{"id":"https://openalex.org/W2123350836","doi":"https://doi.org/10.3390/sym4040581","title":"Hexagonal Inflation Tilings and Planar Monotiles","display_name":"Hexagonal Inflation Tilings and Planar Monotiles","publication_year":2012,"publication_date":"2012-10-22","ids":{"openalex":"https://openalex.org/W2123350836","doi":"https://doi.org/10.3390/sym4040581","mag":"2123350836"},"language":"en","primary_location":{"id":"doi:10.3390/sym4040581","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym4040581","pdf_url":"https://www.mdpi.com/2073-8994/4/4/581/pdf?version=1354285207","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Symmetry","raw_type":"journal-article"},"type":"article","indexed_in":["arxiv","crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://www.mdpi.com/2073-8994/4/4/581/pdf?version=1354285207","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5018078583","display_name":"Michael Baake","orcid":"https://orcid.org/0000-0001-8513-8807"},"institutions":[{"id":"https://openalex.org/I20121455","display_name":"Bielefeld University","ror":"https://ror.org/02hpadn98","country_code":"DE","type":"education","lineage":["https://openalex.org/I20121455"]}],"countries":["DE"],"is_corresponding":false,"raw_author_name":"Michael Baake","raw_affiliation_strings":["Fakult\u00e4t f\u00fcr Mathematik, Universit\u00e4t Bielefeld, Postfach 100131, Bielefeld 33501, Germany"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Fakult\u00e4t f\u00fcr Mathematik, Universit\u00e4t Bielefeld, Postfach 100131, Bielefeld 33501, Germany","institution_ids":["https://openalex.org/I20121455"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5054961181","display_name":"Franz G\u00e4hler","orcid":"https://orcid.org/0000-0002-1012-7692"},"institutions":[{"id":"https://openalex.org/I20121455","display_name":"Bielefeld University","ror":"https://ror.org/02hpadn98","country_code":"DE","type":"education","lineage":["https://openalex.org/I20121455"]}],"countries":["DE"],"is_corresponding":false,"raw_author_name":"Franz G\u00e4hler","raw_affiliation_strings":["Fakult\u00e4t f\u00fcr Mathematik, Universit\u00e4t Bielefeld, Postfach 100131, Bielefeld 33501, Germany"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Fakult\u00e4t f\u00fcr Mathematik, Universit\u00e4t Bielefeld, Postfach 100131, Bielefeld 33501, Germany","institution_ids":["https://openalex.org/I20121455"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5040334898","display_name":"Uwe Grimm","orcid":"https://orcid.org/0000-0001-5492-7467"},"institutions":[{"id":"https://openalex.org/I204136569","display_name":"The Open University","ror":"https://ror.org/05mzfcs16","country_code":"GB","type":"education","lineage":["https://openalex.org/I204136569"]}],"countries":["GB"],"is_corresponding":true,"raw_author_name":"Uwe Grimm","raw_affiliation_strings":["Department of Mathematics and Statistics, The Open University, Walton Hall, Milton Keynes MK7 6AA, UK"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics and Statistics, The Open University, Walton Hall, Milton Keynes MK7 6AA, UK","institution_ids":["https://openalex.org/I204136569"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5040334898"],"corresponding_institution_ids":["https://openalex.org/I204136569"],"apc_list":{"value":2000,"currency":"CHF","value_usd":2165},"apc_paid":{"value":2000,"currency":"CHF","value_usd":2165},"fwci":0.604,"has_fulltext":false,"cited_by_count":16,"citation_normalized_percentile":{"value":0.67123419,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":97},"biblio":{"volume":"4","issue":"4","first_page":"581","last_page":"602"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11887","display_name":"Quasicrystal Structures and Properties","score":0.9995999932289124,"subfield":{"id":"https://openalex.org/subfields/2505","display_name":"Materials Chemistry"},"field":{"id":"https://openalex.org/fields/25","display_name":"Materials Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T11887","display_name":"Quasicrystal Structures and Properties","score":0.9995999932289124,"subfield":{"id":"https://openalex.org/subfields/2505","display_name":"Materials