{"id":"https://openalex.org/W2950992242","doi":"https://doi.org/10.1007/s00454-020-00173-4","title":"Arrangements of Pseudocircles: Triangles and Drawings","display_name":"Arrangements of Pseudocircles: Triangles and Drawings","publication_year":2020,"publication_date":"2020-01-27","ids":{"openalex":"https://openalex.org/W2950992242","doi":"https://doi.org/10.1007/s00454-020-00173-4","mag":"2950992242"},"language":"en","primary_location":{"id":"doi:10.1007/s00454-020-00173-4","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s00454-020-00173-4","pdf_url":"https://link.springer.com/content/pdf/10.1007/s00454-020-00173-4.pdf","source":{"id":"https://openalex.org/S103671469","display_name":"Discrete & Computational Geometry","issn_l":"0179-5376","issn":["0179-5376","1432-0444"],"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":"Discrete &amp; Computational Geometry","raw_type":"journal-article"},"type":"preprint","indexed_in":["arxiv","crossref","datacite"],"open_access":{"is_oa":true,"oa_status":"hybrid","oa_url":"https://link.springer.com/content/pdf/10.1007/s00454-020-00173-4.pdf","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5005468020","display_name":"Stefan Felsner","orcid":"https://orcid.org/0000-0002-6150-1998"},"institutions":[{"id":"https://openalex.org/I4577782","display_name":"Technische Universit\u00e4t Berlin","ror":"https://ror.org/03v4gjf40","country_code":"DE","type":"education","lineage":["https://openalex.org/I4577782"]}],"countries":["DE"],"is_corresponding":true,"raw_author_name":"Stefan Felsner","raw_affiliation_strings":["Institut f\u00fcr Mathematik, Technische Universit\u00e4t, 10623 Berlin, Germany","Technical University of Berlin"],"raw_orcid":"https://orcid.org/0000-0002-6150-1998","affiliations":[{"raw_affiliation_string":"Institut f\u00fcr Mathematik, Technische Universit\u00e4t, 10623 Berlin, Germany","institution_ids":["https://openalex.org/I4577782"]},{"raw_affiliation_string":"Technical University of Berlin","institution_ids":["https://openalex.org/I4577782"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5078050536","display_name":"Manfred Scheucher","orcid":"https://orcid.org/0000-0002-1657-9796"},"institutions":[{"id":"https://openalex.org/I4577782","display_name":"Technische Universit\u00e4t Berlin","ror":"https://ror.org/03v4gjf40","country_code":"DE","type":"education","lineage":["https://openalex.org/I4577782"]}],"countries":["DE"],"is_corresponding":false,"raw_author_name":"Manfred Scheucher","raw_affiliation_strings":["Institut f\u00fcr Mathematik, Technische Universit\u00e4t, 10623 Berlin, Germany","Technical University of Berlin"],"raw_orcid":"https://orcid.org/0000-0002-1657-9796","affiliations":[{"raw_affiliation_string":"Institut f\u00fcr Mathematik, Technische Universit\u00e4t, 10623 Berlin, Germany","institution_ids":["https://openalex.org/I4577782"]},{"raw_affiliation_string":"Technical University of Berlin","institution_ids":["https://openalex.org/I4577782"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5005468020"],"corresponding_institution_ids":["https://openalex.org/I4577782"],"apc_list":{"value":2290,"currency":"EUR","value_usd":2890},"apc_paid":{"value":2290,"currency":"EUR","value_usd":2890},"fwci":0.0,"has_fulltext":true,"cited_by_count":1,"citation_normalized_percentile":{"value":0.01068985,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":94},"biblio":{"volume":"65","issue":"1","first_page":"261","last_page":"278"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10996","display_name":"Computational Geometry and Mesh Generation","score":0.9998000264167786,"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":0.9998000264167786,"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/T12176","display_name":"Optimization and Packing Problems","score":0.9930999875068665,"subfield":{"id":"https://openalex.org/subfields/2209","display_name":"Industrial and Manufacturing