{"id":"https://openalex.org/W2593205548","doi":"https://doi.org/10.1080/00029890.2019.1535735","title":"Random Triangles and Polygons in the Plane","display_name":"Random Triangles and Polygons in the Plane","publication_year":2019,"publication_date":"2019-02-07","ids":{"openalex":"https://openalex.org/W2593205548","doi":"https://doi.org/10.1080/00029890.2019.1535735","mag":"2593205548"},"language":"en","primary_location":{"id":"doi:10.1080/00029890.2019.1535735","is_oa":false,"landing_page_url":"https://doi.org/10.1080/00029890.2019.1535735","pdf_url":null,"source":{"id":"https://openalex.org/S156280010","display_name":"American Mathematical Monthly","issn_l":"0002-9890","issn":["0002-9890","1930-0972"],"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/P4310320547","host_organization_name":"Taylor & Francis","host_organization_lineage":["https://openalex.org/P4310320547"],"host_organization_lineage_names":["Taylor & Francis"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"The American Mathematical Monthly","raw_type":"journal-article"},"type":"article","indexed_in":["arxiv","crossref"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/1702.01027","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5081432563","display_name":"Jason Cantarella","orcid":"https://orcid.org/0000-0002-0613-6131"},"institutions":[{"id":"https://openalex.org/I165733156","display_name":"University of Georgia","ror":"https://ror.org/00te3t702","country_code":"US","type":"education","lineage":["https://openalex.org/I165733156"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Jason Cantarella","raw_affiliation_strings":[",","Mathematics Department, University of Georgia, Athens, GA 30602"],"affiliations":[{"raw_affiliation_string":",","institution_ids":[]},{"raw_affiliation_string":"Mathematics Department, University of Georgia, Athens, GA 30602","institution_ids":["https://openalex.org/I165733156"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5073878869","display_name":"Tom Needham","orcid":null},"institutions":[{"id":"https://openalex.org/I52357470","display_name":"The Ohio State University","ror":"https://ror.org/00rs6vg23","country_code":"US","type":"education","lineage":["https://openalex.org/I52357470"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Tom Needham","raw_affiliation_strings":[",","Mathematics Department, The Ohio State University, Columbus, OH 43210"],"affiliations":[{"raw_affiliation_string":",","institution_ids":[]},{"raw_affiliation_string":"Mathematics Department, The Ohio State University, Columbus, OH 43210","institution_ids":["https://openalex.org/I52357470"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5042389629","display_name":"Clayton Shonkwiler","orcid":"https://orcid.org/0000-0002-4811-8409"},"institutions":[{"id":"https://openalex.org/I92446798","display_name":"Colorado State University","ror":"https://ror.org/03k1gpj17","country_code":"US","type":"education","lineage":["https://openalex.org/I92446798"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Clayton Shonkwiler","raw_affiliation_strings":[",","Department of Mathematics, Colorado State University, 80523, Fort Collins, CO. Email: Clayton.Shohttp://ukcatsapps.uk.corplan.net:8535/PTS/final/processReviewOnlineAdvanceConfirmation.do#nkwiler@colostate.edu"],"affiliations":[{"raw_affiliation_string":",","institution_ids":[]},{"raw_affiliation_string":"Department of Mathematics, Colorado State University, 80523, Fort Collins, CO. Email: Clayton.Shohttp://ukcatsapps.uk.corplan.net:8535/PTS/final/processReviewOnlineAdvanceConfirmation.do#nkwiler@colostate.edu","institution_ids":["https://openalex.org/I92446798"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5005655649","display_name":"Gavin Stewart","orcid":"https://orcid.org/0000-0002-6835-6818"},"institutions":[{"id":"https://openalex.org/I36672615","display_name":"Courant Institute of Mathematical Sciences","ror":"https://ror.org/037tm7f56","country_code":"US","type":"education","lineage":["https://openalex.org/I36672615","https://openalex.org/I57206974"]},{"id":"https://openalex.org/I57206974","display_name":"New York University","ror":"https://ror.org/0190ak572","country_code":"US","type":"education","lineage":["https://openalex.org/I57206974"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Gavin