{"id":"https://openalex.org/W4400197544","doi":"https://doi.org/10.5070/c64163843","title":"An aperiodic monotile","display_name":"An aperiodic monotile","publication_year":2024,"publication_date":"2024-07-01","ids":{"openalex":"https://openalex.org/W4400197544","doi":"https://doi.org/10.5070/c64163843"},"language":"en","primary_location":{"id":"doi:10.5070/c64163843","is_oa":true,"landing_page_url":"https://doi.org/10.5070/c64163843","pdf_url":null,"source":{"id":"https://openalex.org/S4210174179","display_name":"Combinatorial Theory","issn_l":"2766-1334","issn":["2766-1334"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Combinatorial Theory","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"diamond","oa_url":"https://doi.org/10.5070/c64163843","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5013491544","display_name":"David J. Smith","orcid":"https://orcid.org/0000-0002-8810-4386"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"David Smith","raw_affiliation_strings":["Yorkshire, U.K"],"affiliations":[{"raw_affiliation_string":"Yorkshire, U.K","institution_ids":[]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5036551729","display_name":"Joseph Samuel Myers","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Joseph Samuel Myers","raw_affiliation_strings":["Cambridge,"],"affiliations":[{"raw_affiliation_string":"Cambridge,","institution_ids":[]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5082892265","display_name":"Craig S. Kaplan","orcid":"https://orcid.org/0000-0002-3168-6805"},"institutions":[{"id":"https://openalex.org/I151746483","display_name":"University of Waterloo","ror":"https://ror.org/01aff2v68","country_code":"CA","type":"education","lineage":["https://openalex.org/I151746483"]}],"countries":["CA"],"is_corresponding":false,"raw_author_name":"Craig S. Kaplan","raw_affiliation_strings":["School of Computer Science, University of Waterloo, Waterloo, Ontario, Canada"],"affiliations":[{"raw_affiliation_string":"School of Computer Science, University of Waterloo, Waterloo, Ontario, Canada","institution_ids":["https://openalex.org/I151746483"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5079268173","display_name":"Chaim Goodman-Strauss","orcid":null},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Chaim Goodman-Strauss","raw_affiliation_strings":["National Museum of Mathematics, New York, New York, U.S.A"],"affiliations":[{"raw_affiliation_string":"National Museum of Mathematics, New York, New York, U.S.A","institution_ids":[]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":4,"corresponding_author_ids":["https://openalex.org/A5079268173"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":7.3399,"has_fulltext":true,"cited_by_count":34,"citation_normalized_percentile":{"value":0.98310272,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":89,"max":100},"biblio":{"volume":"4","issue":"1","first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11737","display_name":"Advanced Materials and Mechanics","score":0.9182999730110168,"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"}},"topics":[{"id":"https://openalex.org/T11737","display_name":"Advanced Materials and Mechanics","score":0.9182999730110168,"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":0.8302923440933228},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.39759528636932373},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.2912524938583374},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.11932504177093506}],"concepts":[{"id":"https://openalex.org/C104247578","wikidata":"https://www.wikidata.org/wiki/Q4779368","display_name":"Aperiodic graph","level":2,"score":0.8302923440933228},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.39759528636932373},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.2912524938583374},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.11932504177093506}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.5070/c64163843","is_oa":true,"landing_page_url":"https://doi.org/10.5070/c64163843","pdf_url":null,"source":{"id":"https://openalex.org/S4210174179","display_name":"Combinatorial Theory","issn_l":"2766-1334","issn":["2766-1334"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Combinatorial Theory","raw_type":"journal-article"},{"id":"pmh:oai:escholarship.org:ark:/13030/qt3317z9z9","is_oa":true,"landing_page_url":"https://escholarship.org/uc/item/3317z9z9","pdf_url":"https://escholarship.org/content/qt3317z9z9/qt3317z9z9.pdf","source":{"id":"https://openalex.org/S4306400115","display_name":"eScholarship (California Digital Library)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I2801248553","host_organization_name":"California Digital Library","host_organization_lineage":["https://openalex.org/I2801248553"],"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":"Combinatorial Theory, vol 4, iss 1","raw_type":"article"}],"best_oa_location":{"id":"doi:10.5070/c64163843","is_oa":true,"landing_page_url":"https://doi.org/10.5070/c64163843","pdf_url":null,"source":{"id":"https://openalex.org/S4210174179","display_name":"Combinatorial Theory","issn_l":"2766-1334","issn":["2766-1334"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Combinatorial Theory","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[{"id":"https://openalex.org/F4320334670","display_name":"Trinity College, University of Cambridge","ror":null}],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":41,"referenced_works":["https://openalex.org/W1509480829","https://openalex.org/W1833217458","https://openalex.org/W1917615162","https://openalex.org/W1982939827","https://openalex.org/W1986667059","https://openalex.org/W1991511994","https://openalex.org/W2000173223","https://openalex.org/W2004860994","https://openalex.org/W2017025480","https://openalex.org/W2043284104","https://openalex.org/W2043318682","https://openalex.org/W2048190114","https://openalex.org/W2058072255","https://openalex.org/W2073943587","https://openalex.org/W2075873540","https://openalex.org/W2078456288","https://openalex.org/W2080896750","https://openalex.org/W2115103988","https://openalex.org/W2123350836","https://openalex.org/W2284502833","https://openalex.org/W2296088710","https://openalex.org/W2562856664","https://openalex.org/W2740991083","https://openalex.org/W2912957486","https://openalex.org/W2963943941","https://openalex.org/W2964105946","https://openalex.org/W3002510247","https://openalex.org/W3118445680","https://openalex.org/W3123306137","https://openalex.org/W3161412004","https://openalex.org/W3191581046","https://openalex.org/W3204665133","https://openalex.org/W4233539255","https://openalex.org/W4237970627","https://openalex.org/W4256492654","https://openalex.org/W4310510177","https://openalex.org/W4313481967","https://openalex.org/W4378508518","https://openalex.org/W4386875728","https://openalex.org/W4400596988","https://openalex.org/W4402927841"],"related_works":["https://openalex.org/W4391375266","https://openalex.org/W2748952813","https://openalex.org/W2037661079","https://openalex.org/W2339191433","https://openalex.org/W4220811941","https://openalex.org/W4287817191","https://openalex.org/W4298800952","https://openalex.org/W2039582600","https://openalex.org/W1992569127","https://openalex.org/W172284633"],"abstract_inverted_index":{"A":[0],"longstanding":[1],"open":[2],"problem":[3,29],"asks":[4],"for":[5,30,58],"an":[6,12],"aperiodic":[7,41,120],"monotile,":[8],"also":[9],"known":[10],"as":[11],"\"einstein\":":[13],"a":[14,36,47,92,102],"shape":[15],"that":[16,46,81,106],"admits":[17],"tilings":[18,69],"of":[19,38,70,84,87,95],"the":[20,50,66,71,76,107],"plane,":[21,72],"but":[22],"never":[23],"periodic":[24],"tilings.":[25],"We":[26,43,78],"answer":[27],"this":[28],"topological":[31],"disk":[32],"tiles":[33],"by":[34],"exhibiting":[35],"continuum":[37,86],"combinatorially":[39],"equivalent":[40],"polygons.":[42],"first":[44],"show":[45],"representative":[48],"example,":[49],"\"hat\"":[51],"polykite,":[52],"can":[53,62],"form":[54,110],"clusters":[55],"called":[56],"\"metatiles\",":[57],"which":[59],"substitution":[60],"rules":[61],"be":[63],"defined.":[64],"Because":[65],"metatiles":[67],"admit":[68],"so":[73],"too":[74],"does":[75],"hat.":[77],"then":[79],"prove":[80],"generic":[82],"members":[83],"our":[85],"polygons":[88],"are":[89],"aperiodic,":[90],"through":[91],"new":[93],"kind":[94],"geometric":[96],"incommensurability":[97],"argument.":[98],"Separately,":[99],"we":[100],"give":[101],"combinatorial,":[103],"computer-assisted":[104],"proof":[105],"hat":[108],"must":[109],"hierarchical--and":[111],"hence":[112],"aperiodic--tilings.Mathematics":[113],"Subject":[114],"Classifications:":[115],"05B45,":[116],"52C20,":[117],"05B50Keywords:":[118],"Tilings,":[119],"order,":[121],"polyforms":[122]},"counts_by_year":[{"year":2026,"cited_by_count":3},{"year":2025,"cited_by_count":12},{"year":2024,"cited_by_count":18},{"year":2023,"cited_by_count":1}],"updated_date":"2026-03-11T14:59:36.786465","created_date":"2025-10-10T00:00:00"}
