{"id":"https://openalex.org/W3113049957","doi":"https://doi.org/10.1080/00029890.2020.1814673","title":"Tangent Spheres of Tetrahedra and a Theorem of Grace","display_name":"Tangent Spheres of Tetrahedra and a Theorem of Grace","publication_year":2020,"publication_date":"2020-11-25","ids":{"openalex":"https://openalex.org/W3113049957","doi":"https://doi.org/10.1080/00029890.2020.1814673","mag":"3113049957"},"language":"en","primary_location":{"id":"doi:10.1080/00029890.2020.1814673","is_oa":false,"landing_page_url":"https://doi.org/10.1080/00029890.2020.1814673","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":["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/A5078786798","display_name":"Hiroshi Maehara","orcid":"https://orcid.org/0000-0001-9957-8141"},"institutions":[{"id":"https://openalex.org/I32915989","display_name":"University of the Ryukyus","ror":"https://ror.org/02z1n9q24","country_code":"JP","type":"education","lineage":["https://openalex.org/I32915989"]}],"countries":["JP"],"is_corresponding":true,"raw_author_name":"Hiroshi Maehara","raw_affiliation_strings":["Ryukyu University, Nishihara, Okinawa 903\u20130213, Japan;"],"affiliations":[{"raw_affiliation_string":"Ryukyu University, Nishihara, Okinawa 903\u20130213, Japan;","institution_ids":["https://openalex.org/I32915989"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5040806028","display_name":"Horst Martini","orcid":null},"institutions":[{"id":"https://openalex.org/I2610724","display_name":"Chemnitz University of Technology","ror":"https://ror.org/00a208s56","country_code":"DE","type":"education","lineage":["https://openalex.org/I2610724"]}],"countries":["DE"],"is_corresponding":false,"raw_author_name":"Horst Martini","raw_affiliation_strings":["Faculty of Mathematics, Chemnitz University of Technology, 09107 Chemnitz, Germany"],"affiliations":[{"raw_affiliation_string":"Faculty of Mathematics, Chemnitz University of Technology, 09107 Chemnitz, Germany","institution_ids":["https://openalex.org/I2610724"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5078786798"],"corresponding_institution_ids":["https://openalex.org/I32915989"],"apc_list":null,"apc_paid":null,"fwci":0.7795,"has_fulltext":false,"cited_by_count":3,"citation_normalized_percentile":{"value":0.72154274,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":99},"biblio":{"volume":"127","issue":"10","first_page":"897","last_page":"910"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12504","display_name":"Mathematics and Applications","score":0.9998999834060669,"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/T12504","display_name":"Mathematics and Applications","score":0.9998999834060669,"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/T12170","display_name":"History and Theory of Mathematics","score":0.9973999857902527,"subfield":{"id":"https://openalex.org/subfields/2614","display_name":"Theoretical Computer Science"},"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/T10061","display_name":"Algebraic Geometry and Number Theory","score":0.9962000250816345,"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/tetrahedron","display_name":"Tetrahedron","score":0.9139986634254456},{"id":"https://openalex.org/keywords/spheres","display_name":"SPHERES","score":0.7012835144996643},{"id":"https://openalex.org/keywords/tangent","display_name":"Tangent","score":0.6895030736923218},{"id":"https://openalex.org/keywords/tangent-space","display_name":"Tangent space","score":0.6169565320014954},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.5028793215751648},{"id":"https://openalex.org/keywords/face","display_name":"Face (sociological concept)","score":0.4232194125652313},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.39434969425201416},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.38485705852508545},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.3848347067832947},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.32277756929397583},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.26854223012924194}],"concepts":[{"id":"https://openalex.org/C105239961","wikidata":"https://www.wikidata.org/wiki/Q160003","display_name":"Tetrahedron","level":2,"score":0.9139986634254456},{"id":"https://openalex.org/C72422203","wikidata":"https://www.wikidata.org/wiki/Q7392545","display_name":"SPHERES","level":2,"score":0.7012835144996643},{"id":"https://openalex.org/C138187205","wikidata":"https://www.wikidata.org/wiki/Q131251","display_name":"Tangent","level":2,"score":0.6895030736923218},{"id":"https://openalex.org/C157157409","wikidata":"https://www.wikidata.org/wiki/Q909601","display_name":"Tangent