Stewart","raw_affiliation_strings":[",","Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, NY 10012"],"affiliations":[{"raw_affiliation_string":",","institution_ids":[]},{"raw_affiliation_string":"Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, NY 10012","institution_ids":["https://openalex.org/I36672615","https://openalex.org/I57206974"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":4,"corresponding_author_ids":["https://openalex.org/A5081432563"],"corresponding_institution_ids":["https://openalex.org/I165733156"],"apc_list":null,"apc_paid":null,"fwci":1.5176,"has_fulltext":false,"cited_by_count":10,"citation_normalized_percentile":{"value":0.79212818,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":99},"biblio":{"volume":"126","issue":"2","first_page":"113","last_page":"134"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12417","display_name":"Morphological variations and asymmetry","score":0.9958999752998352,"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"}},"topics":[{"id":"https://openalex.org/T12417","display_name":"Morphological variations and asymmetry","score":0.9958999752998352,"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"}},{"id":"https://openalex.org/T12536","display_name":"Topological and Geometric Data Analysis","score":0.9939000010490417,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"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/T10996","display_name":"Computational Geometry and Mesh Generation","score":0.977400004863739,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/grassmannian","display_name":"Grassmannian","score":0.823758065700531},{"id":"https://openalex.org/keywords/quadrilateral","display_name":"Quadrilateral","score":0.8009804487228394},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.6006097793579102},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.57976895570755},{"id":"https://openalex.org/keywords/plane","display_name":"Plane (geometry)","score":0.5365562438964844},{"id":"https://openalex.org/keywords/point","display_name":"Point (geometry)","score":0.49787068367004395},{"id":"https://openalex.org/keywords/moduli-space","display_name":"Moduli space","score":0.44910240173339844},{"id":"https://openalex.org/keywords/space","display_name":"Space (punctuation)","score":0.44532671570777893},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.28983452916145325},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.26116159558296204},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.08573386073112488}],"concepts":[{"id":"https://openalex.org/C162929932","wikidata":"https://www.wikidata.org/wiki/Q129638","display_name":"Grassmannian","level":2,"score":0.823758065700531},{"id":"https://openalex.org/C151756577","wikidata":"https://www.wikidata.org/wiki/Q36810","display_name":"Quadrilateral","level":3,"score":0.8009804487228394},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.6006097793579102},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.57976895570755},{"id":"https://openalex.org/C17825722","wikidata":"https://www.wikidata.org/wiki/Q17285","display_name":"Plane (geometry)","level":2,"score":0.5365562438964844},{"id":"https://openalex.org/C28719098","wikidata":"https://www.wikidata.org/wiki/Q44946","display_name":"Point (geometry)","level":2,"score":0.49787068367004395},{"id":"https://openalex.org/C73373263","wikidata":"https://www.wikidata.org/wiki/Q845677","display_name":"Moduli space","level":2,"score":0.44910240173339844},{"id":"https://openalex.org/C2778572836","wikidata":"https://www.wikidata.org/wiki/Q380933","display_name":"Space (punctuation)","level":2,"score":0.44532671570777893},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.28983452916145325},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.26116159558296204},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.08573386073112488},{"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/C135628077","wikidata":"https://www.wikidata.org/wiki/Q220184","display_name":"Finite element method","level":2,"score":0.0},{"id":"https://openalex.org/C97355855","wikidata":"https://www.wikidata.org/wiki/Q11473","display_name":"Thermodynamics","level":1,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1080/00029890.2019.1535735","is_oa":false,"landing_page_url":"https://doi.org/10.1080/00029890.2019.1535735","pdf_url":null,"source":{"id":"https://openalex.org/S156280010","display_name":"American