{"id":"https://openalex.org/W4311831242","doi":"https://doi.org/10.1007/s10208-022-09601-8","title":"Mathematics of 2-Dimensional Lattices","display_name":"Mathematics of 2-Dimensional Lattices","publication_year":2022,"publication_date":"2022-12-07","ids":{"openalex":"https://openalex.org/W4311831242","doi":"https://doi.org/10.1007/s10208-022-09601-8"},"language":"en","primary_location":{"id":"doi:10.1007/s10208-022-09601-8","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s10208-022-09601-8","pdf_url":"https://link.springer.com/content/pdf/10.1007/s10208-022-09601-8.pdf","source":{"id":"https://openalex.org/S151639445","display_name":"Foundations of Computational Mathematics","issn_l":"1615-3375","issn":["1615-3375","1615-3383"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Foundations of Computational Mathematics","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"hybrid","oa_url":"https://link.springer.com/content/pdf/10.1007/s10208-022-09601-8.pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5067536683","display_name":"Vitaliy Kurlin","orcid":"https://orcid.org/0000-0001-5328-5351"},"institutions":[{"id":"https://openalex.org/I146655781","display_name":"University of Liverpool","ror":"https://ror.org/04xs57h96","country_code":"GB","type":"education","lineage":["https://openalex.org/I146655781"]}],"countries":["GB"],"is_corresponding":true,"raw_author_name":"Vitaliy Kurlin","raw_affiliation_strings":["Computer Science, University of Liverpool, Liverpool, UK"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Computer Science, University of Liverpool, Liverpool, UK","institution_ids":["https://openalex.org/I146655781"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5067536683"],"corresponding_institution_ids":["https://openalex.org/I146655781"],"apc_list":{"value":2390,"currency":"EUR","value_usd":2990},"apc_paid":{"value":2390,"currency":"EUR","value_usd":2990},"fwci":1.281,"has_fulltext":true,"cited_by_count":17,"citation_normalized_percentile":{"value":0.78202069,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":99},"biblio":{"volume":"24","issue":"3","first_page":"805","last_page":"863"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11887","display_name":"Quasicrystal Structures and Properties","score":0.984499990940094,"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":0.984499990940094,"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/T12536","display_name":"Topological and Geometric Data Analysis","score":0.9718999862670898,"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/T12285","display_name":"Surface Chemistry and Catalysis","score":0.9611999988555908,"subfield":{"id":"https://openalex.org/subfields/2204","display_name":"Biomedical 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/mathematics","display_name":"Mathematics","score":0.8151050209999084},{"id":"https://openalex.org/keywords/integer-lattice","display_name":"Integer lattice","score":0.6307944059371948},{"id":"https://openalex.org/keywords/lattice","display_name":"Lattice (music)","score":0.6072142720222473},{"id":"https://openalex.org/keywords/euclidean-geometry","display_name":"Euclidean geometry","score":0.577913224697113},{"id":"https://openalex.org/keywords/congruence","display_name":"Congruence (geometry)","score":0.5564647912979126},{"id":"https://openalex.org/keywords/euclidean-space","display_name":"Euclidean space","score":0.5011391639709473},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.4642397165298462},{"id":"https://openalex.org/keywords/isometry","display_name":"Isometry (Riemannian geometry)","score":0.45780113339424133},{"id":"https://openalex.org/keywords/metric-space","display_name":"Metric space","score":0.43991559743881226},{"id":"https://openalex.org/keywords/scaling","display_name":"Scaling","score":0.43080177903175354},{"id":"https://openalex.org/keywords/tetrahedron","display_name":"Tetrahedron","score":0.4166172444820404},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.3399679958820343},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.2653580904006958}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8151050209999084},{"id":"https://openalex.org/C142885752","wikidata":"https://www.wikidata.org/wiki/Q1252145","display_name":"Integer lattice","level":3,"score":0.6307944059371948},{"id":"https://openalex.org/C2781204021","wikidata":"https://www.wikidata.org/wiki/Q6497091","display_name":"Lattice (music)","level":2,"score":0.6072142720222473},{"id":"https://openalex.org/C129782007","wikidata":"https://www.wikidata.org/wiki/Q162886","display_name":"Euclidean geometry","level":2,"score":0.577913224697113},{"id":"https://openalex.org/C132074034","wikidata":"https://www.wikidata.org/wiki/Q154210","display_name":"Congruence (geometry)","level":2,"score":0.5564647912979126},{"id":"https://openalex.org/C186450821","wikidata":"https://www.wikidata.org/wiki/Q17295","display_name":"Euclidean space","level":2,"score":0.5011391639709473},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.4642397165298462},{"id":"https://openalex.org/C82457910","wikidata":"https://www.wikidata.org/wiki/Q740207","display_name":"Isometry (Riemannian geometry)","level":2,"score":0.45780113339424133},{"id":"https://openalex.org/C198043062","wikidata":"https://www.wikidata.org/wiki/Q180953","display_name":"Metric