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Trans. R. Soc. A."],"published-print":{"date-parts":[[2024,9,9]]},"abstract":"\n Many deployable structures in nature, as well as human-made mechanisms, preserve symmetry as their configurations evolve. Examples in nature include blooming flowers, dilation of the iris within the human eye, viral capsid maturation and molecular and bacterial motors. Engineered examples include opening umbrellas, elongating scissor jacks, variable apertures in cameras, expanding Hoberman spheres and some kinds of morphing origami structures. In these cases, the structures either preserve a discrete symmetry group or are described as an evolution from one discrete symmetry group to another of the same type as the structure deploys. Likewise, elastic metamaterials built from lattice structures can also preserve symmetry type while passively deforming and changing lattice parameters. A mathematical formulation of such transitions\/deployments is articulated here. It is shown that if\n \n \n \n X<\/mml:mi>\n <\/mml:semantics>\n <\/mml:math>\n <\/jats:inline-formula>\n is Euclidean space,\n \n \n \n G<\/mml:mi>\n <\/mml:semantics>\n <\/mml:math>\n <\/jats:inline-formula>\n is a continuous group of motions of Euclidean space and\n \n \n \n \u0393<\/mml:mi>\n <\/mml:semantics>\n <\/mml:math>\n <\/jats:inline-formula>\n is the type of the discrete subgroup of\n \n \n \n G<\/mml:mi>\n <\/mml:semantics>\n <\/mml:math>\n <\/jats:inline-formula>\n describing the symmetries of the deploying structure, then the symmetry of the evolving structure can be described by time-dependent subgroups of\n \n \n \n G<\/mml:mi>\n <\/mml:semantics>\n <\/mml:math>\n <\/jats:inline-formula>\n of the form\n \n \n \n \n \n \u0393<\/mml:mi>\n \n \n \u03b1<\/mml:mi>\n t<\/mml:mi>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:msub>\n :<\/mml:mo>\n =<\/mml:mo>\n \n \u03b1<\/mml:mi>\n t<\/mml:mi>\n <\/mml:msub>\n \u0393<\/mml:mi>\n \n \u03b1<\/mml:mi>\n t<\/mml:mi>\n \n \u2212<\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msubsup>\n <\/mml:mrow>\n <\/mml:semantics>\n <\/mml:math>\n <\/jats:inline-formula>\n , where\n \n \n \n \n \n \u03b1<\/mml:mi>\n t<\/mml:mi>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:semantics>\n <\/mml:math>\n <\/jats:inline-formula>\n is a time-dependent affine transformation. Then, instead of considering the whole structure in\n \n \n \n X<\/mml:mi>\n <\/mml:semantics>\n <\/mml:math>\n <\/jats:inline-formula>\n , a \u2018sector\u2019 of it that lives in the orbit space\n \n \n \n \n \n \u0393<\/mml:mi>\n \n \n \u03b1<\/mml:mi>\n t<\/mml:mi>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:msub>\n \\<\/mml:mo>\n X<\/mml:mi>\n <\/mml:mrow>\n <\/mml:semantics>\n <\/mml:math>\n <\/jats:inline-formula>\n can be considered at each instant in time, and instead of considering all motions in\n \n \n \n G<\/mml:mi>\n <\/mml:semantics>\n <\/mml:math>\n <\/jats:inline-formula>\n , only representatives from right cosets in the space\n \n \n \n \n \n \u0393<\/mml:mi>\n \n \n \u03b1<\/mml:mi>\n t<\/mml:mi>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:msub>\n \\<\/mml:mo>\n G<\/mml:mi>\n <\/mml:mrow>\n <\/mml:semantics>\n <\/mml:math>\n <\/jats:inline-formula>\n need to be considered.\n <\/jats:p>\n This article is part of the theme issue 'Current developments in elastic and acoustic metamaterials science (Part 1)'.<\/jats:p>","DOI":"10.1098\/rsta.2023.0352","type":"journal-article","created":{"date-parts":[[2024,7,29]],"date-time":"2024-07-29T03:05:10Z","timestamp":1722222310000},"update-policy":"https:\/\/doi.org\/10.1098\/crossmark-policy","source":"Crossref","is-referenced-by-count":2,"title":["Group-theoretic analysis of symmetry-preserving deployable structures and metamaterials"],"prefix":"10.1098","volume":"382","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0542-9028","authenticated-orcid":true,"given":"Gregory S.","family":"Chirikjian","sequence":"first","affiliation":[{"name":"Department of Mechanical Engineering, National University of Singapore , Singapore, Singapore"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"175","published-online":{"date-parts":[[2024,7,29]]},"reference":[{"key":"e_1_3_8_2_2","doi-asserted-by":"publisher","DOI":"10.1080\/01691864.2015.1052848"},{"key":"e_1_3_8_3_2","doi-asserted-by":"publisher","DOI":"10.1115\/1.1901706"},{"key":"e_1_3_8_4_2","doi-asserted-by":"publisher","DOI":"10.1038\/nature08145"},{"key":"e_1_3_8_5_2","doi-asserted-by":"publisher","DOI":"10.1098\/rstb.2000.0593"},{"key":"e_1_3_8_6_2","doi-asserted-by":"publisher","DOI":"10.1016\/S1047-8477(03)00126-6"},{"key":"e_1_3_8_7_2","doi-asserted-by":"publisher","DOI":"10.1038\/24409"},{"key":"e_1_3_8_8_2","doi-asserted-by":"publisher","DOI":"10.1107\/S0021889806019297"},{"key":"e_1_3_8_9_2","doi-asserted-by":"publisher","DOI":"10.1017\/S0263574715001046"},{"key":"e_1_3_8_10_2","doi-asserted-by":"publisher","DOI":"10.1115\/1.1480413"},{"key":"e_1_3_8_11_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.mechmachtheory.2015.07.009"},{"key":"e_1_3_8_12_2","doi-asserted-by":"publisher","DOI":"10.1098\/rspa.2009.0676"},{"key":"e_1_3_8_13_2","doi-asserted-by":"publisher","DOI":"10.1098\/rspa.2015.0846"},{"key":"e_1_3_8_14_2","doi-asserted-by":"publisher","DOI":"10.1177\/0266351117711290"},{"key":"e_1_3_8_15_2","doi-asserted-by":"publisher","DOI":"10.1016\/0045-7949(89)90354-4"},{"key":"e_1_3_8_16_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.mechmachtheory.2023.105293"},{"key":"e_1_3_8_17_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.mechmachtheory.2007.04.011"},{"key":"e_1_3_8_18_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.mechmachtheory.2016.07.019"},{"key":"e_1_3_8_19_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.mechmachtheory.2019.103663"},{"key":"e_1_3_8_20_2","doi-asserted-by":"publisher","DOI":"10.1038\/s41586-021-03407-4"},{"key":"e_1_3_8_21_2","first-page":"87","article-title":"The science of Miura-ori: a review","volume":"4","author":"Miura K","year":"2009","unstructured":"Miura K , Lang RJ . 2009 The science of Miura-ori: a review. 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