{"id":"https://openalex.org/W4313432842","doi":"https://doi.org/10.3390/e25010092","title":"Voronoi Tessellations and the Shannon Entropy of the Pentagonal Tilings","display_name":"Voronoi Tessellations and the Shannon Entropy of the Pentagonal Tilings","publication_year":2023,"publication_date":"2023-01-02","ids":{"openalex":"https://openalex.org/W4313432842","doi":"https://doi.org/10.3390/e25010092","pmid":"https://pubmed.ncbi.nlm.nih.gov/36673233"},"language":"en","primary_location":{"id":"doi:10.3390/e25010092","is_oa":true,"landing_page_url":"https://doi.org/10.3390/e25010092","pdf_url":"https://www.mdpi.com/1099-4300/25/1/92/pdf?version=1672659756","source":{"id":"https://openalex.org/S195231649","display_name":"Entropy","issn_l":"1099-4300","issn":["1099-4300"],"is_oa":true,"is_in_doaj":true,"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":"Entropy","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj","pubmed"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://www.mdpi.com/1099-4300/25/1/92/pdf?version=1672659756","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5088635445","display_name":"Edward Bormashenko","orcid":"https://orcid.org/0000-0003-1356-2486"},"institutions":[{"id":"https://openalex.org/I52170813","display_name":"Ariel University","ror":"https://ror.org/03nz8qe97","country_code":"IL","type":"education","lineage":["https://openalex.org/I52170813"]}],"countries":["IL"],"is_corresponding":true,"raw_author_name":"Edward Bormashenko","raw_affiliation_strings":["Chemical Engineering Department, Engineering Faculty, Ariel University, P.O.B. 3, Ariel 407000, Israel"],"raw_orcid":"https://orcid.org/0000-0003-1356-2486","affiliations":[{"raw_affiliation_string":"Chemical Engineering Department, Engineering Faculty, Ariel University, P.O.B. 3, Ariel 407000, Israel","institution_ids":["https://openalex.org/I52170813"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5016034099","display_name":"Irina Legchenkova","orcid":"https://orcid.org/0000-0002-6384-7570"},"institutions":[{"id":"https://openalex.org/I52170813","display_name":"Ariel University","ror":"https://ror.org/03nz8qe97","country_code":"IL","type":"education","lineage":["https://openalex.org/I52170813"]}],"countries":["IL"],"is_corresponding":false,"raw_author_name":"Irina Legchenkova","raw_affiliation_strings":["Chemical Engineering Department, Engineering Faculty, Ariel University, P.O.B. 3, Ariel 407000, Israel"],"raw_orcid":"https://orcid.org/0000-0002-6384-7570","affiliations":[{"raw_affiliation_string":"Chemical Engineering Department, Engineering Faculty, Ariel University, P.O.B. 3, Ariel 407000, Israel","institution_ids":["https://openalex.org/I52170813"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5034655838","display_name":"Mark Frenkel","orcid":"https://orcid.org/0000-0003-3025-6130"},"institutions":[{"id":"https://openalex.org/I52170813","display_name":"Ariel University","ror":"https://ror.org/03nz8qe97","country_code":"IL","type":"education","lineage":["https://openalex.org/I52170813"]}],"countries":["IL"],"is_corresponding":false,"raw_author_name":"Mark Frenkel","raw_affiliation_strings":["Chemical Engineering Department, Engineering Faculty, Ariel University, P.O.B. 3, Ariel 407000, Israel"],"raw_orcid":"https://orcid.org/0000-0003-3025-6130","affiliations":[{"raw_affiliation_string":"Chemical Engineering Department, Engineering Faculty, Ariel University, P.O.B. 3, Ariel 407000, Israel","institution_ids":["https://openalex.org/I52170813"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5079590419","display_name":"Nir Shvalb","orcid":"https://orcid.org/0000-0001-8246-3727"},"institutions":[{"id":"https://openalex.org/I52170813","display_name":"Ariel University","ror":"https://ror.org/03nz8qe97","country_code":"IL","type":"education","lineage":["https://openalex.org/I52170813"]}],"countries":["IL"],"is_corresponding":false,"raw_author_name":"Nir Shvalb","raw_affiliation_strings":["Department of Mechanical Engineering & Mechatronics, Faculty of Engineering, Ariel University, P.O.B. 3, Ariel 407000, Israel"],"raw_orcid":"https://orcid.org/0000-0001-8246-3727","affiliations":[{"raw_affiliation_string":"Department of Mechanical Engineering & Mechatronics, Faculty of Engineering, Ariel University, P.O.B. 3, Ariel 407000, Israel","institution_ids":["https://openalex.org/I52170813"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5066005912","display_name":"Shraga Shoval","orcid":"https://orcid.org/0000-0002-0582-4821"},"institutions":[{"id":"https://openalex.org/I52170813","display_name":"Ariel University","ror":"https://ror.org/03nz8qe97","country_code":"IL","type":"education","lineage":["https://openalex.org/I52170813"]}],"countries":["IL"],"is_corresponding":false,"raw_author_name":"Shraga Shoval","raw_affiliation_strings":["Department of Industrial Engineering and Management, Faculty of Engineering, Ariel University, P.O.B. 3, Ariel 