"given":"Mason N.","family":"Dean","sequence":"additional","affiliation":[{"name":"Max Planck Institute of Colloids and Interfaces Potsdam Science Park  Am M\u00fchlenberg 1 OT Golm 14476 Potsdam Germany"},{"name":"Department of Infectious Diseases &amp; Public Health City University of Hong Kong  Hong Kong China"},{"name":"Centre for Nature Inspired Engineering City University of Hong Kong  Hong Kong China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"311","published-online":{"date-parts":[[2024,11,7]]},"reference":[{"key":"e_1_2_12_1_1","doi-asserted-by":"publisher","DOI":"10.1002\/adma.202304166"},{"key":"e_1_2_12_1_2","doi-asserted-by":"publisher","DOI":"10.1038\/srep03024"},{"key":"e_1_2_12_1_3","doi-asserted-by":"publisher","DOI":"10.1016\/j.ijsolstr.2022.111498"},{"key":"e_1_2_12_2_1","first-page":"37","volume-title":"Sharks, Skates and Rays","author":"Applegate S. 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