{"id":"https://openalex.org/W7221816618","doi":"https://doi.org/10.48550/arxiv.2610.13950","title":"Platonic geodesic trees in the $K-$Space","display_name":"Platonic geodesic trees in the $K-$Space","publication_year":2026,"publication_date":"2026-08-05","ids":{"openalex":"https://openalex.org/W7221816618","doi":"https://doi.org/10.48550/arxiv.2610.13950"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2610.13950","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.13950","pdf_url":"https://arxiv.org/pdf/2610.13950","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"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"},"volume":null,"type":"preprint","is_traditional":true,"indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2610.13950","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5044127481","display_name":"Anastasios N. Zachos","orcid":"https://orcid.org/0000-0002-7137-3096","observed_orcids":["https://orcid.org/0000-0002-7137-3096"]},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Zachos, Anastasios N.","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5044127481"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"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/T12504","display_name":"Mathematics and Applications","score":0.9199000000953674,"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":0.9199000000953674,"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/T13080","display_name":"Advanced Differential Geometry Research","score":0.026900000870227814,"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/T10229","display_name":"Geometric Analysis and Curvature Flows","score":0.01850000023841858,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/k-space","display_name":"k-space","score":0.7049999833106995},{"id":"https://openalex.org/keywords/steiner-points","display_name":"Steiner points","score":0.7860000133514404},{"id":"https://openalex.org/keywords/fermat-point","display_name":"Fermat point","score":0.9449999928474426},{"id":"https://openalex.org/keywords/cartesian-coordinates","display_name":"Cartesian coordinates","score":0.9490000009536743}],"concepts":[{"id":"https://openalex.org/C165818556","wikidata":"https://www.wikidata.org/wiki/Q213488","display_name":"Geodesic","level":2,"score":0.7940000295639038},{"id":"https://openalex.org/C80899671","wikidata":"https://www.wikidata.org/wiki/Q1304193","display_name":"Vertex (graph theory)","level":3,"score":0.7634000182151794},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6875},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.6069999933242798},{"id":"https://openalex.org/C113174947","wikidata":"https://www.wikidata.org/wiki/Q2859736","display_name":"Tree (set theory)","level":2,"score":0.5041999816894531},{"id":"https://openalex.org/C62354387","wikidata":"https://www.wikidata.org/wiki/Q875399","display_name":"Boundary (topology)","level":2,"score":0.5031999945640564},{"id":"https://openalex.org/C162269530","wikidata":"https://www.wikidata.org/wiki/Q17100560","display_name":"Geodesic map","level":3,"score":0.49549999833106995},{"id":"https://openalex.org/C2780009758","wikidata":"https://www.wikidata.org/wiki/Q6804172","display_name":"Measure (data warehouse)","level":2,"score":0.4726000130176544},{"id":"https://openalex.org/C87670640","wikidata":"https://www.wikidata.org/wiki/Q132469","display_name":"Fermat's Last Theorem","level":2,"score":0.43639999628067017},{"id":"https://openalex.org/C80551277","wikidata":"https://www.wikidata.org/wiki/Q11210","display_name":"Coordinate system","level":2,"score":0.39649999141693115},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.2793999910354614}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2610.13950","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.13950","pdf_url":"https://arxiv.org/pdf/2610.13950","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"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"},{"id":"doi:10.48550/arxiv.2610.13950","doi":"https://doi.org/10.48550/arxiv.2610.13950","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2610.13950","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2610.13950","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.13950","pdf_url":"https://arxiv.org/pdf/2610.13950","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"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":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7221816618.pdf","grobid_xml":"https://content.openalex.org/works/W7221816618.grobid-xml"},"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W7099850952","https://openalex.org/W2513684333","https://openalex.org/W4225921218","https://openalex.org/W7214999321","https://openalex.org/W7017036041","https://openalex.org/W204824005","https://openalex.org/W7161260877","https://openalex.org/W52154682","https://openalex.org/W7214379933","https://openalex.org/W3125327031"],"abstract_inverted_index":{"In":[0],"this":[1],"paper,":[2],"We":[3],"prove":[4],"that:":[5],"(A)":[6],"If":[7,43,70,98,135,166],"$m\\ge":[8,136,167],"4,$":[9,105],"there":[10,45,72,106,138,169],"exists":[11,46,73,107],"a":[12,38,47,74,108,130,142,161,173,206],"Platonic":[13,48,75,109,143,174],"geodesic":[14,49,67,110,144,175,214],"tree":[15,50,76,111,145,176],"having":[16,87,112,146,177],"$m$":[17,23,99,113,147,178],"edges":[18,114,148],"(geodesic":[19],"segments),":[20],"which":[21,57,90,116,150,180,239],"form":[22,117,151,181],"platonic":[24,31,118,153,182,213],"angles":[25,32,119,154,183],"with":[26,33,51,83,120,155,184,216],"measure":[27,34,156,187],"$120^{o}$":[28,157],"and":[29,54,77,102,126,194,201,221],"$(\\frac{m(m-1)}{2}-m)$":[30],"$90^{o}.$":[35],"(Existence":[36,128],"of":[37,129,160,205,212],"Fermat":[39,85,162,207],"Coordinate":[40],"system)":[41],"(B)":[42],"$m=2,$":[44],"three":[52,88],"vertices":[53],"2":[55],"edges,":[56,89,179],"meet":[58,91],"at":[59,92],"$120^{\\circ}$":[60,93,193],"or":[61,63,189,196],"$150^{\\circ}$":[62,195,202],"$180^{\\circ}$":[64],"(Minimal":[65],"spanning":[66],"tree).":[68],"(C)":[69],"$m=3,$":[71],"the":[78,84,227],"Steiner":[79],"point":[80],"is":[81,100,229],"identical":[82],"point,":[86],"(Hexagonal":[94],"coordinate":[95,132,163,208],"system).":[96,133,164,209],"(D)":[97],"even":[101],"$m":[103],"\\ge":[104],",":[115,149],"only":[121,152,185],"two":[122,190,197],"distinct":[123,191,198],"measures":[124,192,199],"$90^{o}$":[125],"$180^{\\circ}.$":[127],"Cartesian":[131],"(E)":[134],"4$,":[137],"does":[139,170],"not":[140,171],"exist":[141,172],"(Non":[158,203],"existence":[159,204],"(F)":[165],"3,$":[168],"one":[186],"$150^{o},$":[188],"$90^{\\circ}$":[200],"A":[210],"classification":[211],"trees":[215],"boundary":[217],"vertex":[218,223,242],"set":[219,224],"$V$":[220],"added":[222],"$Q$":[225],"in":[226,243],"$K-$Space":[228],"given":[230],"by":[231],"conditions":[232],"(A),":[233],"(B),":[234],"(C),":[235],"(D),":[236],"(E),":[237],"(F),":[238],"characterize":[240],"each":[241],"$V\\cup":[244],"Q.$":[245]},"counts_by_year":[],"updated_date":"2026-10-28T10:33:24.094632","created_date":"2026-10-15T00:00:00"}