407000, Israel"],"raw_orcid":"https://orcid.org/0000-0002-0582-4821","affiliations":[{"raw_affiliation_string":"Department of Industrial Engineering and Management, Faculty of Engineering, Ariel University, P.O.B. 3, Ariel 407000, Israel","institution_ids":["https://openalex.org/I52170813"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5088635445"],"corresponding_institution_ids":["https://openalex.org/I52170813"],"apc_list":{"value":2000,"currency":"CHF","value_usd":2165},"apc_paid":{"value":2000,"currency":"CHF","value_usd":2165},"fwci":0.6717,"has_fulltext":true,"cited_by_count":8,"citation_normalized_percentile":{"value":0.60407605,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":96,"max":98},"biblio":{"volume":"25","issue":"1","first_page":"92","last_page":"92"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11887","display_name":"Quasicrystal Structures and Properties","score":0.9973999857902527,"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":0.9973999857902527,"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/T10306","display_name":"Liquid Crystal Research Advancements","score":0.9223999977111816,"subfield":{"id":"https://openalex.org/subfields/2504","display_name":"Electronic, Optical and Magnetic Materials"},"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/T10474","display_name":"Material Dynamics and Properties","score":0.9003999829292297,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/voronoi-diagram","display_name":"Voronoi diagram","score":0.9539356827735901},{"id":"https://openalex.org/keywords/centroidal-voronoi-tessellation","display_name":"Centroidal Voronoi tessellation","score":0.8306429386138916},{"id":"https://openalex.org/keywords/substitution-tiling","display_name":"Substitution tiling","score":0.5630952715873718},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.558508038520813},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.5275922417640686},{"id":"https://openalex.org/keywords/entropy","display_name":"Entropy (arrow of time)","score":0.5112106800079346},{"id":"https://openalex.org/keywords/tessellation","display_name":"Tessellation (computer graphics)","score":0.5010466575622559},{"id":"https://openalex.org/keywords/regular-polygon","display_name":"Regular polygon","score":0.4533277750015259},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.32459867000579834},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.11540234088897705}],"concepts":[{"id":"https://openalex.org/C24881265","wikidata":"https://www.wikidata.org/wiki/Q757267","display_name":"Voronoi diagram","level":2,"score":0.9539356827735901},{"id":"https://openalex.org/C205672865","wikidata":"https://www.wikidata.org/wiki/Q5062961","display_name":"Centroidal Voronoi tessellation","level":3,"score":0.8306429386138916},{"id":"https://openalex.org/C21550011","wikidata":"https://www.wikidata.org/wiki/Q7632143","display_name":"Substitution tiling","level":2,"score":0.5630952715873718},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.558508038520813},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.5275922417640686},{"id":"https://openalex.org/C106301342","wikidata":"https://www.wikidata.org/wiki/Q4117933","display_name":"Entropy (arrow of time)","level":2,"score":0.5112106800079346},{"id":"https://openalex.org/C43817857","wikidata":"https://www.wikidata.org/wiki/Q9357547","display_name":"Tessellation (computer graphics)","level":2,"score":0.5010466575622559},{"id":"https://openalex.org/C112680207","wikidata":"https://www.wikidata.org/wiki/Q714886","display_name":"Regular polygon","level":2,"score":0.4533277750015259},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.32459867000579834},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.11540234088897705},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0}],"mesh":[],"locations_count":5,"locations":[{"id":"doi:10.3390/e25010092","is_oa":true,"landing_page_url":"https://doi.org/10.3390/e25010092","pdf_url":"https://www.mdpi.com/1099-4300/25/1/92/pdf?version=1672659756","source":{"id":"https://openalex.org/S195231649","display_name":"Entropy","issn_l":"1099-4300","issn":["1099-4300"],"is_oa":true,"is_in_doaj":true,"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":"Entropy","raw_type":"journal-article"},{"id":"pmid:36673233","is_oa":false,"landing_page_url":"https://pubmed.ncbi.nlm.nih.gov/36673233","pdf_url":null,"source":{"id":"https://openalex.org/S4306525036","display_name":"PubMed","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I1299303238","host_organization_name":"National Institutes of Health","host_organization_lineage":["https://openalex.org/I1299303238"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Entropy (Basel, Switzerland)","raw_type":null},{"id":"pmh:oai:pubmedcentral.nih.gov:9857938","is_oa":true,"landing_page_url":"https://www.ncbi.nlm.nih.gov/pmc/articles/9857938","pdf_url":null,"source":{"id":"https://openalex.org/S2764455111","display_name":"PubMed