Chemistry"},"field":{"id":"https://openalex.org/fields/25","display_name":"Materials Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T12162","display_name":"Cellular Automata and Applications","score":0.9991000294685364,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11737","display_name":"Advanced Materials and Mechanics","score":0.9944999814033508,"subfield":{"id":"https://openalex.org/subfields/2210","display_name":"Mechanical Engineering"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/aperiodic-graph","display_name":"Aperiodic graph","score":0.9717564582824707},{"id":"https://openalex.org/keywords/substitution-tiling","display_name":"Substitution tiling","score":0.9079207181930542},{"id":"https://openalex.org/keywords/hexagonal-tiling","display_name":"Hexagonal tiling","score":0.6375077962875366},{"id":"https://openalex.org/keywords/planar","display_name":"Planar","score":0.5909090042114258},{"id":"https://openalex.org/keywords/penrose-tiling","display_name":"Penrose tiling","score":0.5777373313903809},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.49797844886779785},{"id":"https://openalex.org/keywords/hexagonal-crystal-system","display_name":"Hexagonal crystal system","score":0.48847854137420654},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.4650641083717346},{"id":"https://openalex.org/keywords/set","display_name":"Set (abstract data type)","score":0.45807430148124695},{"id":"https://openalex.org/keywords/topology","display_name":"Topology (electrical circuits)","score":0.34761908650398254},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.34488505125045776},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.30286911129951477},{"id":"https://openalex.org/keywords/quasicrystal","display_name":"Quasicrystal","score":0.2820148468017578},{"id":"https://openalex.org/keywords/crystallography","display_name":"Crystallography","score":0.13201913237571716},{"id":"https://openalex.org/keywords/computer-graphics","display_name":"Computer graphics (images)","score":0.07853072881698608}],"concepts":[{"id":"https://openalex.org/C104247578","wikidata":"https://www.wikidata.org/wiki/Q4779368","display_name":"Aperiodic graph","level":2,"score":0.9717564582824707},{"id":"https://openalex.org/C21550011","wikidata":"https://www.wikidata.org/wiki/Q7632143","display_name":"Substitution tiling","level":2,"score":0.9079207181930542},{"id":"https://openalex.org/C35908053","wikidata":"https://www.wikidata.org/wiki/Q3063631","display_name":"Hexagonal tiling","level":3,"score":0.6375077962875366},{"id":"https://openalex.org/C134786449","wikidata":"https://www.wikidata.org/wiki/Q3391255","display_name":"Planar","level":2,"score":0.5909090042114258},{"id":"https://openalex.org/C76513542","wikidata":"https://www.wikidata.org/wiki/Q585688","display_name":"Penrose tiling","level":3,"score":0.5777373313903809},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.49797844886779785},{"id":"https://openalex.org/C128765274","wikidata":"https://www.wikidata.org/wiki/Q663314","display_name":"Hexagonal crystal system","level":2,"score":0.48847854137420654},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.4650641083717346},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.45807430148124695},{"id":"https://openalex.org/C184720557","wikidata":"https://www.wikidata.org/wiki/Q7825049","display_name":"Topology (electrical circuits)","level":2,"score":0.34761908650398254},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.34488505125045776},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.30286911129951477},{"id":"https://openalex.org/C203101518","wikidata":"https://www.wikidata.org/wiki/Q263214","display_name":"Quasicrystal","level":2,"score":0.2820148468017578},{"id":"https://openalex.org/C8010536","wikidata":"https://www.wikidata.org/wiki/Q160398","display_name":"Crystallography","level":1,"score":0.13201913237571716},{"id":"https://openalex.org/C121684516","wikidata":"https://www.wikidata.org/wiki/Q7600677","display_name":"Computer graphics (images)","level":1,"score":0.07853072881698608},{"id":"https://openalex.org/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","level":1,"score":0.0},{"id":"https://openalex.org/C187691185","wikidata":"https://www.wikidata.org/wiki/Q2020720","display_name":"Grid","level":2,"score":0.0},{"id":"https://openalex.org/C185592680","wikidata":"https://www.wikidata.org/wiki/Q2329","display_name":"Chemistry","level":0,"score":0.0}],"mesh":[],"locations_count":10,"locations":[{"id":"doi:10.3390/sym4040581","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym4040581","pdf_url":"https://www.mdpi.com/2073-8994/4/4/581/pdf?version=1354285207","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Symmetry","raw_type":"journal-article"},{"id":"pmh:oai:oro.open.ac.uk:34803","is_oa":false,"landing_page_url":null,"pdf_url":null,"source":{"id":"https://openalex.org/S4306401187","display_name":"Open Research Online (The Open University)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I204136569","host_organization_name":"The Open University","host_organization_lineage":["https://openalex.org/I204136569"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"acceptedVersion","is_accepted":true,"is_published":false,"raw_source_name":"","raw_type":"Journal Item"},{"id":"pmh:oai:arXiv.org:1210.3967","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1210.3967","pdf_url":"https://arxiv.org/pdf/1210.3967","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.384.8185","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.384.8185","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://oro.open.ac.uk/34803/1/monotil.pdf","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.387.3238","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.387.3238","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.mdpi.com/2073-8994/4/4/581/pdf-vor/","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.387.7971","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.387.7971","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.mdpi.com/2073-8994/4/4/581/pdf/","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.763.5023","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.763.5023","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://arxiv.org/pdf/1210.3967.pdf","raw_type":"text"},{"id":"pmh:oai:doaj.org/article:ce50687baddf42548fe0f20ce4670744","is_oa":true,"landing_page_url":"https://doaj.org/article/ce50687baddf42548fe0f20ce4670744","pdf_url":null,"source":{"id":"https://openalex.org/S4306401280","display_name":"DOAJ (DOAJ: Directory of Open Access Journals)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by-sa","license_id":"https://openalex.org/licenses/cc-by-sa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Symmetry, Vol 4, Iss 4, Pp 581-602 (2012)","raw_type":"article"},{"id":"pmh:oai:mdpi.com:/2073-8994/4/4/581/","is_oa":true,"landing_page_url":"https://dx.doi.org/10.3390/sym4040581","pdf_url":null,"source":{"id":"https://openalex.org/S4306400947","display_name":"MDPI (MDPI AG)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I4210097602","host_organization_name":"Multidisciplinary Digital Publishing Institute (Switzerland)","host_organization_lineage":["https://openalex.org/I4210097602"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Symmetry; Volume 4; Issue 4; Pages: 581-602","raw_type":"Text"},{"id":"pmh:oai:pub.librecat.org:2541297","is_oa":true,"landing_page_url":"https://pub.uni-bielefeld.de/record/2541297","pdf_url":null,"source":{"id":"https://openalex.org/S4306401671","display_name":"PUB \u2013 Publications at Bielefeld University (Bielefeld University)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I20121455","host_organization_name":"Bielefeld University","host_organization_lineage":["https://openalex.org/I20121455"],"host_organization_lineage_names":[],"type":"repository"},"license":"other-oa","license_id":"https://openalex.org/licenses/other-oa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Baake M, G\u00e4hler F, Grimm U. Hexagonal Inflation Tilings and Planar Monotiles. <em>Symmetry</em>. 