@prefix dbo: .
@prefix dbr: .
dbr:Edmund_Husserl dbo:wikiPageWikiLink dbr:Non-Euclidean_geometry .
dbo:wikiPageWikiLink dbr:Non-Euclidean_geometry .
@prefix rdfs: .
dbr:Omar_Khayyam rdfs:seeAlso dbr:Non-Euclidean_geometry ;
dbo:wikiPageWikiLink dbr:Non-Euclidean_geometry .
dbr:Relativity_priority_dispute dbo:wikiPageWikiLink dbr:Non-Euclidean_geometry .
dbo:wikiPageWikiLink dbr:Non-Euclidean_geometry .
dbo:wikiPageWikiLink dbr:Non-Euclidean_geometry .
dbr:Timeline_of_geometry dbo:wikiPageWikiLink dbr:Non-Euclidean_geometry .
dbr:La_Femme_au_Cheval dbo:wikiPageWikiLink dbr:Non-Euclidean_geometry .
dbr:The_Dreams_in_the_Witch_House dbo:wikiPageWikiLink dbr:Non-Euclidean_geometry .
dbr:Mathematics_of_general_relativity dbo:wikiPageWikiLink dbr:Non-Euclidean_geometry .
dbr:Philosophical_Problems_of_Space_and_Time dbo:wikiPageWikiLink dbr:Non-Euclidean_geometry .
dbr:Wormholes_in_fiction dbo:wikiPageWikiLink dbr:Non-Euclidean_geometry .
dbr:Relationship_between_mathematics_and_physics dbo:wikiPageWikiLink dbr:Non-Euclidean_geometry .
dbo:wikiPageWikiLink dbr:Non-Euclidean_geometry .
dbo:wikiPageWikiLink dbr:Non-Euclidean_geometry .
dbr:List_of_geometers dbo:wikiPageWikiLink dbr:Non-Euclidean_geometry .
dbr:Geometry_of_Complex_Numbers dbo:wikiPageWikiLink dbr:Non-Euclidean_geometry .
dbr:Timeline_of_manifolds dbo:wikiPageWikiLink dbr:Non-Euclidean_geometry .
dbr:Timeline_of_mathematics dbo:wikiPageWikiLink dbr:Non-Euclidean_geometry .
dbo:wikiPageWikiLink dbr:Non-Euclidean_geometry .
dbr:Mathematics_in_Nazi_Germany dbo:wikiPageWikiLink dbr:Non-Euclidean_geometry .
dbo:wikiPageWikiLink dbr:Non-Euclidean_geometry .
dbr:Hyperbolic_sector dbo:wikiPageWikiLink dbr:Non-Euclidean_geometry .
dbr:Arthur_Buchheim dbo:wikiPageWikiLink dbr:Non-Euclidean_geometry .
dbr:Victor_Schlegel dbo:wikiPageWikiLink dbr:Non-Euclidean_geometry .
dbr:Unifying_theories_in_mathematics dbo:wikiPageWikiLink dbr:Non-Euclidean_geometry .
dbr:John_Wesley_Young dbo:wikiPageWikiLink dbr:Non-Euclidean_geometry .
dbr:Margaret_Wertheim dbo:wikiPageWikiLink dbr:Non-Euclidean_geometry .
dbr:Riemannian_geometry dbo:wikiPageWikiLink dbr:Non-Euclidean_geometry .
dbr:Non-Euclidean_surface_growth dbo:wikiPageWikiLink dbr:Non-Euclidean_geometry .
dbo:wikiPageWikiLink dbr:Non-Euclidean_geometry .
dbr:Laguerre_transformations dbo:wikiPageWikiLink dbr:Non-Euclidean_geometry .
dbr:Euclidean dbo:wikiPageWikiLink dbr:Non-Euclidean_geometry ;
dbo:wikiPageDisambiguates dbr:Non-Euclidean_geometry .
dbr:List_of_mathematics_history_topics dbo:wikiPageWikiLink dbr:Non-Euclidean_geometry .
dbr:Non-Euclidean dbo:wikiPageWikiLink dbr:Non-Euclidean_geometry ;
dbo:wikiPageRedirects dbr:Non-Euclidean_geometry .
dbr:Metamathematics rdfs:seeAlso dbr:Non-Euclidean_geometry .
dbr:Hyperbolic_motion dbo:wikiPageWikiLink dbr:Non-Euclidean_geometry .
dbr:Erlangen_program dbo:wikiPageWikiLink dbr:Non-Euclidean_geometry .
dbo:wikiPageWikiLink dbr:Non-Euclidean_geometry .
