{"id":"https://openalex.org/W7225223732","doi":"https://doi.org/10.5281/zenodo.23938908","title":"Gauss-Bonnet Theorem: Unifying Curvature, Topology, and Polytope Combinatorics \u2014 E8 Intelligence Research","display_name":"Gauss-Bonnet Theorem: Unifying Curvature, Topology, and Polytope Combinatorics \u2014 E8 Intelligence Research","publication_year":2026,"publication_date":"2026-11-04","ids":{"openalex":"https://openalex.org/W7225223732","doi":"https://doi.org/10.5281/zenodo.23938908"},"language":"en","primary_location":{"id":"doi:10.5281/zenodo.23938908","doi":"https://doi.org/10.5281/zenodo.23938908","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.23938908","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","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"},"volume":null,"type":"preprint","indexed_in":["datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://doi.org/10.5281/zenodo.23938908","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5153285007","display_name":"Andrew Stewart Caldin","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Caldin, Andrew Stewart","raw_affiliation_strings":["Independent Researcher, United Kingdom"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Independent Researcher, United Kingdom","institution_ids":[]}]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5153285007"],"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/T12504","display_name":"Mathematics and Applications","score":0.483599990606308,"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":0.483599990606308,"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/T10304","display_name":"Geometric and Algebraic Topology","score":0.15189999341964722,"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":0.1281999945640564,"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/gauss-bonnet-theorem","display_name":"Gauss-Bonnet theorem","score":1.0},{"id":"https://openalex.org/keywords/euler-characteristic","display_name":"Euler characteristic","score":0.8389999866485596},{"id":"https://openalex.org/keywords/polyhedral-topology","display_name":"polyhedral topology","score":0.7149999737739563},{"id":"https://openalex.org/keywords/regular-polytopes","display_name":"regular polytopes","score":0.9520000219345093},{"id":"https://openalex.org/keywords/platonic-solids","display_name":"Platonic solids","score":0.906000018119812},{"id":"https://openalex.org/keywords/chern-gauss-bonnet-theorem","display_name":"Gauss-Bonnet-Chern theorem","score":0.9269999861717224},{"id":"https://openalex.org/keywords/pfaffian","display_name":"Pfaffian","score":0.8410000205039978},{"id":"https://openalex.org/keywords/topological-invariants","display_name":"topological invariants","score":0.6019999980926514},{"id":"https://openalex.org/keywords/e8","display_name":"E8","score":0.5839999914169312}],"concepts":[{"id":"https://openalex.org/C145691206","wikidata":"https://www.wikidata.org/wiki/Q747980","display_name":"Polytope","level":2,"score":0.7534999847412109},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6855000257492065},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.6072999835014343},{"id":"https://openalex.org/C2775956649","wikidata":"https://www.wikidata.org/wiki/Q1189744","display_name":"Pfaffian","level":2,"score":0.5849999785423279},{"id":"https://openalex.org/C80899671","wikidata":"https://www.wikidata.org/wiki/Q1304193","display_name":"Vertex (graph theory)","level":3,"score":0.5113999843597412},{"id":"https://openalex.org/C190470478","wikidata":"https://www.wikidata.org/wiki/Q2370229","display_name":"Invariant (physics)","level":2,"score":0.4494999945163727},{"id":"https://openalex.org/C105239961","wikidata":"https://www.wikidata.org/wiki/Q160003","display_name":"Tetrahedron","level":2,"score":0.44020000100135803},{"id":"https://openalex.org/C170315498","wikidata":"https://www.wikidata.org/wiki/Q5282039","display_name":"Discrete Morse theory","level":3,"score":0.43230000138282776},{"id":"https://openalex.org/C62884695","wikidata":"https://www.wikidata.org/wiki/Q184871","display_name":"Euler's formula","level":2,"score":0.4271000027656555},{"id":"https://openalex.org/C195065555","wikidata":"https://www.wikidata.org/wiki/Q214881","display_name":"Curvature","level":2,"score":0.4171999990940094},{"id":"https://openalex.org/C177148314","wikidata":"https://www.wikidata.org/wiki/Q170084","display_name":"Generalization","level":2,"score":0.41609999537467957},{"id":"https://openalex.org/C196433757","wikidata":"https://www.wikidata.org/wiki/Q852973","display_name":"Euler characteristic","level":2,"score":0.40299999713897705},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.33250001072883606},{"id":"https://openalex.org/C156135169","wikidata":"https://www.wikidata.org/wiki/Q5535522","display_name":"Geometry and topology","level":2,"score":0.3276999890804291},{"id":"https://openalex.org/C4107886","wikidata":"https://www.wikidata.org/wiki/Q906377","display_name":"Discrete geometry","level":2,"score":0.32429999113082886},{"id":"https://openalex.org/C184720557","wikidata":"https://www.wikidata.org/wiki/Q7825049","display_name":"Topology (electrical circuits)","level":2,"score":0.3005000054836273},{"id":"https://openalex.org/C181576044","wikidata":"https://www.wikidata.org/wiki/Q4129926","display_name":"Computational