{"id":"https://openalex.org/W7225705740","doi":"https://doi.org/10.5281/zenodo.24122019","title":"DICTO 1:2 Truncated Octahedron: The 4D Vacuus of the Permutohedron","display_name":"DICTO 1:2 Truncated Octahedron: The 4D Vacuus of the Permutohedron","publication_year":2026,"publication_date":"2026-11-11","ids":{"openalex":"https://openalex.org/W7225705740","doi":"https://doi.org/10.5281/zenodo.24122019"},"language":"en","primary_location":{"id":"doi:10.5281/zenodo.24122019","doi":"https://doi.org/10.5281/zenodo.24122019","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.24122019","pdf_url":null,"source":{"id":"https://openalex.org/S4306400562","display_name":"Zenodo (CERN European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by-sa","license_id":"https://openalex.org/licenses/cc-by-sa","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"},"volume":null,"type":"preprint","is_traditional":true,"indexed_in":["datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://doi.org/10.5281/zenodo.24122019","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5158495986","display_name":"DICTO","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"DICTO","raw_affiliation_strings":["Independent artist, Osaka, Japan"],"raw_orcid":"https://orcid.org/0009-0001-2620-2036","affiliations":[{"raw_affiliation_string":"Independent artist, Osaka, Japan","institution_ids":[]}]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5158495986"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":false,"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/T10948","display_name":"Advanced Combinatorial Mathematics","score":0.9955999851226807,"subfield":{"id":"https://openalex.org/subfields/2607","display_name":"Discrete Mathematics and Combinatorics"},"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/T10948","display_name":"Advanced Combinatorial Mathematics","score":0.9955999851226807,"subfield":{"id":"https://openalex.org/subfields/2607","display_name":"Discrete Mathematics and Combinatorics"},"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/T12504","display_name":"Mathematics and Applications","score":0.0026000000070780516,"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/T10849","display_name":"Finite Group Theory Research","score":0.00019999999494757503,"subfield":{"id":"https://openalex.org/subfields/2607","display_name":"Discrete Mathematics and Combinatorics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[],"concepts":[{"id":"https://openalex.org/C68693459","wikidata":"https://www.wikidata.org/wiki/Q657586","display_name":"Hyperplane","level":2,"score":0.48890000581741333},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.4830999970436096},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.45680001378059387},{"id":"https://openalex.org/C182306322","wikidata":"https://www.wikidata.org/wiki/Q1779371","display_name":"Order (exchange)","level":2,"score":0.45660001039505005},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.37220001220703125},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.35910001397132874},{"id":"https://openalex.org/C149364088","wikidata":"https://www.wikidata.org/wiki/Q185917","display_name":"Translation (biology)","level":4,"score":0.3569999933242798},{"id":"https://openalex.org/C51364203","wikidata":"https://www.wikidata.org/wiki/Q1563532","display_name":"HERO","level":2,"score":0.3278999924659729},{"id":"https://openalex.org/C129782007","wikidata":"https://www.wikidata.org/wiki/Q162886","display_name":"Euclidean geometry","level":2,"score":0.3041999936103821},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.29910001158714294},{"id":"https://openalex.org/C2776291640","wikidata":"https://www.wikidata.org/wiki/Q2912517","display_name":"Value (mathematics)","level":2,"score":0.28049999475479126},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.2709999978542328},{"id":"https://openalex.org/C33676613","wikidata":"https://www.wikidata.org/wiki/Q13415176","display_name":"Dimension (graph theory)","level":2,"score":0.2678999900817871},{"id":"https://openalex.org/C138885662","wikidata":"https://www.wikidata.org/wiki/Q5891","display_name":"Philosophy","level":0,"score":0.25760000944137573},{"id":"https://openalex.org/C2987642246","wikidata":"https://www.wikidata.org/wiki/Q5452196","display_name":"First