{"id":"https://openalex.org/W2082617679","doi":"https://doi.org/10.1145/2582112.2582113","title":"The Opaque Square","display_name":"The Opaque Square","publication_year":2014,"publication_date":"2014-05-02","ids":{"openalex":"https://openalex.org/W2082617679","doi":"https://doi.org/10.1145/2582112.2582113","mag":"2082617679"},"language":"en","primary_location":{"id":"doi:10.1145/2582112.2582113","is_oa":false,"landing_page_url":"https://doi.org/10.1145/2582112.2582113","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Proceedings of the thirtieth annual symposium on Computational geometry","raw_type":"proceedings-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5091313409","display_name":"Adrian Dumitrescu","orcid":"https://orcid.org/0000-0002-1118-0321"},"institutions":[{"id":"https://openalex.org/I43579087","display_name":"University of Wisconsin\u2013Milwaukee","ror":"https://ror.org/031q21x57","country_code":"US","type":"education","lineage":["https://openalex.org/I43579087"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Adrian Dumitrescu","raw_affiliation_strings":["Dept. of Computer Science, Univ. of Wisconsin-Milwaukee, WI 53201-0784, USA"],"affiliations":[{"raw_affiliation_string":"Dept. of Computer Science, Univ. of Wisconsin-Milwaukee, WI 53201-0784, USA","institution_ids":["https://openalex.org/I43579087"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5080563649","display_name":"Minghui Jiang","orcid":"https://orcid.org/0000-0003-1843-9292"},"institutions":[{"id":"https://openalex.org/I121980950","display_name":"Utah State University","ror":"https://ror.org/00h6set76","country_code":"US","type":"education","lineage":["https://openalex.org/I121980950"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Minghui Jiang","raw_affiliation_strings":["Dept. of Computer Science, Utah State University, Logan, UT 84322-4205, USA","Department of Computer Science, Utah State University, Logan, UT 84322-4205 USA"],"affiliations":[{"raw_affiliation_string":"Dept. of Computer Science, Utah State University, Logan, UT 84322-4205, USA","institution_ids":["https://openalex.org/I121980950"]},{"raw_affiliation_string":"Department of Computer Science, Utah State University, Logan, UT 84322-4205 USA","institution_ids":["https://openalex.org/I121980950"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5091313409"],"corresponding_institution_ids":["https://openalex.org/I43579087"],"apc_list":null,"apc_paid":null,"fwci":1.3899,"has_fulltext":false,"cited_by_count":5,"citation_normalized_percentile":{"value":0.80155642,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":94},"biblio":{"volume":null,"issue":null,"first_page":"529","last_page":"538"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11830","display_name":"Point processes and geometric inequalities","score":0.9991000294685364,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T11830","display_name":"Point processes and geometric inequalities","score":0.9991000294685364,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10996","display_name":"Computational Geometry and Mesh Generation","score":0.9977999925613403,"subfield":{"id":"https://openalex.org/subfields/1704","display_name":"Computer Graphics and Computer-Aided Design"},"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/T12056","display_name":"Markov Chains and Monte Carlo Methods","score":0.9872999787330627,"subfield":{"id":"https://openalex.org/subfields/2613","display_name":"Statistics and Probability"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/unit-square","display_name":"Unit square","score":0.7163088917732239},{"id":"https://openalex.org/keywords/square","display_name":"Square (algebra)","score":0.7031757831573486},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.6413914561271667},{"id":"https://openalex.org/keywords/upper-and-lower-bounds","display_name":"Upper and lower bounds","score":0.6174371242523193},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.5828620791435242},{"id":"https://openalex.org/keywords/line","display_name":"Line (geometry)","score":0.553321897983551},{"id":"https://openalex.org/keywords/disjoint-sets","display_name":"Disjoint sets","score":0.5474888682365417},{"id":"https://openalex.org/keywords/regular-polygon","display_name":"Regular polygon","score":0.5468302965164185},{"id":"https://openalex.org/keywords/opacity","display_name":"Opacity","score":0.48973798751831055},{"id":"https://openalex.org/keywords/block","display_name":"Block (permutation group theory)","score":0.4161340296268463},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.3642765283584595},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.2184397280216217},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.14447328448295593},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.12875708937644958}],"concepts":[{"id":"https://openalex.org/C7356347","wikidata":"https://www.wikidata.org/wiki/Q1307838","display_name":"Unit square","level":2,"score":0.7163088917732239},{"id":"https://openalex.org/C135692309","wikidata":"https://www.wikidata.org/wiki/Q111124","display_name":"Square (algebra)","level":2,"score":0.7031757831573486},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.6413914561271667},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.6174371242523193},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5828620791435242},{"id":"https://openalex.org/C198352243","wikidata":"https://www.wikidata.org/wiki/Q37105","display_name":"Line (geometry)","level":2,"score":0.553321897983551},{"id":"https://openalex.org/C45340560","wikidata":"https://www.wikidata.org/wiki/Q215382","display_name":"Disjoint sets","level":2,"score":0.5474888682365417},{"id":"https://openalex.org/C112680207","wikidata":"https://www.wikidata.org/wiki/Q714886","display_name":"Regular polygon","level":2,"score":0.5468302965164185},{"id":"https://openalex.org/C60056205","wikidata":"https://www.wikidata.org/wiki/Q691914","display_name":"Opacity","level":2,"score":0.48973798751831055},{"id":"https://openalex.org/C2777210771","wikidata":"https://www.wikidata.org/wiki/Q4927124","display_name":"Block (permutation group theory)","level":2,"score":0.4161340296268463},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.3642765283584595},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.2184397280216217},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.14447328448295593},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.12875708937644958},{"id":"https://openalex.org/C120665830","wikidata":"https://www.wikidata.org/wiki/Q14620","display_name":"Optics","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1145/2582112.2582113","is_oa":false,"landing_page_url":"https://doi.org/10.1145/2582112.2582113","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Proceedings of the thirtieth annual symposium on Computational geometry","raw_type":"proceedings-article"}],"best_oa_location":null,"sustainable_development_goals":[{"display_name":"Sustainable cities and communities","score":0.6399999856948853,"id":"https://metadata.un.org/sdg/11"}],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":29,"referenced_works":["https://openalex.org/W12111610","https://openalex.org/W113788050","https://openalex.org/W1024677470","https://openalex.org/W1541206082","https://openalex.org/W1572708899","https://openalex.org/W1974114050","https://openalex.org/W1974171274","https://openalex.org/W1975892719","https://openalex.org/W1985018595","https://openalex.org/W1996931690","https://openalex.org/W1998571970","https://openalex.org/W2022906393","https://openalex.org/W2030428607","https://openalex.org/W2058215634","https://openalex.org/W2075885162","https://openalex.org/W2076863991","https://openalex.org/W2084120415","https://openalex.org/W2084665183","https://openalex.org/W2088558457","https://openalex.org/W2132402436","https://openalex.org/W2159560597","https://openalex.org/W2312749223","https://openalex.org/W2321017396","https://openalex.org/W2325472777","https://openalex.org/W2564817534","https://openalex.org/W2978215457","https://openalex.org/W4231808319","https://openalex.org/W4235556280","https://openalex.org/W4295634057"],"related_works":["https://openalex.org/W3161375928","https://openalex.org/W222989450","https://openalex.org/W2022929595","https://openalex.org/W2798172748","https://openalex.org/W4386395423","https://openalex.org/W2949900893","https://openalex.org/W4366562581","https://openalex.org/W2110938502","https://openalex.org/W2886094488","https://openalex.org/W1987762223"],"abstract_inverted_index":{"The":[0,36,47],"problem":[1],"of":[2,55,63,109,133,147,156,168,184,210],"finding":[3,130],"small":[4],"sets":[5],"that":[6,187,204],"block":[7],"every":[8],"line":[9],"passing":[10],"through":[11],"a":[12,25,31,56,97,126,199,211],"unit":[13,153],"square":[14,90],"was":[15],"first":[16],"considered":[17],"by":[18,114,179],"Mazurkiewicz":[19],"in":[20,72,88,92,125,172],"1916.":[21],"We":[22],"call":[23],"such":[24],"set":[26,29],"an":[27],"opaque":[28],"or":[30,91],"barrier":[32,39,60,83,150,214],"for":[33,52,106,142,151,181,201,215],"the":[34,53,65,82,89,120,131,134,145,152,169,182,208,216],"square.":[35,217],"shortest":[37,135],"known":[38,79],"has":[40],"length":[41,54,132,146],"\u221a2":[42],"+":[43,116,165],"\u221a6/2":[44],"=":[45],"2.6389....":[46],"current":[48],"best":[49],"lower":[50,76,104],"bound":[51,77,139],"(not":[57],"necessarily":[58],"connected)":[59],"is":[61,78,84,140,161],"2,":[62],"which":[64,118],"earliest":[66],"record":[67],"dates":[68],"back":[69],"to":[70,86,207],"Jones":[71],"1964.":[73],"No":[74],"better":[75],"even":[80,195],"if":[81],"restricted":[85],"lie":[87],"its":[93],"close":[94],"vicinity.":[95],"Under":[96],"suitable":[98],"locality":[99],"assumption,":[100],"we":[101],"replace":[102],"this":[103],"bound,":[105],"barriers":[107],"consisting":[108,155],"finitely":[110,157],"many":[111,158],"straight-line":[112,159],"segments,":[113,160],"2":[115,164],"10--12,":[117],"represents":[119],"first,":[121],"albeit":[122],"small,":[123],"step":[124],"long":[127],"time":[128],"toward":[129],"barrier.":[136],"A":[137],"sharper":[138],"obtained":[141],"interior":[143,149],"barriers:":[144],"any":[148],"square,":[154],"at":[162],"least":[163],"10--5.":[166],"Two":[167],"key":[170],"elements":[171],"our":[173],"proofs":[174],"are:":[175],"(i)":[176],"formulas":[177],"established":[178],"Sylvester":[180],"measure":[183],"all":[185],"lines":[186,203],"meet":[188],"two":[189],"disjoint":[190],"planar":[191],"convex":[192],"bodies,":[193],"and,":[194],"more":[196],"significant,":[197],"(ii)":[198],"procedure":[200],"detecting":[202],"are":[205],"witness":[206],"invalidity":[209],"short":[212],"bogus":[213]},"counts_by_year":[{"year":2024,"cited_by_count":1},{"year":2019,"cited_by_count":1},{"year":2016,"cited_by_count":1},{"year":2015,"cited_by_count":1},{"year":2014,"cited_by_count":1}],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2016-06-24T00:00:00"}