Engineering"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T12923","display_name":"Digital Image Processing Techniques","score":0.9779999852180481,"subfield":{"id":"https://openalex.org/subfields/1707","display_name":"Computer Vision and Pattern Recognition"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.8350540399551392},{"id":"https://openalex.org/keywords/pairwise-comparison","display_name":"Pairwise comparison","score":0.8280191421508789},{"id":"https://openalex.org/keywords/conjecture","display_name":"Conjecture","score":0.7348151206970215},{"id":"https://openalex.org/keywords/simple","display_name":"Simple (philosophy)","score":0.7145897150039673},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6655275225639343},{"id":"https://openalex.org/keywords/upper-and-lower-bounds","display_name":"Upper and lower bounds","score":0.5637699365615845},{"id":"https://openalex.org/keywords/section","display_name":"Section (typography)","score":0.4295293688774109},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.12583929300308228},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.06065261363983154},{"id":"https://openalex.org/keywords/statistics","display_name":"Statistics","score":0.043753743171691895}],"concepts":[{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.8350540399551392},{"id":"https://openalex.org/C184898388","wikidata":"https://www.wikidata.org/wiki/Q1435712","display_name":"Pairwise comparison","level":2,"score":0.8280191421508789},{"id":"https://openalex.org/C2780990831","wikidata":"https://www.wikidata.org/wiki/Q319141","display_name":"Conjecture","level":2,"score":0.7348151206970215},{"id":"https://openalex.org/C2780586882","wikidata":"https://www.wikidata.org/wiki/Q7520643","display_name":"Simple (philosophy)","level":2,"score":0.7145897150039673},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6655275225639343},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.5637699365615845},{"id":"https://openalex.org/C2780129039","wikidata":"https://www.wikidata.org/wiki/Q1931107","display_name":"Section (typography)","level":2,"score":0.4295293688774109},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.12583929300308228},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.06065261363983154},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.043753743171691895},{"id":"https://openalex.org/C138885662","wikidata":"https://www.wikidata.org/wiki/Q5891","display_name":"Philosophy","level":0,"score":0.0},{"id":"https://openalex.org/C111919701","wikidata":"https://www.wikidata.org/wiki/Q9135","display_name":"Operating system","level":1,"score":0.0},{"id":"https://openalex.org/C111472728","wikidata":"https://www.wikidata.org/wiki/Q9471","display_name":"Epistemology","level":1,"score":0.0}],"mesh":[],"locations_count":7,"locations":[{"id":"doi:10.1007/s00454-020-00173-4","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s00454-020-00173-4","pdf_url":"https://link.springer.com/content/pdf/10.1007/s00454-020-00173-4.pdf","source":{"id":"https://openalex.org/S103671469","display_name":"Discrete & Computational Geometry","issn_l":"0179-5376","issn":["0179-5376","1432-0444"],"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":"Discrete &amp; Computational Geometry","raw_type":"journal-article"},{"id":"pmh:oai:arXiv.org:1708.06449","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1708.06449","pdf_url":"https://arxiv.org/pdf/1708.06449","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":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"text"},{"id":"mag:2950992242","is_oa":true,"landing_page_url":"https://arxiv.org/pdf/1708.06449","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":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"arXiv (Cornell