@prefix rdf: .
@prefix yago: .
dbr:Non-Euclidean_geometry rdf:type yago:YagoLegalActorGeo ,
yago:YagoGeoEntity ,
yago:YagoPermanentlyLocatedEntity ,
yago:PhysicalEntity100001930 .
@prefix owl: .
dbr:Non-Euclidean_geometry rdf:type owl:Thing ,
yago:Object100002684 ,
yago:Location100027167 ,
yago:Tract108673395 ,
yago:Region108630985 ,
yago:GeographicalArea108574314 ,
yago:WikicatFieldsOfMathematics ,
yago:Field108569998 ;
rdfs:label "\u0647\u0646\u062F\u0633\u0629 \u0644\u0627\u0625\u0642\u0644\u064A\u062F\u064A\u0629"@ar ,
"Geometria no euclidiana"@ca ,
"Icke-euklidisk geometri"@sv ,
"Niet-euclidische meetkunde"@nl ,
"Neeukleidovsk\u00E1 geometrie"@cs ,
"\u041D\u0435\u0435\u0432\u043A\u043B\u0438\u0434\u043E\u0432\u0430 \u0433\u0435\u043E\u043C\u0435\u0442\u0440\u0438\u044F"@ru ,
"Geometria ez-euklidear"@eu ,
"Nee\u016Dklidaj geometrioj"@eo ,
"\u039C\u03B7 \u03B5\u03C5\u03BA\u03BB\u03B5\u03AF\u03B4\u03B5\u03B9\u03B5\u03C2 \u03B3\u03B5\u03C9\u03BC\u03B5\u03C4\u03C1\u03AF\u03B5\u03C2"@el ,
"Geometria nieeuklidesowa"@pl ,
"Geoim\u00E9adrachta\u00ED neamh-Eoicl\u00EDd\u00E9acha"@ga ,
"\u041D\u0435\u0435\u0432\u043A\u043B\u0456\u0434\u043E\u0432\u0430 \u0433\u0435\u043E\u043C\u0435\u0442\u0440\u0456\u044F"@uk ,
"Non-Euclidean geometry"@en ,
"G\u00E9om\u00E9trie non euclidienne"@fr ,
"\u975E\u30E6\u30FC\u30AF\u30EA\u30C3\u30C9\u5E7E\u4F55\u5B66"@ja ,
"\uBE44\uC720\uD074\uB9AC\uB4DC \uAE30\uD558\uD559"@ko ,
"Nichteuklidische Geometrie"@de ,
"Geometria n\u00E3o euclidiana"@pt ,
"Geometria non euclidea"@it ,
"\u975E\u6B27\u51E0\u91CC\u5F97\u51E0\u4F55"@zh ,
"Geometr\u00EDa no euclidiana"@es ,
"Geometri non-Euklides"@in ;
rdfs:comment "Niet-euclidische meetkunde is meetkunde waarbij het vijfde postulaat van Euclides (het parallellenpostulaat) niet wordt aangenomen. Euclides ging bij zijn meetkunde uit van een aantal postulaten (axioma's). De meeste daarvan zijn eenvoudig, maar het vijfde vormt een uitzondering. Het postulaat heeft diverse vormen, maar de bekendste is waarschijnlijk \"Gegeven een rechte l en een punt P dat niet op l ligt, dan is er in het vlak door l en P maar \u00E9\u00E9n rechte door P die l niet snijdt.\" (Euclides' oorspronkelijke vorm was gecompliceerder.) Er zijn twee typen niet-euclidische meetkunde:"@nl ,
"\u041D\u0435\u0435\u0432\u043A\u043B\u0438\u0434\u043E\u0432\u0430 \u0433\u0435\u043E\u043C\u0435\u0442\u0440\u0438\u044F \u2014 \u0432 \u0431\u0443\u043A\u0432\u0430\u043B\u044C\u043D\u043E\u043C \u043F\u043E\u043D\u0438\u043C\u0430\u043D\u0438\u0438 \u2014 \u043B\u044E\u0431\u0430\u044F \u0433\u0435\u043E\u043C\u0435\u0442\u0440\u0438\u0447\u0435\u0441\u043A\u0430\u044F \u0441\u0438\u0441\u0442\u0435\u043C\u0430, \u043A\u043E\u0442\u043E\u0440\u0430\u044F \u043E\u0442\u043B\u0438\u0447\u0430\u0435\u0442\u0441\u044F \u043E\u0442 \u0433\u0435\u043E\u043C\u0435\u0442\u0440\u0438\u0438 \u0415\u0432\u043A\u043B\u0438\u0434\u0430; \u043E\u0434\u043D\u0430\u043A\u043E \u0442\u0440\u0430\u0434\u0438\u0446\u0438\u043E\u043D\u043D\u043E \u0442\u0435\u0440\u043C\u0438\u043D \u00AB\u043D\u0435\u0435\u0432\u043A\u043B\u0438\u0434\u043E\u0432\u0430 \u0433\u0435\u043E\u043C\u0435\u0442\u0440\u0438\u044F\u00BB \u043F\u0440\u0438\u043C\u0435\u043D\u044F\u0435\u0442\u0441\u044F \u0432 \u0431\u043E\u043B\u0435\u0435 \u0443\u0437\u043A\u043E\u043C \u0441\u043C\u044B\u0441\u043B\u0435 \u0438 \u043E\u0442\u043D\u043E\u0441\u0438\u0442\u0441\u044F \u0442\u043E\u043B\u044C\u043A\u043E \u043A \u0434\u0432\u0443\u043C \u0433\u0435\u043E\u043C\u0435\u0442\u0440\u0438\u0447\u0435\u0441\u043A\u0438\u043C \u0441\u0438\u0441\u0442\u0435\u043C\u0430\u043C: \u0433\u0435\u043E\u043C\u0435\u0442\u0440\u0438\u0438 \u041B\u043E\u0431\u0430\u0447\u0435\u0432\u0441\u043A\u043E\u0433\u043E \u0438 \u0441\u0444\u0435\u0440\u0438\u0447\u0435\u0441\u043A\u043E\u0439 \u0433\u0435\u043E\u043C\u0435\u0442\u0440\u0438\u0438 (\u0438\u043B\u0438 \u0441\u0445\u043E\u0436\u0435\u0439 \u0441 \u043D\u0435\u0439 \u0433\u0435\u043E\u043C\u0435\u0442\u0440\u0438\u0438 \u0420\u0438\u043C\u0430\u043D\u0430)."@ru ,
"Geoim\u00E9adrachta\u00ED a saothra\u00EDodh \u00F3n 18\u00FA c\u00E9ad anuas, bunaithe ar 5\u00FA aics\u00EDm Eoicl\u00EDd\u00E9is a thr\u00E9igean ar bheala\u00ED \u00E9ags\u00FAla. Ba \u00ED an aics\u00EDm sin nach f\u00E9idir ach l\u00EDne dh\u00EDreach amh\u00E1in a tharraingt tr\u00ED phointe ar leith at\u00E1 comhthreomhar le l\u00EDne dh\u00EDreach ar leith. Mar shampla, m\u00E1 ghlactar mar aics\u00EDm nua ina hionad sin gur f\u00E9idir n\u00EDos m\u00F3 n\u00E1 l\u00EDne dh\u00EDreach amh\u00E1in a tharraingt tr\u00ED phointe ar leith at\u00E1 comhthreomhar le l\u00EDne dh\u00EDreach ar leith, forbra\u00EDtear geoim\u00E9adrachta\u00ED hipearb\u00F3ileacha, mar at\u00E1 d\u00E9anta ag Gau\u00DF is Lobachevsky. Ba \u00ED seo an ch\u00E9ad gheoim\u00E9adracht fhisici\u00FAil shochreidte mar mhalairt ar gheoim\u00E9adracht Eoicl\u00EDd\u00E9is. Mar shampla eile, m\u00E1 ghlactar mar aisc\u00EDm eile nach f\u00E9idir aon l\u00EDne dh\u00EDreach a tharraingt tr\u00ED phointe ar leith at\u00E1 comhthreomhar le l\u00EDne dh\u00EDreach ar leith, forbra\u00EDtear geoim\u00E9adracht \u00E9ilipseach"@ga ,
"In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those that specify Euclidean geometry. As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean geometry arises by either replacing the parallel postulate with an alternative, or relaxing the metric requirement. In the former case, one obtains hyperbolic geometry and elliptic geometry, the traditional non-Euclidean geometries. When the metric requirement is relaxed, then there are affine planes associated with the , which give rise to that have also been called non-Euclidean geometry."@en ,
