{"id":"https://openalex.org/W3197151903","doi":"https://doi.org/10.3390/sym13091659","title":"Voronoi Entropy vs. Continuous Measure of Symmetry of the Penrose Tiling: Part I. Analysis of the Voronoi Diagrams","display_name":"Voronoi Entropy vs. Continuous Measure of Symmetry of the Penrose Tiling: Part I. Analysis of the Voronoi Diagrams","publication_year":2021,"publication_date":"2021-09-08","ids":{"openalex":"https://openalex.org/W3197151903","doi":"https://doi.org/10.3390/sym13091659","mag":"3197151903"},"language":"en","primary_location":{"id":"doi:10.3390/sym13091659","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym13091659","pdf_url":"https://www.mdpi.com/2073-8994/13/9/1659/pdf?version=1631107743","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":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://www.mdpi.com/2073-8994/13/9/1659/pdf?version=1631107743","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":null,"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":null,"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":3.3425,"has_fulltext":false,"cited_by_count":19,"citation_normalized_percentile":{"value":0.91995331,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":94,"max":99},"biblio":{"volume":"13","issue":"9","first_page":"1659","last_page":"1659"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12496","display_name":"Color perception and design","score":0.987500011920929,"subfield":{"id":"https://openalex.org/subfields/3207","display_name":"Social Psychology"},"field":{"id":"https://openalex.org/fields/32","display_name":"Psychology"},"domain":{"id":"https://openalex.org/domains/2","display_name":"Social Sciences"}},"topics":[{"id":"https://openalex.org/T12496","display_name":"Color perception and design","score":0.987500011920929,"subfield":{"id":"https://openalex.org/subfields/3207","display_name":"Social Psychology"},"field":{"id":"https://openalex.org/fields/32","display_name":"Psychology"},"domain":{"id":"https://openalex.org/domains/2","display_name":"Social Sciences"}},{"id":"https://openalex.org/T11887","display_name":"Quasicrystal Structures and Properties","score":0.9872000217437744,"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/T11666","display_name":"Color Science and Applications","score":0.9656999707221985,"subfield":{"id":"https://openalex.org/subfields/3107","display_name":"Atomic and Molecular Physics, and Optics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/voronoi-diagram","display_name":"Voronoi diagram","score":0.9383432865142822},{"id":"https://openalex.org/keywords/penrose-tiling","display_name":"Penrose tiling","score":0.8151626586914062},{"id":"https://openalex.org/keywords/centroidal-voronoi-tessellation","display_name":"Centroidal Voronoi tessellation","score":0.655221164226532},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.5884591341018677},{"id":"https://openalex.org/keywords/entropy","display_name":"Entropy (arrow of time)","score":0.554844319820404},{"id":"https://openalex.org/keywords/measure","display_name":"Measure (data warehouse)","score":0.5297965407371521},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.5185754299163818},{"id":"https://openalex.org/keywords/substitution-tiling","display_name":"Substitution tiling","score":0.4847569167613983},{"id":"https://openalex.org/keywords/weighted-voronoi-diagram","display_name":"Weighted Voronoi diagram","score":0.4601050615310669},{"id":"https://openalex.org/keywords/tessellation","display_name":"Tessellation (computer graphics)","score":0.4263232350349426},{"id":"https://openalex.org/keywords/symmetry","display_name":"Symmetry (geometry)","score":0.4252602159976959},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.33328408002853394},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.2633677124977112},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.16858887672424316},{"id":"https://openalex.org/keywords/quasicrystal","display_name":"Quasicrystal","score":0.11445257067680359},{"id":"https://openalex.org/keywords/quantum-mechanics","display_name":"Quantum mechanics","score":0.10996085405349731},{"id":"https://openalex.org/keywords/data-mining","display_name":"Data mining","score":0.09231346845626831}],"concepts":[{"id":"https://openalex.org/C24881265","wikidata":"https://www.wikidata.org/wiki/Q757267","display_name":"Voronoi diagram","level":2,"score":0.9383432865142822},{"id":"https://openalex.org/C76513542","wikidata":"https://www.wikidata.org/wiki/Q585688","display_name":"Penrose tiling","level":3,"score":0.8151626586914062},{"id":"https://openalex.org/C205672865","wikidata":"https://www.wikidata.org/wiki/Q5062961","display_name":"Centroidal Voronoi tessellation","level":3,"score":0.655221164226532},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5884591341018677},{"id":"https://openalex.org/C106301342","wikidata":"https://www.wikidata.org/wiki/Q4117933","display_name":"Entropy (arrow of time)","level":2,"score":0.554844319820404},{"id":"https://openalex.org/C2780009758","wikidata":"https://www.wikidata.org/wiki/Q6804172","display_name":"Measure (data warehouse)","level":2,"score":0.5297965407371521},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.5185754299163818},{"id":"https://openalex.org/C21550011","wikidata":"https://www.wikidata.org/wiki/Q7632143","display_name":"Substitution tiling","level":2,"score":0.4847569167613983},{"id":"https://openalex.org/C20468658","wikidata":"https://www.wikidata.org/wiki/Q17130624","display_name":"Weighted Voronoi diagram","level":3,"score":0.4601050615310669},{"id":"https://openalex.org/C43817857","wikidata":"https://www.wikidata.org/wiki/Q9357547","display_name":"Tessellation (computer graphics)","level":2,"score":0.4263232350349426},{"id":"https://openalex.org/C2779886137","wikidata":"https://www.wikidata.org/wiki/Q21030012","display_name":"Symmetry (geometry)","level":2,"score":0.4252602159976959},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.33328408002853394},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.2633677124977112},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.16858887672424316},{"id":"https://openalex.org/C203101518","wikidata":"https://www.wikidata.org/wiki/Q263214","display_name":"Quasicrystal","level":2,"score":0.11445257067680359},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.10996085405349731},{"id":"https://openalex.org/C124101348","wikidata":"https://www.wikidata.org/wiki/Q172491","display_name":"Data mining","level":1,"score":0.09231346845626831}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.3390/sym13091659","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym13091659","pdf_url":"https://www.mdpi.com/2073-8994/13/9/1659/pdf?version=1631107743","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:doaj.org/article:b082a9d34a9b484dab226d95d77ebc6a","is_oa":true,"landing_page_url":"https://doaj.org/article/b082a9d34a9b484dab226d95d77ebc6a","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 13, Iss 9, p 1659 (2021)","raw_type":"article"},{"id":"pmh:oai:mdpi.com:/2073-8994/13/9/1659/","is_oa":true,"landing_page_url":"https://dx.doi.org/10.3390/sym13091659","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","raw_type":"Text"}],"best_oa_location":{"id":"doi:10.3390/sym13091659","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym13091659","pdf_url":"https://www.mdpi.com/2073-8994/13/9/1659/pdf?version=1631107743","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":[{"display_name":"Life in Land","score":0.44999998807907104,"id":"https://metadata.un.org/sdg/15"}],"awards":[{"id":"https://openalex.org/G5352162587","display_name":null,"funder_award_id":"19-19-00076","funder_id":"https://openalex.org/F4320324099","funder_display_name":"Russian Science Foundation"}],"funders":[{"id":"https://openalex.org/F4320324099","display_name":"Russian Science Foundation","ror":"https://ror.org/03y2gwe85"}],"has_content":{"grobid_xml":false,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W3197151903.pdf"},"referenced_works_count":40,"referenced_works":["https://openalex.org/W1520127302","https://openalex.org/W1592283523","https://openalex.org/W1963679001","https://openalex.org/W1988619801","https://openalex.org/W1995482580","https://openalex.org/W1997439464","https://openalex.org/W2001099600","https://openalex.org/W2001354866","https://openalex.org/W2005358242","https://openalex.org/W2007551675","https://openalex.org/W2037066848","https://openalex.org/W2042642397","https://openalex.org/W2046945477","https://openalex.org/W2047263344","https://openalex.org/W2061504435","https://openalex.org/W2073622316","https://openalex.org/W2090238112","https://openalex.org/W2115103988","https://openalex.org/W2124579751","https://openalex.org/W2135255496","https://openalex.org/W2330383299","https://openalex.org/W2532399514","https://openalex.org/W2612207348","https://openalex.org/W2652717287","https://openalex.org/W2766892668","https://openalex.org/W2902610165","https://openalex.org/W2911967961","https://openalex.org/W2943224956","https://openalex.org/W2948581074","https://openalex.org/W2949643913","https://openalex.org/W2990437741","https://openalex.org/W2999543410","https://openalex.org/W3026676619","https://openalex.org/W3037085755","https://openalex.org/W3086424822","https://openalex.org/W3093693011","https://openalex.org/W3121259400","https://openalex.org/W3123492901","https://openalex.org/W3123982061","https://openalex.org/W3145655529"],"related_works":["https://openalex.org/W4379365430","https://openalex.org/W2063059075","https://openalex.org/W2157142397","https://openalex.org/W4390436779","https://openalex.org/W2371724110","https://openalex.org/W2315038554","https://openalex.org/W4206423751","https://openalex.org/W4242194342","https://openalex.org/W4302348895","https://openalex.org/W1968872938"],"abstract_inverted_index":{"A":[0,22],"continuous":[1,72,114,142],"measure":[2,74,115,143],"of":[3,9,31,42,45,64,79,83,100,103,116,128,131,134,144,146,156],"symmetry":[4,62,73,117,145],"and":[5,39,75,118],"the":[6,17,32,36,40,43,46,52,60,65,71,76,80,87,98,101,104,110,113,119,141,163],"Voronoi":[7,13,33,57,77,94,120,126,151],"entropy":[8,78,121,127],"2D":[10],"patterns":[11],"representing":[12],"diagrams":[14,34,58,95,152],"emerging":[15,96],"from":[16,97,162],"Penrose":[18,24,47,66,88,105,157,165],"tiling":[19,25,132],"were":[20],"calculated.":[21],"given":[23],"gives":[26],"rise":[27],"to":[28],"a":[29],"diversity":[30],"when":[35],"centers,":[37],"vertices,":[38],"centers":[41,99],"edges":[44,102],"rhombs":[48],"are":[49,122,160],"taken":[50],"as":[51],"seed":[53],"points":[54],"(or":[55],"nuclei).":[56],"keep":[59],"initial":[61],"group":[63],"tiling.":[67],"We":[68],"demonstrate":[69],"that":[70,112],"studied":[81],"sets":[82],"points,":[84],"generated":[85],"by":[86],"tiling,":[89,158],"do":[90],"not":[91],"necessarily":[92],"correlate.":[93],"rhombs,":[106],"considered":[107],"nuclei,":[108],"deny":[109],"hypothesis":[111],"always":[123],"correlated.":[124],"The":[125],"this":[129,147],"kind":[130],"built":[133],"asymmetric":[135],"convex":[136],"quadrangles":[137],"equals":[138],"zero,":[139],"whereas":[140],"pattern":[148],"is":[149],"high.":[150],"generate":[153],"new":[154],"types":[155],"which":[159],"different":[161],"classical":[164],"tessellation.":[166]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2025,"cited_by_count":2},{"year":2024,"cited_by_count":2},{"year":2023,"cited_by_count":4},{"year":2022,"cited_by_count":7},{"year":2021,"cited_by_count":3}],"updated_date":"2026-05-22T06:13:13.366637","created_date":"2025-10-10T00:00:00"}