order","level":2,"score":0.25609999895095825}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.5281/zenodo.24122019","doi":"https://doi.org/10.5281/zenodo.24122019","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.24122019","pdf_url":null,"source":{"id":"https://openalex.org/S4306400562","display_name":"Zenodo (CERN European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by-sa","license_id":"https://openalex.org/licenses/cc-by-sa","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"doi:10.5281/zenodo.24122019","doi":"https://doi.org/10.5281/zenodo.24122019","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.24122019","pdf_url":null,"source":{"id":"https://openalex.org/S4306400562","display_name":"Zenodo (CERN European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by-sa","license_id":"https://openalex.org/licenses/cc-by-sa","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":4,"referenced_works":["https://openalex.org/W3165960980","https://openalex.org/W7221765657","https://openalex.org/W7223099137","https://openalex.org/W7223562089"],"related_works":[],"abstract_inverted_index":{"Version":[0],"2":[1,142,179],"adds":[2],"the":[3,7,18,25,29,39,64,74,77,84,123,128,153,163,166,203,216,220,229],"hero":[4],"shots":[5],"from":[6],"page":[8],"The":[9,95,118,157,188],"4D":[10,78,81],"Permutohedron":[11],"(cells":[12],"pulled":[13],"apart),":[14],"a":[15,149,183],"figure":[16],"of":[17,24,28,66,76,86,134,159,165,168,193,205,252],"orderings":[19],"on":[20],"every":[21,51],"corner,":[22],"cutaways":[23],"heart":[26],"and":[27,37,53,70,141,256,264,268],"DICTO":[30,129,221,230],"1:2":[31,130,222],"Truncated":[32,131,223,232],"Octahedron's":[33],"own":[34],"3D":[35],"Vacuus,":[36],"names":[38],"cells":[40],"as":[41],"permutohedra.":[42],"No":[43],"result":[44],"changes.":[45],"Write":[46],"1":[47,140],"to":[48,180],"5":[49],"in":[50,100,105,143,148],"order":[52],"link":[54],"two":[55,61,146],"orders":[56],"that":[57,89,121],"differ":[58],"by":[59,248],"exchanging":[60],"neighbouring":[62],"values:":[63],"web":[65],"exchange,":[67],"120":[68],"knots":[69],"240":[71],"exchanges,":[72],"is":[73,83,228],"skeleton":[75],"permutohedron.":[79],"Its":[80],"Vacuus":[82],"set":[85],"central":[87,103,160],"slices":[88,120],"carry":[90],"its":[91],"faces.":[92],"Theorem":[93,116,155,186],"1.":[94],"150":[96],"two-dimensional":[97],"faces":[98],"lie":[99],"exactly":[101],"65":[102],"slices,":[104],"four":[106,206,253],"congruence":[107],"classes":[108],"(30,":[109],"20,":[110],"10,":[111],"5),":[112],"each":[113],"described":[114],"exactly.":[115],"2.":[117],"five":[119],"hold":[122],"middle":[124],"value":[125],"fixed":[126],"are":[127,202],"Octahedron":[132],"(squares":[133],"side":[135],"1,":[136],"hexagons":[137],"with":[138,182,270],"sides":[139],"turn);":[144],"any":[145],"cross":[147],"regular":[150],"hexagon":[151],"through":[152,162,219],"centre.":[154],"3.":[156],"number":[158],"hyperplanes":[161],"ridges":[164],"permutohedron":[167],"1..n:":[169],"3,":[170],"15,":[171],"65,":[172],"225,":[173],"770,":[174],"2443,":[175],"7761":[176],"for":[177],"dimensions":[178],"8,":[181],"closed":[184],"formula.":[185],"4.":[187],"semi-uniform":[189],"truncated":[190],"octahedra":[191],"T(x)":[192],"DICTONIC's":[194],"golden":[195],"bond":[196],"(Koca":[197],"et":[198],"al.":[199],"2015;":[200],"#48)":[201],"permutohedra":[204],"values":[207],"placed":[208],"symmetrically":[209],"about":[210],"zero;":[211],"at":[212],"x":[213],"=":[214],"1/2":[215],"slider":[217],"passes":[218],"Octahedron,":[224],"whose":[225],"space-filling":[226],"partner":[227],"2:1":[231],"Octahedron.":[233],"Classical":[234],"parts":[235],"credited":[236],"(the":[237],"weak":[238],"order,":[239],"reduced":[240],"words,":[241],"majorization,":[242],"translation":[243],"tilings).":[244],"Checked":[245],"before":[246],"release":[247],"an":[249],"in-house":[250],"panel":[251],"automated":[254],"referees":[255],"one":[257],"independent":[258],"attack.":[259],"DICTO:":[260],"ideas,":[261],"names,":[262],"constructions":[263],"direction.":[265],"Computation,":[266],"verification":[267],"drafting":[269],"Claude":[271],"(Anthropic).":[272]},"counts_by_year":[],"updated_date":"2026-11-11T12:16:26.678958","created_date":"2026-11-11T00:00:00"}