space","level":2,"score":0.6169565320014954},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5028793215751648},{"id":"https://openalex.org/C2779304628","wikidata":"https://www.wikidata.org/wiki/Q3503480","display_name":"Face (sociological concept)","level":2,"score":0.4232194125652313},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.39434969425201416},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.38485705852508545},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.3848347067832947},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.32277756929397583},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.26854223012924194},{"id":"https://openalex.org/C36289849","wikidata":"https://www.wikidata.org/wiki/Q34749","display_name":"Social science","level":1,"score":0.0},{"id":"https://openalex.org/C1276947","wikidata":"https://www.wikidata.org/wiki/Q333","display_name":"Astronomy","level":1,"score":0.0},{"id":"https://openalex.org/C144024400","wikidata":"https://www.wikidata.org/wiki/Q21201","display_name":"Sociology","level":0,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1080/00029890.2020.1814673","is_oa":false,"landing_page_url":"https://doi.org/10.1080/00029890.2020.1814673","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"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":21,"referenced_works":["https://openalex.org/W28370716","https://openalex.org/W114372112","https://openalex.org/W564773951","https://openalex.org/W1510237842","https://openalex.org/W1531499523","https://openalex.org/W1701606820","https://openalex.org/W1995930239","https://openalex.org/W2006188145","https://openalex.org/W2014524739","https://openalex.org/W2029179868","https://openalex.org/W2066200953","https://openalex.org/W2066770030","https://openalex.org/W2072715215","https://openalex.org/W2084098115","https://openalex.org/W2114859109","https://openalex.org/W2118000153","https://openalex.org/W2768364248","https://openalex.org/W4230822413","https://openalex.org/W4233317155","https://openalex.org/W4253914708","https://openalex.org/W4285719527"],"related_works":["https://openalex.org/W4235659780","https://openalex.org/W1568242700","https://openalex.org/W2018291641","https://openalex.org/W2951237747","https://openalex.org/W2594233670","https://openalex.org/W2036174934","https://openalex.org/W2361271799","https://openalex.org/W47544861","https://openalex.org/W2577818139","https://openalex.org/W123898865"],"abstract_inverted_index":{"A":[0],"sphere":[1,23,60],"that":[2,42,63,78,103,111],"is":[3,19,57,86,106,123,152],"tangent":[4,22,28,48,89,96],"to":[5,90,97,124,142],"all":[6],"four":[7],"face-planes":[8],"(i.e.,":[9],"the":[10,14,25,68,72,76,135,154],"affine":[11],"hulls":[12],"of":[13,16,24,46,53,71,120,131,137,156,161],"faces)":[15],"a":[17,21,44,54,58,138,157],"tetrahedron":[18,55,73],"called":[20,31],"tetrahedron.":[26],"Two":[27],"spheres":[29,49],"are":[30],"neighboring":[32,47],"if":[33],"exactly":[34],"one":[35],"face-plane":[36,77],"separates":[37,79],"them.":[38],"Grace\u2019s":[39,162],"theorem":[40,105,133],"states":[41],"for":[43],"pair":[45],"S":[50,80],"and":[51,81,83,110,128,141],"T":[52,93],"there":[56],"unique":[59],"\u0398":[61,65,85],"such":[62],"(1)":[64],"passes":[66],"through":[67],"three":[69],"vertices":[70],"lying":[74],"on":[75],"T,":[82],"(2)":[84],"either":[87],"externally":[88],"both":[91,98],"S,":[92,99],"or":[94],"internally":[95],"T.":[100],"It":[101],"seems":[102],"this":[104,121,132,148],"not":[107],"widely":[108],"known,":[109],"no":[112],"elementary":[113,127],"proof":[114,130],"has":[115],"been":[116],"given.":[117],"The":[118],"purpose":[119],"article":[122],"present":[125],"an":[126],"direct":[129],"in":[134,147],"case":[136],"trirectangular":[139],"tetrahedron,":[140],"obtain":[143],"several":[144],"further":[145],"results":[146],"direction.":[149],"Among":[150],"them":[151],"also":[153],"confirmation":[155],"slightly":[158],"generalized":[159],"form":[160],"theorem.":[163]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2023,"cited_by_count":1},{"year":2020,"cited_by_count":1}],"updated_date":"2026-03-10T16:38:18.471706","created_date":"2025-10-10T00:00:00"}