2012;4(4):581-602.","raw_type":"info:eu-repo/semantics/article"}],"best_oa_location":{"id":"doi:10.3390/sym4040581","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym4040581","pdf_url":"https://www.mdpi.com/2073-8994/4/4/581/pdf?version=1354285207","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Symmetry","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2123350836.pdf","grobid_xml":"https://content.openalex.org/works/W2123350836.grobid-xml"},"referenced_works_count":45,"referenced_works":["https://openalex.org/W56268668","https://openalex.org/W627762779","https://openalex.org/W630641102","https://openalex.org/W836513070","https://openalex.org/W1497212726","https://openalex.org/W1506222223","https://openalex.org/W1509103970","https://openalex.org/W1610769795","https://openalex.org/W1632804639","https://openalex.org/W1940546594","https://openalex.org/W1964179968","https://openalex.org/W1970792778","https://openalex.org/W1977183350","https://openalex.org/W1986891133","https://openalex.org/W1991511994","https://openalex.org/W2000366473","https://openalex.org/W2019658507","https://openalex.org/W2028433580","https://openalex.org/W2039654286","https://openalex.org/W2043318682","https://openalex.org/W2045297465","https://openalex.org/W2053097304","https://openalex.org/W2063558317","https://openalex.org/W2076355659","https://openalex.org/W2098493855","https://openalex.org/W2132894592","https://openalex.org/W2170867015","https://openalex.org/W2487593418","https://openalex.org/W2501263083","https://openalex.org/W2503294867","https://openalex.org/W2756884159","https://openalex.org/W2949477170","https://openalex.org/W3100307238","https://openalex.org/W3100589988","https://openalex.org/W3101430836","https://openalex.org/W3103597113","https://openalex.org/W3103915183","https://openalex.org/W3105136967","https://openalex.org/W3106525194","https://openalex.org/W4205965692","https://openalex.org/W4211163171","https://openalex.org/W4233937866","https://openalex.org/W4250300094","https://openalex.org/W4300594881","https://openalex.org/W6666258365"],"related_works":["https://openalex.org/W4380551899","https://openalex.org/W3206739003","https://openalex.org/W4286901852","https://openalex.org/W4300876896","https://openalex.org/W4312570297","https://openalex.org/W4286858168","https://openalex.org/W3105321552","https://openalex.org/W4320931267","https://openalex.org/W3213562055","https://openalex.org/W2123350836"],"abstract_inverted_index":{"Aperiodic":[0],"tilings":[1,29,70],"with":[2,37,71,83,97,117],"a":[3,34,46,54,84,110],"small":[4],"number":[5],"of":[6,9,50,62,127],"prototiles":[7],"are":[8,31],"particular":[10],"interest,":[11],"both":[12],"theoretically":[13],"and":[14,93,114,124],"for":[15,53],"applications":[16],"in":[17,91,109],"crystallography.":[18],"In":[19,102],"this":[20],"direction,":[21],"many":[22],"people":[23],"have":[24],"tried":[25],"to":[26,67],"construct":[27],"aperiodic":[28,77],"that":[30,80],"built":[32],"from":[33],"single":[35],"prototile":[36],"nearest":[38],"neighbour":[39],"matching":[40],"rules,":[41],"which":[42],"is":[43],"then":[44],"called":[45],"monotile.":[47],"One":[48],"strand":[49],"the":[51,98,122,125,128],"search":[52],"planar":[55],"monotile":[56],"has":[57],"focused":[58],"on":[59,121],"hexagonal":[60],"analogues":[61],"Wang":[63],"tiles.":[64],"This":[65],"led":[66],"two":[68],"inflation":[69],"interesting":[72],"structural":[73],"details.":[74],"Both":[75],"possess":[76],"local":[78],"rules":[79],"define":[81],"hulls":[82],"model":[85],"set":[86],"structure.":[87],"We":[88],"review":[89],"them":[90,116],"comparison,":[92],"clarify":[94],"their":[95],"relation":[96],"classic":[99],"half-hex":[100],"tiling.":[101],"particular,":[103],"we":[104],"formulate":[105],"various":[106],"known":[107],"results":[108,120],"more":[111],"comparative":[112],"way,":[113],"augment":[115],"some":[118],"new":[119],"geometry":[123],"topology":[126],"underlying":[129],"tiling":[130],"spaces.":[131]},"counts_by_year":[{"year":2024,"cited_by_count":2},{"year":2023,"cited_by_count":1},{"year":2021,"cited_by_count":4},{"year":2020,"cited_by_count":2},{"year":2019,"cited_by_count":1},{"year":2018,"cited_by_count":1},{"year":2017,"cited_by_count":1},{"year":2014,"cited_by_count":2},{"year":2013,"cited_by_count":1},{"year":2012,"cited_by_count":1}],"updated_date":"2026-05-06T08:25:59.206177","created_date":"2025-10-10T00:00:00"}