University)","raw_type":null},{"id":"pmh:oai:depositonce.tu-berlin.de:11303/12810","is_oa":true,"landing_page_url":"https://depositonce.tu-berlin.de/handle/11303/12810","pdf_url":null,"source":{"id":"https://openalex.org/S4406922277","display_name":"DepositOnce","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","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"publishedVersion"},{"id":"pmh:oai:europepmc.org:6706481","is_oa":true,"landing_page_url":"http://europepmc.org/pmc/articles/PMC7779420","pdf_url":null,"source":{"id":"https://openalex.org/S4306400806","display_name":"Europe PMC (PubMed Central)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I1303153112","host_organization_name":"European Bioinformatics Institute","host_organization_lineage":["https://openalex.org/I1303153112"],"host_organization_lineage_names":[],"type":"repository"},"license":"other-oa","license_id":"https://openalex.org/licenses/other-oa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"Text"},{"id":"doi:10.48550/arxiv.1708.06449","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.1708.06449","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":"article"},{"id":"doi:10.14279/depositonce-11610","is_oa":true,"landing_page_url":"https://doi.org/10.14279/depositonce-11610","pdf_url":null,"source":{"id":"https://openalex.org/S7407052985","display_name":"Technische Universit\u00e4t Berlin \u2013 Universit\u00e4tsbibliothek","issn_l":null,"issn":[],"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","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":"article-journal"}],"best_oa_location":{"id":"doi:10.1007/s00454-020-00173-4","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s00454-020-00173-4","pdf_url":"https://link.springer.com/content/pdf/10.1007/s00454-020-00173-4.pdf","source":{"id":"https://openalex.org/S103671469","display_name":"Discrete & Computational Geometry","issn_l":"0179-5376","issn":["0179-5376","1432-0444"],"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":"Discrete &amp; Computational Geometry","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G7644750246","display_name":null,"funder_award_id":"FE 340/11-1 and FE 340/12-1","funder_id":"https://openalex.org/F4320320879","funder_display_name":"Deutsche Forschungsgemeinschaft"}],"funders":[{"id":"https://openalex.org/F4320310430","display_name":"TU Graz, Internationale Beziehungen und Mobilit\u00e4tsprogramme","ror":"https://ror.org/00d7xrm67"},{"id":"https://openalex.org/F4320320879","display_name":"Deutsche Forschungsgemeinschaft","ror":"https://ror.org/018mejw64"},{"id":"https://openalex.org/F4320334678","display_name":"European Research Council","ror":"https://ror.org/0472cxd90"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2950992242.pdf","grobid_xml":"https://content.openalex.org/works/W2950992242.grobid-xml"},"referenced_works_count":10,"referenced_works":["https://openalex.org/W650639416","https://openalex.org/W1500063450","https://openalex.org/W1667649711","https://openalex.org/W1967603179","https://openalex.org/W2163913133","https://openalex.org/W2404367485","https://openalex.org/W2911399207","https://openalex.org/W2963179494","https://openalex.org/W2971330320","https://openalex.org/W4206475567"],"related_works":["https://openalex.org/W2746347573","https://openalex.org/W2948180810","https://openalex.org/W2767208609","https://openalex.org/W3090353283","https://openalex.org/W2163913133","https://openalex.org/W2184800459","https://openalex.org/W2087146362","https://openalex.org/W2087679745","https://openalex.org/W3022117908","https://openalex.org/W3138908823","https://openalex.org/W2155151614","https://openalex.org/W2207462936","https://openalex.org/W1673739447","https://openalex.org/W2963021779","https://openalex.org/W1986278299","https://openalex.org/W3094507641","https://openalex.org/W2130006374","https://openalex.org/W1972485581","https://openalex.org/W3152864601","https://openalex.org/W2185983738"],"abstract_inverted_index":{"Abstract":[0],"A":[1],"pseudocircle":[2],"is":[3,65,172,320],"a":[4,87,102,215,386],"simple":[5,31,177],"closed":[6,32],"curve":[7],"on":[8,91],"the":[9,13,44,146,258,396,405,413],"sphere":[10],"or":[11],"in":[12,37,57,263