Central","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I1299303238","host_organization_name":"National Institutes of Health","host_organization_lineage":["https://openalex.org/I1299303238"],"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":"Entropy (Basel)","raw_type":"Text"},{"id":"pmh:oai:doaj.org/article:b0a56917245e44469add498f72594d27","is_oa":true,"landing_page_url":"https://doaj.org/article/b0a56917245e44469add498f72594d27","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":"Entropy, Vol 25, Iss 1, p 92 (2023)","raw_type":"article"},{"id":"pmh:oai:mdpi.com:/1099-4300/25/1/92/","is_oa":true,"landing_page_url":"https://dx.doi.org/10.3390/e25010092","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":"Entropy; Volume 25; Issue 1; Pages: 92","raw_type":"Text"}],"best_oa_location":{"id":"doi:10.3390/e25010092","is_oa":true,"landing_page_url":"https://doi.org/10.3390/e25010092","pdf_url":"https://www.mdpi.com/1099-4300/25/1/92/pdf?version=1672659756","source":{"id":"https://openalex.org/S195231649","display_name":"Entropy","issn_l":"1099-4300","issn":["1099-4300"],"is_oa":true,"is_in_doaj":true,"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":"Entropy","raw_type":"journal-article"},"sustainable_development_goals":[{"display_name":"Life in Land","score":0.46000000834465027,"id":"https://metadata.un.org/sdg/15"}],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4313432842.pdf"},"referenced_works_count":34,"referenced_works":["https://openalex.org/W1520127302","https://openalex.org/W1970796230","https://openalex.org/W1972199661","https://openalex.org/W1978018076","https://openalex.org/W1988619801","https://openalex.org/W1989756836","https://openalex.org/W1997851568","https://openalex.org/W2001354866","https://openalex.org/W2007551675","https://openalex.org/W2024552670","https://openalex.org/W2041508287","https://openalex.org/W2042253782","https://openalex.org/W2094791143","https://openalex.org/W2101287554","https://openalex.org/W2109343827","https://openalex.org/W2317678240","https://openalex.org/W2652717287","https://openalex.org/W2768605006","https://openalex.org/W2806129833","https://openalex.org/W2811366195","https://openalex.org/W2902610165","https://openalex.org/W2949643913","https://openalex.org/W2963259283","https://openalex.org/W2964017204","https://openalex.org/W2993322147","https://openalex.org/W3037085755","https://openalex.org/W3083421372","https://openalex.org/W3121259400","https://openalex.org/W3195645775","https://openalex.org/W3197151903","https://openalex.org/W3197738924","https://openalex.org/W3201444856","https://openalex.org/W4297973239","https://openalex.org/W6619246739"],"related_works":["https://openalex.org/W1978501764","https://openalex.org/W2371724110","https://openalex.org/W1997141012","https://openalex.org/W2005041861","https://openalex.org/W2053576657","https://openalex.org/W4206423751","https://openalex.org/W1516845492","https://openalex.org/W2059217232","https://openalex.org/W4390436779","https://openalex.org/W2013420653"],"abstract_inverted_index":{"We":[0],"used":[1],"the":[2,11,19,31,37,66,91,112,152,157,161,172,175],"complete":[3],"set":[4],"of":[5,25,30,36,48,54,79,106,114,132,145,160,174],"convex":[6],"pentagons":[7,124,133],"to":[8,45,142],"enable":[9],"filing":[10],"plane":[12],"without":[13],"any":[14],"overlaps":[15],"or":[16],"gaps":[17],"(including":[18],"Marjorie":[20],"Rice":[21],"tiles)":[22],"as":[23],"generators":[24],"Voronoi":[26,49,62,93,98,128,146,166],"tessellations.":[27,50],"Shannon":[28,52,77],"entropy":[29,53,78],"tessellations":[32,56,99,167],"was":[33,74,134],"calculated.":[34],"Some":[35],"basic":[38,67,137],"mosaics":[39,139],"are":[40,100,123,149],"flexible":[41],"and":[42,85,117,125],"give":[43,140],"rise":[44,141],"a":[46,59,143],"diversity":[47,144],"The":[51,95,108,119],"these":[55],"varied":[57],"in":[58,90],"broad":[60],"range.":[61],"tessellation,":[63],"emerging":[64],"from":[65,71,102],"pentagonal":[68,138],"tiling":[69,81,129],"built":[70,101,130],"hexagons":[72],"only,":[73],"revealed":[75],"(the":[76],"this":[80],"is":[82,111,163],"zero).":[83],"Decagons":[84],"hendecagon":[86],"did":[87],"not":[88],"appear":[89],"studied":[92],"diagrams.":[94],"most":[96,109,120],"abundant":[97,121],"three":[103],"different":[104],"kinds":[105],"polygons.":[107],"widespread":[110],"combination":[113],"pentagons,":[115],"hexagons,":[116],"heptagons.":[118],"polygons":[122],"hexagons.":[126],"No":[127],"only":[131],"registered.":[135],"Flexible":[136],"tessellations,":[147],"which":[148],"characterized":[150],"by":[151],"same":[153],"symmetry":[154],"group.":[155],"However,":[156],"coordination":[158],"number":[159],"vertices":[162],"variable.":[164],"These":[165],"may":[168],"be":[169],"useful":[170],"for":[171],"interpretation":[173],"iso-symmetrical":[176],"phase":[177],"transitions.":[178]},"counts_by_year":[{"year":2024,"cited_by_count":3},{"year":2023,"cited_by_count":5}],"updated_date":"2026-05-21T06:26:12.895304","created_date":"2025-10-10T00:00:00"}