Mathematical Monthly","issn_l":"0002-9890","issn":["0002-9890","1930-0972"],"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/P4310320547","host_organization_name":"Taylor & Francis","host_organization_lineage":["https://openalex.org/P4310320547"],"host_organization_lineage_names":["Taylor & Francis"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"The American Mathematical Monthly","raw_type":"journal-article"},{"id":"pmh:oai:arXiv.org:1702.01027","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1702.01027","pdf_url":"https://arxiv.org/pdf/1702.01027","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":null,"raw_type":"text"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:1702.01027","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1702.01027","pdf_url":"https://arxiv.org/pdf/1702.01027","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":null,"raw_type":"text"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":36,"referenced_works":["https://openalex.org/W1571839244","https://openalex.org/W1896893915","https://openalex.org/W1927956358","https://openalex.org/W1966647070","https://openalex.org/W1966856967","https://openalex.org/W1993570538","https://openalex.org/W2001786791","https://openalex.org/W2025449352","https://openalex.org/W2036161092","https://openalex.org/W2045336066","https://openalex.org/W2054740189","https://openalex.org/W2121658906","https://openalex.org/W2135233944","https://openalex.org/W2168243811","https://openalex.org/W2258126206","https://openalex.org/W2265184865","https://openalex.org/W2313894639","https://openalex.org/W2315812153","https://openalex.org/W2334426548","https://openalex.org/W2621358153","https://openalex.org/W2950555822","https://openalex.org/W2963350927","https://openalex.org/W2963970580","https://openalex.org/W3099824818","https://openalex.org/W3100100815","https://openalex.org/W3104844908","https://openalex.org/W3106066603","https://openalex.org/W3169774611","https://openalex.org/W4235484205","https://openalex.org/W4240041676","https://openalex.org/W4240063553","https://openalex.org/W4241909813","https://openalex.org/W4243628866","https://openalex.org/W4250818842","https://openalex.org/W4254672552","https://openalex.org/W6797011485"],"related_works":["https://openalex.org/W4225381306","https://openalex.org/W1968844089","https://openalex.org/W2953041517","https://openalex.org/W4297210420","https://openalex.org/W2491011773","https://openalex.org/W2964120036","https://openalex.org/W2371953206","https://openalex.org/W2361023359","https://openalex.org/W2349543044","https://openalex.org/W3007915326"],"abstract_inverted_index":{"In":[0,49],"1997,":[1],"Jean-Claude":[2],"Hausmann":[3],"and":[4,10,16,33,46,106],"Allen":[5],"Knutson":[6],"introduced":[7],"a":[8,29,34,77,84],"natural":[9,30,35],"beautiful":[11],"correspondence":[12],"between":[13],"planar":[14,97],"n-gons":[15],"the":[17,64,74,93,109],"Grassmann":[18],"manifold":[19],"of":[20,60,67,96,112],"2-planes":[21],"in":[22,43],"real":[23],"n-space.":[24],"This":[25],"construction":[26],"leads":[27],"to":[28,57,72,102],"probability":[31,75],"distribution":[32],"metric":[36],"on":[37],"polygons":[38],"which":[39,82],"has":[40],"been":[41],"used":[42],"shape":[44],"classification":[45],"computer":[47],"vision.":[48],"this":[50,58,71],"paper,":[51],"we":[52],"provide":[53],"an":[54,100],"accessible":[55],"introduction":[56],"circle":[59],"ideas":[61],"by":[62,87],"explaining":[63],"Grassmannian":[65,94],"geometry":[66,95],"triangles.":[68],"We":[69,90],"use":[70],"find":[73],"that":[76],"random":[78],"triangle":[79],"is":[80],"obtuse,":[81],"was":[83],"question":[85],"raised":[86],"Lewis":[88],"Carroll.":[89],"then":[91],"explore":[92],"quadrilaterals,":[98],"providing":[99],"answer":[101],"Sylvester\u2019s":[103],"four-point":[104],"problem,":[105],"describing":[107],"explicitly":[108],"moduli":[110],"space":[111],"unordered":[113],"quadrilaterals.":[114]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2024,"cited_by_count":1},{"year":2023,"cited_by_count":1},{"year":2021,"cited_by_count":1},{"year":2020,"cited_by_count":1},{"year":2019,"cited_by_count":2},{"year":2018,"cited_by_count":2},{"year":2017,"cited_by_count":1}],"updated_date":"2026-04-04T16:13:02.066488","created_date":"2025-10-10T00:00:00"}