],"plane.":[14],"The":[15,376],"study":[16],"of":[17,19,30,46,60,104,188,218,261,267,328,332,381,383,389,412],"arrangements":[18,59,107,187,210,266,331,382],"pseudocircles":[20,64,96,334,384],"was":[21,392],"initiated":[22],"by":[23,400],"Gr\u00fcnbaum,":[24],"who":[25],"defined":[26],"them":[27],"as":[28],"collections":[29],"curves":[33],"that":[34,43,145,184,239,271],"pairwise":[35,62,190,208,264,329],"intersect":[36],"exactly":[38],"two":[39],"crossings.":[40],"Gr\u00fcnbaum":[41],"conjectured":[42],"number":[45,260],"triangular":[47],"cells":[48],"$$p_3$$":[49],"<mml:math":[50,69,115,153,197,222,243,278,342],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">":[51,70,116,154,198,223,244,279,343],"<mml:msub>":[52,118,156,225,246,281,345],"<mml:mi>p</mml:mi>":[53,119,157,226,247,282,346],"<mml:mn>3</mml:mn>":[54,120,158,169,227,233,248,283,288,347,352],"</mml:msub>":[55,121,159,228,249,284,348],"</mml:math>":[56,77,141,171,205,235,255,317,374],"digon-free":[58,106,186],"n":[61,189,333],"intersecting":[63,105,191,209,265,330],"at":[66,194],"least":[67,195],"$$2n-4$$":[68,196],"<mml:mrow>":[71,117,122,155,160,199,224,245,280,291,300,311,344,355,364],"<mml:mn>2</mml:mn>":[72,200,230,302,366],"<mml:mi>n</mml:mi>":[73,128,167,201,231,251,295,313,359],"<mml:mo>-</mml:mo>":[74,202,252],"<mml:mn>4</mml:mn>":[75,166,203,287,351],"</mml:mrow>":[76,126,140,164,170,204,234,254,303,307,315,316,367,371,373],".":[78,142,256,318,375],"We":[79,143],"present":[80],"examples":[81],"to":[82],"disprove":[83],"this":[84],"conjecture.":[85],"With":[86],"recursive":[88],"construction":[89],"based":[90],"an":[92],"example":[93],"with":[94,108,211,335],"12":[95],"and":[97,237,385],"16":[98],"triangles":[99,262],"we":[100,213,269,408],"obtain":[101],"family":[103],"$$p_3({\\mathscr":[109,149],"{A}})/n":[110],"\\rightarrow":[111],"16/11":[112],"=":[113,337],"1.\\overline{45}$$":[114],"<mml:mo>(</mml:mo>":[123,161,312],"<mml:mi>A</mml:mi>":[124,162],"<mml:mo>)</mml:mo>":[125,163,314],"<mml:mo>/</mml:mo>":[127,131,168,232],"<mml:mo>\u2192</mml:mo>":[129],"<mml:mn>16</mml:mn>":[130],"<mml:mn>11</mml:mn>":[132],"<mml:mo>=</mml:mo>":[133,349],"<mml:mn>1</mml:mn>":[134,253],"<mml:mo>.</mml:mo>":[135],"<mml:mover>":[136],"<mml:mn>45</mml:mn>":[137],"<mml:mo>\u00af</mml:mo>":[138],"</mml:mover>":[139],"expect":[144],"lower":[147,216],"bound":[148,217],"{A}})":[150],"\\ge":[151,220,241],"4n/3$$":[152],"<mml:mo>\u2265</mml:mo>":[165,229,250],"tight":[173],"for":[174],"infinitely":[175],"many":[176,379],"arrangements.":[178],"It":[179],"may":[180],"however":[181],"be":[182],"true":[183],"all":[185],"circles":[192],"have":[193,214],"triangles.":[206],"For":[207],"digons":[212],"$$p_3":[219,240,272,336],"2n/3$$":[221],",":[236],"conjecture":[238],"n-1$$":[242],"Concerning":[257],"maximum":[259],"pseudocircles,":[268],"show":[270],"\\le":[273],"\\frac{4}{3}\\left(":[274,338],"{\\begin{array}{c}n\\\\":[275,339],"2\\end{array}}\\right)":[276,340],"+O(n)$$":[277],"<mml:mo>\u2264</mml:mo>":[285],"<mml:mfrac>":[286,350],"</mml:mfrac>":[289,353],"<mml:mfenced>":[290,354],"<mml:mtable>":[292,356],"<mml:mtr>":[293,298,357,362],"<mml:mtd>":[294,299,358,363],"</mml:mtd>":[296,304,360,368],"</mml:mtr>":[297,305,361,369],"<mml:mrow/>":[301,365],"</mml:mtable>":[306,370],"</mml:mfenced>":[308,372],"<mml:mo>+</mml:mo>":[309],"<mml:mi>O</mml:mi>":[310],"This":[319],"essentially":[321],"best":[322],"possible":[323],"because":[324],"there":[325],"are":[326],"families":[327],"$$":[341],"paper":[377],"contains":[378],"drawings":[380,391],"good":[387],"fraction":[388],"these":[390],"produced":[393,399],"automatically":[394],"from":[395],"combinatorial":[397],"data":[398],"our":[401],"generation":[402],"algorithm.":[403,415],"In":[404],"final":[406],"section":[407],"describe":[409],"some":[410],"aspects":[411],"drawing":[414]},"counts_by_year":[{"year":2018,"cited_by_count":1}],"updated_date":"2026-05-21T06:26:12.895304","created_date":"2025-10-10T00:00:00"}