space","level":2,"score":0.43991559743881226},{"id":"https://openalex.org/C99844830","wikidata":"https://www.wikidata.org/wiki/Q102441924","display_name":"Scaling","level":2,"score":0.43080177903175354},{"id":"https://openalex.org/C105239961","wikidata":"https://www.wikidata.org/wiki/Q160003","display_name":"Tetrahedron","level":2,"score":0.4166172444820404},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.3399679958820343},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.2653580904006958},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0},{"id":"https://openalex.org/C24890656","wikidata":"https://www.wikidata.org/wiki/Q82811","display_name":"Acoustics","level":1,"score":0.0},{"id":"https://openalex.org/C56863137","wikidata":"https://www.wikidata.org/wiki/Q743139","display_name":"Half-integer","level":2,"score":0.0},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1007/s10208-022-09601-8","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s10208-022-09601-8","pdf_url":"https://link.springer.com/content/pdf/10.1007/s10208-022-09601-8.pdf","source":{"id":"https://openalex.org/S151639445","display_name":"Foundations of Computational Mathematics","issn_l":"1615-3375","issn":["1615-3375","1615-3383"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Foundations of Computational Mathematics","raw_type":"journal-article"}],"best_oa_location":{"id":"doi:10.1007/s10208-022-09601-8","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s10208-022-09601-8","pdf_url":"https://link.springer.com/content/pdf/10.1007/s10208-022-09601-8.pdf","source":{"id":"https://openalex.org/S151639445","display_name":"Foundations of Computational Mathematics","issn_l":"1615-3375","issn":["1615-3375","1615-3383"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Foundations of Computational Mathematics","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[{"id":"https://openalex.org/F4320320005","display_name":"Royal Academy of Engineering","ror":"https://ror.org/0526snb40"},{"id":"https://openalex.org/F4320334627","display_name":"Engineering and Physical Sciences Research Council","ror":"https://ror.org/0439y7842"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4311831242.pdf","grobid_xml":"https://content.openalex.org/works/W4311831242.grobid-xml"},"referenced_works_count":27,"referenced_works":["https://openalex.org/W21857574","https://openalex.org/W610273277","https://openalex.org/W1972922764","https://openalex.org/W1993789015","https://openalex.org/W2012589104","https://openalex.org/W2013155090","https://openalex.org/W2023699509","https://openalex.org/W2044622624","https://openalex.org/W2065550084","https://openalex.org/W3008244982","https://openalex.org/W3162861102","https://openalex.org/W3214916886","https://openalex.org/W4205366009","https://openalex.org/W4241950961","https://openalex.org/W4252813291","https://openalex.org/W4281888261","https://openalex.org/W4287284240","https://openalex.org/W4308457847","https://openalex.org/W4310054370","https://openalex.org/W4312534869","https://openalex.org/W4312835742","https://openalex.org/W4312955973","https://openalex.org/W4384825949","https://openalex.org/W6602534768","https://openalex.org/W6604431491","https://openalex.org/W6607592198","https://openalex.org/W6636251970"],"related_works":["https://openalex.org/W3118783749","https://openalex.org/W2002583644","https://openalex.org/W3119989344","https://openalex.org/W2072413983","https://openalex.org/W4297872901","https://openalex.org/W1982933349","https://openalex.org/W102694430","https://openalex.org/W2005721943","https://openalex.org/W2065991554","https://openalex.org/W2014898497"],"abstract_inverted_index":{"Abstract":[0],"A":[1],"periodic":[2],"lattice":[3,20,101,113],"in":[4],"Euclidean":[5,55],"space":[6],"is":[7],"the":[8,81,122],"infinite":[9],"set":[10],"of":[11,16,50,100,125],"all":[12,95],"integer":[13],"linear":[14],"combinations":[15],"basis":[17],"vectors.":[18],"Any":[19],"can":[21],"be":[22],"generated":[23],"by":[24],"infinitely":[25],"many":[26],"different":[27],"bases.":[28,102],"This":[29,44],"ambiguity":[30],"was":[31],"partially":[32],"resolved,":[33],"but":[34],"standard":[35],"reductions":[36],"remain":[37],"discontinuous":[38],"under":[39],"perturbations":[40],"modelling":[41],"atomic":[42],"displacements.":[43],"paper":[45],"completes":[46],"a":[47],"continuous":[48],"classification":[49],"2-dimensional":[51],"lattices":[52,77],"up":[53,79,86],"to":[54,80,87,106],"isometry":[56],"(or":[57],"congruence),":[58],"rigid":[59,88],"motion":[60,89],"(without":[61],"reflections),":[62],"and":[63,93,128],"similarity":[64],"(with":[65],"uniform":[66],"scaling).":[67],"The":[68,84,118],"new":[69],"homogeneous":[70],"invariants":[71],"allow":[72],"easily":[73],"computable":[74],"metrics":[75,85,104],"on":[76,98],"considered":[78],"equivalences":[82],"above.":[83],"are":[90],"especially":[91],"non-trivial":[92],"settle":[94],"remaining":[96],"questions":[97],"(dis)continuity":[99],"These":[103],"lead":[105],"real-valued":[107],"chiral":[108],"distances":[109],"that":[110],"continuously":[111],"measure":[112],"deviations":[114],"from":[115],"higher-symmetry":[116],"neighbours.":[117],"geometric":[119],"methods":[120],"extend":[121],"past":[123],"work":[124],"Delone,":[126],"Conway,":[127],"Sloane.":[129]},"counts_by_year":[{"year":2025,"cited_by_count":5},{"year":2024,"cited_by_count":3},{"year":2023,"cited_by_count":8},{"year":2022,"cited_by_count":1}],"updated_date":"2026-05-21T06:26:12.895304","created_date":"2025-10-10T00:00:00"}