topology","level":3,"score":0.2969000041484833},{"id":"https://openalex.org/C54829058","wikidata":"https://www.wikidata.org/wiki/Q172937","display_name":"Polyhedron","level":2,"score":0.29319998621940613},{"id":"https://openalex.org/C88230418","wikidata":"https://www.wikidata.org/wiki/Q131476","display_name":"Graph theory","level":2,"score":0.2816999852657318},{"id":"https://openalex.org/C129621563","wikidata":"https://www.wikidata.org/wiki/Q429593","display_name":"Betti number","level":2,"score":0.2815000116825104},{"id":"https://openalex.org/C202652594","wikidata":"https://www.wikidata.org/wiki/Q2982669","display_name":"Tropical geometry","level":2,"score":0.2809000015258789},{"id":"https://openalex.org/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"score":0.2775000035762787},{"id":"https://openalex.org/C2779304628","wikidata":"https://www.wikidata.org/wiki/Q3503480","display_name":"Face (sociological concept)","level":2,"score":0.2720000147819519},{"id":"https://openalex.org/C110943637","wikidata":"https://www.wikidata.org/wiki/Q4129097","display_name":"Convex polytope","level":5,"score":0.26969999074935913},{"id":"https://openalex.org/C156340839","wikidata":"https://www.wikidata.org/wiki/Q2704791","display_name":"Enumeration","level":2,"score":0.2696000039577484},{"id":"https://openalex.org/C174248988","wikidata":"https://www.wikidata.org/wiki/Q8348314","display_name":"Polyhedral combinatorics","level":5,"score":0.2624000012874603},{"id":"https://openalex.org/C122041747","wikidata":"https://www.wikidata.org/wiki/Q838611","display_name":"Ball (mathematics)","level":2,"score":0.2615000009536743},{"id":"https://openalex.org/C110202963","wikidata":"https://www.wikidata.org/wiki/Q1783542","display_name":"Convex geometry","level":5,"score":0.26100000739097595}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.5281/zenodo.23938908","doi":"https://doi.org/10.5281/zenodo.23938908","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.23938908","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","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"},{"id":"doi:10.5281/zenodo.23938909","doi":"https://doi.org/10.5281/zenodo.23938909","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.23938909","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","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":"doi:10.5281/zenodo.23938908","doi":"https://doi.org/10.5281/zenodo.23938908","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.23938908","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","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W7225048850","https://openalex.org/W7202375548","https://openalex.org/W4230033328","https://openalex.org/W4403677749","https://openalex.org/W7052233445","https://openalex.org/W1990737004","https://openalex.org/W2900842205","https://openalex.org/W1839950852","https://openalex.org/W1623505430","https://openalex.org/W2750272506"],"abstract_inverted_index":{"FINDING:":[0],"The":[1,131],"Gauss-Bonnet":[2],"theorem":[3],"unifies":[4],"local":[5],"curvature":[6,127],"with":[7],"global":[8],"topology;":[9],"its":[10],"discrete":[11,138],"polyhedral":[12],"form":[13],"(Descartes'":[14],"defect-angle":[15],"theorem)":[16],"is":[17,122,136,145],"the":[18,123,126,137,141,176],"combinatorial":[19],"skeleton":[20],"underlying":[21],"all":[22,81,148],"regular":[23,156],"polytopes":[24],"and":[25,144],"their":[26],"higher-dimensional":[27],"analogues.":[28],"|":[29,129],"MATH:":[30],"Continuous":[31],"form:":[32,51],"\u222b_M":[33,44,115],"K":[34,45,96],"dA":[35,46,100,109],"+":[36,78],"\u222e_\u2202M":[37],"k_g":[38],"ds":[39],"=":[40,47,54,59,74,86,91,93,97,101,105,110,117],"2\u03c0\u03c7(M).":[41,48],"Closed":[42],"surface:":[43],"Discrete":[49],"(polyhedral)":[50],"\u03a3_v":[52],"\u03b4_v":[53,58],"2\u03c0\u03c7,":[55],"where":[56,120],"defect":[57,90,135],"2\u03c0":[60],"\u2212":[61,76],"\u03a3_i":[62],"\u03b8_i":[63],"(sum":[64],"of":[65,125,140,175],"face":[66],"angles":[67],"at":[68],"vertex":[69,158],"v).":[70],"Euler":[71],"characteristic":[72],"\u03c7":[73,85,104],"V":[75],"E":[77],"F.":[79],"For":[80,155],"five":[82],"Platonic":[83],"solids":[84],"2":[87],"\u2192":[88,107],"total":[89,133],"4\u03c0":[92],"720\u00b0.":[94],"Sphere:":[95],"1/R\u00b2,":[98],"\u222bK":[99,108],"4\u03c0.":[102],"Torus:":[103],"0":[106],"0.":[111],"Higher-dim":[112],"generalization":[113],"(Chern-Gauss-Bonnet):":[114],"Pf(\u03a9)":[116,121],"(2\u03c0)^n":[118],"\u03c7(M),":[119],"Pfaffian":[124],"2-form.":[128],"CONNECTION:":[130],"720\u00b0":[132],"angular":[134],"signature":[139],"2-sphere's":[142],"topology":[143],"invariant":[146],"under":[147],"subdivisions":[149],"\u2014":[150],"a":[151],"topological":[152],"conservation":[153],"law.":[154],"polytopes,":[157],"figures":[159],"obey":[160],"crystallographic":[161],"angle":[162],"constraints:":[163],"tetrahedron":[164],"(3":[165],"triang":[166],"Author:":[167],"Andrew":[168],"Stewart":[169],"Caldin,":[170],"Independent":[171],"Researcher,":[172],"UK.":[173],"Part":[174],"E8":[177],"Intelligence":[178],"Research":[179],"series.":[180],"Platform:":[181],"e8intelligence.com":[182]},"counts_by_year":[],"updated_date":"2026-11-06T12:05:46.655746","created_date":"2026-11-05T00:00:00"}
