{"id":"https://openalex.org/W7160523678","doi":"https://doi.org/10.48550/arxiv.2605.04111","title":"Optimally Covering Large Triangles with Homothetic Unit Triangles","display_name":"Optimally Covering Large Triangles with Homothetic Unit Triangles","publication_year":2026,"publication_date":"2026-05-05","ids":{"openalex":"https://openalex.org/W7160523678","doi":"https://doi.org/10.48550/arxiv.2605.04111"},"language":null,"primary_location":{"id":"doi:10.48550/arxiv.2605.04111","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2605.04111","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"},"type":"preprint","indexed_in":["datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://doi.org/10.48550/arxiv.2605.04111","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5063312563","display_name":"John M. Boyer","orcid":"https://orcid.org/0000-0002-4755-5535"},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Boyer, John M.","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5063312563"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":false,"cited_by_count":0,"citation_normalized_percentile":null,"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10996","display_name":"Computational Geometry and Mesh Generation","score":0.322299987077713,"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"}},"topics":[{"id":"https://openalex.org/T10996","display_name":"Computational Geometry and Mesh Generation","score":0.322299987077713,"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/T11830","display_name":"Point processes and geometric inequalities","score":0.31709998846054077,"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/T12504","display_name":"Mathematics and Applications","score":0.10180000215768814,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"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/homothetic-transformation","display_name":"Homothetic transformation","score":0.9488000273704529},{"id":"https://openalex.org/keywords/upper-and-lower-bounds","display_name":"Upper and lower bounds","score":0.6144999861717224},{"id":"https://openalex.org/keywords/cover","display_name":"Cover (algebra)","score":0.5877000093460083},{"id":"https://openalex.org/keywords/unit","display_name":"Unit (ring theory)","score":0.5320000052452087},{"id":"https://openalex.org/keywords/unit-square","display_name":"Unit square","score":0.3231000006198883}],"concepts":[{"id":"https://openalex.org/C91709909","wikidata":"https://www.wikidata.org/wiki/Q583960","display_name":"Homothetic transformation","level":2,"score":0.9488000273704529},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7376999855041504},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.6635000109672546},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.6144999861717224},{"id":"https://openalex.org/C2780428219","wikidata":"https://www.wikidata.org/wiki/Q16952335","display_name":"Cover (algebra)","level":2,"score":0.5877000093460083},{"id":"https://openalex.org/C122637931","wikidata":"https://www.wikidata.org/wiki/Q118084","display_name":"Unit (ring theory)","level":2,"score":0.5320000052452087},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.32440000772476196},{"id":"https://openalex.org/C7356347","wikidata":"https://www.wikidata.org/wiki/Q1307838","display_name":"Unit square","level":2,"score":0.3231000006198883},{"id":"https://openalex.org/C47177299","wikidata":"https://www.wikidata.org/wiki/Q2390323","display_name":"Unit disk","level":2,"score":0.2953999936580658},{"id":"https://openalex.org/C128765274","wikidata":"https://www.wikidata.org/wiki/Q663314","display_name":"Hexagonal crystal system","level":2,"score":0.27219998836517334},{"id":"https://openalex.org/C98653994","wikidata":"https://www.wikidata.org/wiki/Q875937","display_name":"Isosceles triangle","level":2,"score":0.26100000739097595}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.48550/arxiv.2605.04111","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2605.04111","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"doi:10.48550/arxiv.2605.04111","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2605.04111","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":0,"referenced_works":[],"related_works":[],"abstract_inverted_index":{"We":[0,155],"answer":[1],"an":[2,133,201],"open":[3,134],"problem":[4,135],"in":[5],"the":[6,49,117,125,136,144,157,173,186,195,209,217],"\\emph{American":[7],"Mathematical":[8],"Monthly}":[9],"about":[10],"covering":[11,183],"large":[12],"triangles.":[13,97,221],"Given":[14],"a":[15,23,58,86,110,213],"triangle":[16,59,182],"$T$":[17,38],"of":[18,52,109,191,208],"any":[19],"triangular":[20],"shape":[21],"with":[22,43,60,67,91,216],"selected":[24,50,61],"side":[25,51,62],"length":[26,53,63],"between":[27],"$n":[28,64],"\\in":[29,69],"\\mathbb{N}$":[30],"and":[31,34,73,114,120,129,159,170,188],"$n+1$,":[32],"Baek":[33,72,128,158],"Lee":[35,74,130,160],"proved":[36],"that":[37,101,193,205],"could":[39],"not":[40],"be":[41,89],"covered":[42,90],"$n^2+1$":[44],"homothetic":[45,95,219],"unit":[46,96,220],"triangles":[47],"(with":[48],"1).":[54],"Letting":[55],"$T_{n+d}$":[56,87,215],"denote":[57],"+":[65,146],"d$":[66],"$d":[68],"(0,":[70],"1)$,":[71],"extended":[75],"their":[76,121],"proof":[77,161],"to":[78,138,162],"establish":[79,163],"upper":[80,141,164,174,196],"bounds":[81,103,142,165,175],"for":[82,116,124,143,149,166,185],"$d$":[83],"above":[84],"which":[85],"cannot":[88],"$n^2+2$":[92,118],"or":[93],"$n^2+3$":[94,126],"Then,":[98],"they":[99],"showed":[100],"these":[102],"are":[104,176],"tight":[105,140,177],"based":[106],"on":[107],"analyses":[108],"method":[111,123,204],"by":[112,178],"Conway":[113],"Soifer":[115],"case":[119],"own":[122],"case.":[127],"stated":[131],"as":[132,198,200],"need":[137],"find":[139],"$n^2":[145],"k$":[147],"cases":[148,190],"$4":[150],"\\le":[151,153],"k":[152],"2n$.":[154],"extend":[156],"those":[167],"higher":[168],"cases,":[169],"we":[171],"show":[172],"presenting":[179],"two":[180,210],"new":[181],"methods":[184],"odd":[187],"even":[189],"$k$":[192],"meet":[194],"bounds,":[197],"well":[199],"optimal":[202],"consolidated":[203],"uses":[206],"whichever":[207],"will":[211],"cover":[212],"given":[214],"fewest":[218]},"counts_by_year":[],"updated_date":"2026-07-01T06:00:48.157686","created_date":"2026-05-08T00:00:00"}
