{"id":"https://openalex.org/W4392042250","doi":"https://doi.org/10.1007/s00454-024-00628-y","title":"On Compact Packings of Euclidean Space with Spheres of Finitely Many Sizes","display_name":"On Compact Packings of Euclidean Space with Spheres of Finitely Many Sizes","publication_year":2024,"publication_date":"2024-02-22","ids":{"openalex":"https://openalex.org/W4392042250","doi":"https://doi.org/10.1007/s00454-024-00628-y"},"language":"en","primary_location":{"id":"doi:10.1007/s00454-024-00628-y","is_oa":true,"landing_page_url":"http://dx.doi.org/10.1007/s00454-024-00628-y","pdf_url":"https://link.springer.com/content/pdf/10.1007/s00454-024-00628-y.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":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"hybrid","oa_url":"https://link.springer.com/content/pdf/10.1007/s00454-024-00628-y.pdf","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5071335946","display_name":"Miek Messerschmidt","orcid":"https://orcid.org/0000-0003-0162-3229"},"institutions":[{"id":"https://openalex.org/I69552723","display_name":"University of Pretoria","ror":"https://ror.org/00g0p6g84","country_code":"ZA","type":"education","lineage":["https://openalex.org/I69552723"]}],"countries":["ZA"],"is_corresponding":true,"raw_author_name":"Miek Messerschmidt","raw_affiliation_strings":["Department of Mathematics and Applied Mathematics, University of Pretoria, Hatfield, Private Bag X20, 0028, Pretoria, South Africa"],"raw_orcid":"https://orcid.org/0000-0003-0162-3229","affiliations":[{"raw_affiliation_string":"Department of Mathematics and Applied Mathematics, University of Pretoria, Hatfield, Private Bag X20, 0028, Pretoria, South Africa","institution_ids":["https://openalex.org/I69552723"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5014433071","display_name":"Eder Kikianty","orcid":"https://orcid.org/0000-0003-4355-0339"},"institutions":[{"id":"https://openalex.org/I69552723","display_name":"University of Pretoria","ror":"https://ror.org/00g0p6g84","country_code":"ZA","type":"education","lineage":["https://openalex.org/I69552723"]}],"countries":["ZA"],"is_corresponding":false,"raw_author_name":"Eder Kikianty","raw_affiliation_strings":["Department of Mathematics and Applied Mathematics, University of Pretoria, Hatfield, Private Bag X20, 0028, Pretoria, South Africa"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics and Applied Mathematics, University of Pretoria, Hatfield, Private Bag X20, 0028, Pretoria, South Africa","institution_ids":["https://openalex.org/I69552723"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5071335946"],"corresponding_institution_ids":["https://openalex.org/I69552723"],"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":0.03711847,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":94},"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":0.9958999752998352,"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":0.9958999752998352,"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":0.9853000044822693,"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/T10304","display_name":"Geometric and Algebraic Topology","score":0.9778000116348267,"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/spheres","display_name":"SPHERES","score":0.7882490754127502},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7636605501174927},{"id":"https://openalex.org/keywords/euclidean-geometry","display_name":"Euclidean geometry","score":0.6269649863243103},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.5674670934677124},{"id":"https://openalex.org/keywords/euclidean-space","display_name":"Euclidean space","score":0.5241132974624634},{"id":"https://openalex.org/keywords/space","display_name":"Space (punctuation)","score":0.4846973419189453},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.39572674036026},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.2672135531902313},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.09943202137947083},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.07838019728660583}],"concepts":[{"id":"https://openalex.org/C72422203","wikidata":"https://www.wikidata.org/wiki/Q7392545","display_name":"SPHERES","level":2,"score":0.7882490754127502},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7636605501174927},{"id":"https://openalex.org/C129782007","wikidata":"https://www.wikidata.org/wiki/Q162886","display_name":"Euclidean geometry","level":2,"score":0.6269649863243103},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.5674670934677124},{"id":"https://openalex.org/C186450821","wikidata":"https://www.wikidata.org/wiki/Q17295","display_name":"Euclidean space","level":2,"score":0.5241132974624634},{"id":"https://openalex.org/C2778572836","wikidata":"https://www.wikidata.org/wiki/Q380933","display_name":"Space (punctuation)","level":2,"score":0.4846973419189453},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.39572674036026},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.2672135531902313},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.09943202137947083},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.07838019728660583},{"id":"https://openalex.org/C1276947","wikidata":"https://www.wikidata.org/wiki/Q333","display_name":"Astronomy","level":1,"score":0.0},{"id":"https://openalex.org/C111919701","wikidata":"https://www.wikidata.org/wiki/Q9135","display_name":"Operating system","level":1,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1007/s00454-024-00628-y","is_oa":true,"landing_page_url":"http://dx.doi.org/10.1007/s00454-024-00628-y","pdf_url":"https://link.springer.com/content/pdf/10.1007/s00454-024-00628-y.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:repository.up.ac.za:2263/98964","is_oa":true,"landing_page_url":"http://hdl.handle.net/2263/98964","pdf_url":null,"source":{"id":"https://openalex.org/S4306401870","display_name":"UpSpace Institutional Repository (University of Pretoria)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I69552723","host_organization_name":"University of Pretoria","host_organization_lineage":["https://openalex.org/I69552723"],"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":"Article"}],"best_oa_location":{"id":"doi:10.1007/s00454-024-00628-y","is_oa":true,"landing_page_url":"http://dx.doi.org/10.1007/s00454-024-00628-y","pdf_url":"https://link.springer.com/content/pdf/10.1007/s00454-024-00628-y.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":[],"funders":[{"id":"https://openalex.org/F4320320692","display_name":"University of Pretoria","ror":"https://ror.org/00g0p6g84"}],"has_content":{"pdf":true,"grobid_xml":false},"content_urls":{"pdf":"https://content.openalex.org/works/W4392042250.pdf"},"referenced_works_count":15,"referenced_works":["https://openalex.org/W1534416612","https://openalex.org/W1993296756","https://openalex.org/W2007467834","https://openalex.org/W2070511479","https://openalex.org/W2486762503","https://openalex.org/W2754356277","https://openalex.org/W2892689348","https://openalex.org/W2950362087","https://openalex.org/W2962682143","https://openalex.org/W2979643164","https://openalex.org/W3165228583","https://openalex.org/W3210473564","https://openalex.org/W4292408570","https://openalex.org/W4327639789","https://openalex.org/W4390098957"],"related_works":["https://openalex.org/W2522712462","https://openalex.org/W4294294347","https://openalex.org/W2145396827","https://openalex.org/W2279239548","https://openalex.org/W115379796","https://openalex.org/W63976795","https://openalex.org/W2404658402","https://openalex.org/W2610298398","https://openalex.org/W4281992692","https://openalex.org/W2105228010"],"abstract_inverted_index":{"Abstract":[0],"For":[1,92,212],"$$d\\in":[2],"{\\mathbb":[3,94,239],"{N}}$$":[4,95,240],"<mml:math":[5,23,40,76,96,109,127,171,241,254,305],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">":[6,24,41,77,97,110,128,172,242,255,306],"<mml:mrow>":[7,26,43,79,98,111,129,146,174,243,256,308],"<mml:mi>d</mml:mi>":[8,29,46,82,99,112,177,244,257,311],"<mml:mo>\u2208</mml:mo>":[9,102,247],"<mml:mi>N</mml:mi>":[10,103,248],"</mml:mrow>":[11,28,45,81,104,117,150,154,176,249,262,310],"</mml:math>":[12,31,48,84,105,118,155,179,250,263,313],",":[13,119],"a":[14,33,65,164,213,220,266,290],"compact":[15,165,221,291,299],"sphere":[16,166,222,300,320],"packing":[17,59,167,223,292],"of":[18,35,57,64,73,123,161,168,186,195,198,206,209,273,276,287,302,319],"Euclidean":[19],"space":[20],"$${\\mathbb":[21,38,74,169,303],"{R}}^{d}$$":[22,39,75,170,304],"<mml:msup>":[25,42,78,173,307],"<mml:mi>R</mml:mi>":[27,44,80,175,309],"</mml:msup>":[30,47,83,178,312],"is":[32,60,293],"set":[34,230],"spheres":[36,162,200,288],"in":[37,163,289],"with":[49,106,251],"disjoint":[50],"interiors":[51],"so":[52],"that":[53,70,270,281],"the":[54,58,61,90,159,204,228,271,285,326],"contact":[55],"hypergraph":[56],"vertex":[62],"scheme":[63],"homogeneous":[66],"simplicial":[67],"d":[68],"-complex":[69],"covers":[71],"all":[72,274,298],".":[85],"We":[86,188,233],"are":[87,323],"motivated":[88],"by":[89],"question:":[91],"$$d,n\\in":[93,238],"<mml:mo>,</mml:mo>":[100,113,245,258],"<mml:mi>n</mml:mi>":[101,114,147,246,259],"$$d,n\\ge":[107,252],"2$$":[108,253],"<mml:mo>\u2265</mml:mo>":[115,260],"<mml:mn>2</mml:mn>":[116,261],"how":[120,219],"many":[121],"configurations":[122,275],"numbers":[124,280],"$$0&lt;r_{0}&lt;r_{1}&lt;\\cdots":[125],"&lt;r_{n-1}=1$$":[126],"<mml:mn>0</mml:mn>":[130,134],"<mml:mo>&lt;</mml:mo>":[131,136,141,143],"<mml:msub>":[132,137,144],"<mml:mi>r</mml:mi>":[133,138,145],"</mml:msub>":[135,140,151],"<mml:mn>1</mml:mn>":[139,149,153],"<mml:mo>\u22ef</mml:mo>":[142],"<mml:mo>-</mml:mo>":[148],"<mml:mo>=</mml:mo>":[152],"can":[156,282],"occur":[157,182,283],"as":[158,284],"radii":[160,286],"wherein":[180],"there":[181],"exactly":[183,316],"n":[184,277,317],"sizes":[185,318],"sphere?":[187],"introduce":[189],"what":[190],"we":[191,202,217],"call":[192],"\u2018heteroperturbative":[193],"sets\u2019":[194],"labeled":[196],"triangulations":[197],"unit":[199],"and":[201,264,321],"discuss":[203,218],"existence":[205],"non-trivial":[207],"examples":[208],"heteroperturbative":[210,215,229,268,328],"sets.":[211],"fixed":[214,267,327],"set,":[216,269],"may":[224],"be":[225],"associated":[226,324],"to":[227,235,325],"or":[231],"not.":[232],"proceed":[234],"show,":[236],"for":[237,265],"collection":[272],"distinct":[278],"positive":[279],"finite,":[294],"when":[295],"taken":[296],"over":[297],"packings":[301],"which":[314,322],"have":[315],"set.":[329]},"counts_by_year":[{"year":2023,"cited_by_count":1}],"updated_date":"2026-05-21T06:26:12.895304","created_date":"2025-10-10T00:00:00"}