"Se denomina geometr\u00EDa no euclidiana o no eucl\u00EDdea, a cualquier sistema formal de geometr\u00EDa cuyos postulados y proposiciones difieren en alg\u00FAn asunto de los establecidos por Euclides en su tratado Elementos. No existe un solo sistema de geometr\u00EDa no eucl\u00EDdea, sino muchos, aunque si se restringe la discusi\u00F3n a espacios homog\u00E9neos, en los que la curvatura del espacio es la misma en cada punto, en los que los puntos del espacio son indistinguibles, pueden distinguirse tres formulaciones\u200B de geometr\u00EDas:"@es ,
"Na matem\u00E1tica, uma geometria n\u00E3o euclidiana \u00E9 uma geometria baseada num sistema axiom\u00E1tico distinto da geometria euclidiana. Modificando o axioma das paralelas, que postula que por um ponto exterior a uma reta passa exatamente uma reta paralela \u00E0 inicial, obt\u00EAm-se as geometrias el\u00EDptica e hiperb\u00F3lica (geometria de Lobachevsky). Na geometria el\u00EDptica n\u00E3o h\u00E1 nenhuma reta paralela \u00E0 inicial, enquanto que na geometria hiperb\u00F3lica existe uma infinidade de rectas paralelas \u00E0 inicial que passam no mesmo ponto. Na geometria el\u00EDptica a soma dos \u00E2ngulos internos de um triangulo \u00E9 maior que dois \u00E2ngulos retos, enquanto na geometria hiperb\u00F3lica esta soma \u00E9 menor que dois \u00E2ngulos retos. Na el\u00EDptica, temos que a circunfer\u00EAncia de um c\u00EDrculo \u00E9 menor do que PI vezes o seu di\u00E2metro, enquanto na hiperb\u00F3lica"@pt ,
"La g\u00E9om\u00E9trie non euclidienne (GNE) est, en math\u00E9matiques, une th\u00E9orie g\u00E9om\u00E9trique ayant recours aux axiomes et postulats pos\u00E9s par Euclide dans les \u00C9l\u00E9ments, sauf le postulat des parall\u00E8les. Les diff\u00E9rentes g\u00E9om\u00E9tries non euclidiennes sont issues de la volont\u00E9 de d\u00E9montrer le cinqui\u00E8me postulat (le postulat d'Euclide, qui semble peu satisfaisant car trop complexe \u00E9tait peut-\u00EAtre redondant)."@fr ,
"\u041D\u0435\u0435\u0432\u043A\u043B\u0456\u0434\u043E\u0432\u0430 \u0433\u0435\u043E\u043C\u0435\u0442\u0440\u0456\u044F \u2014 \u0443 \u0431\u0443\u043A\u0432\u0430\u043B\u044C\u043D\u043E\u043C\u0443 \u0440\u043E\u0437\u0443\u043C\u0456\u043D\u043D\u0456 \u2014 \u0431\u0443\u0434\u044C-\u044F\u043A\u0430 \u0433\u0435\u043E\u043C\u0435\u0442\u0440\u0438\u0447\u043D\u0430 \u0441\u0438\u0441\u0442\u0435\u043C\u0430, \u0432\u0456\u0434\u043C\u0456\u043D\u043D\u0430 \u0432\u0456\u0434 \u0433\u0435\u043E\u043C\u0435\u0442\u0440\u0456\u0457 \u0415\u0432\u043A\u043B\u0456\u0434\u0430; \u043F\u0440\u043E\u0442\u0435 \u0442\u0440\u0430\u0434\u0438\u0446\u0456\u0439\u043D\u043E \u0442\u0435\u0440\u043C\u0456\u043D \u00AB\u041D\u0435\u0435\u0432\u043A\u043B\u0456\u0434\u043E\u0432\u0430 \u0433\u0435\u043E\u043C\u0435\u0442\u0440\u0456\u044F\u00BB \u0437\u0430\u0441\u0442\u043E\u0441\u043E\u0432\u0443\u0454\u0442\u044C\u0441\u044F \u0443 \u0432\u0443\u0436\u0447\u043E\u043C\u0443 \u0441\u0435\u043D\u0441\u0456 \u0439 \u0441\u0442\u043E\u0441\u0443\u0454\u0442\u044C\u0441\u044F \u043B\u0438\u0448\u0435 \u0434\u0432\u043E\u0445 \u0433\u0435\u043E\u043C\u0435\u0442\u0440\u0438\u0447\u043D\u0438\u0445 \u0441\u0438\u0441\u0442\u0435\u043C: \u0433\u0456\u043F\u0435\u0440\u0431\u043E\u043B\u0456\u0447\u043D\u043E\u0457 \u0433\u0435\u043E\u043C\u0435\u0442\u0440\u0456\u0457 \u0439 \u0441\u0444\u0435\u0440\u0438\u0447\u043D\u043E\u0457 \u0433\u0435\u043E\u043C\u0435\u0442\u0440\u0456\u0457. \u042F\u043A \u0456 \u0435\u0432\u043A\u043B\u0456\u0434\u043E\u0432\u0430 \u0446\u0456 \u0433\u0435\u043E\u043C\u0435\u0442\u0440\u0456\u0457 \u043D\u0430\u043B\u0435\u0436\u0430\u0442\u044C \u0434\u043E \u043C\u0435\u0442\u0440\u0438\u0447\u043D\u0438\u0445 \u0433\u0435\u043E\u043C\u0435\u0442\u0440\u0456\u0439 \u0442\u0440\u0438\u0432\u0438\u043C\u0456\u0440\u043D\u043E\u0433\u043E \u043F\u0440\u043E\u0441\u0442\u043E\u0440\u0443 \u043F\u043E\u0441\u0442\u0456\u0439\u043D\u043E\u0457 \u0441\u0435\u043A\u0446\u0456\u0439\u043D\u043E\u0457 \u043A\u0440\u0438\u0432\u0438\u043D\u0438. \u041D\u0443\u043B\u044C\u043E\u0432\u0430 \u043A\u0440\u0438\u0432\u0438\u043D\u0430 \u0432\u0456\u0434\u043F\u043E\u0432\u0456\u0434\u0430\u0454 \u0435\u0432\u043A\u043B\u0456\u0434\u043E\u0432\u0456\u0439 \u0433\u0435\u043E\u043C\u0435\u0442\u0440\u0456\u0457, \u0434\u043E\u0434\u0430\u0442\u043D\u0430 \u2014 \u0441\u0444\u0435\u0440\u0438\u0447\u043D\u0456\u0439, \u0432\u0456\u0434'\u0454\u043C\u043D\u0430 \u2014 \u0433\u0456\u043F\u0435\u0440\u0431\u043E\u043B\u0456\u0447\u043D\u0456\u0439 \u0433\u0435\u043E\u043C\u0435\u0442\u0440\u0456\u0457."@uk ,
"Una geometria non euclidea \u00E8 una geometria costruita negando o non accettando alcuni postulati euclidei.Viene detta anche metageometria."@it ,
"Neeukleidovsk\u00E1 geometrie je obecn\u00E9 ozna\u010Den\u00ED pro takov\u00E9 geometrie (tj. syst\u00E9my spl\u0148uj\u00EDc\u00ED prvn\u00ED \u010Dty\u0159i Eukleidovy postul\u00E1ty), kter\u00E9 nespl\u0148uj\u00ED p\u00E1t\u00FD Eukleid\u016Fv postul\u00E1t. Jej\u00EDmi nejd\u016Fle\u017Eit\u011Bj\u0161\u00EDmi p\u0159\u00EDpady jsou hyperbolick\u00E1 geometrie, (a jej\u00ED zvl\u00E1\u0161tn\u00ED p\u0159\u00EDpad sf\u00E9rick\u00E1 geometrie, tedy geometrie kuloplochy), Riemannova geometrie a . Geometrie spl\u0148uj\u00EDc\u00ED i p\u00E1t\u00FD postul\u00E1t se naz\u00FDv\u00E1 eukleidovsk\u00E1."@cs ,
"\uBE44\uC720\uD074\uB9AC\uB4DC \uAE30\uD558\uD559(non-Euclidean geometry)\uC740 \uC9C1\uC120 \uBC16\uC758 \uD55C \uC810\uC5D0\uC11C \uC9C1\uC120\uC5D0 \uD3C9\uD589\uD55C \uC9C1\uC120\uC744 \uB450 \uAC1C \uC774\uC0C1 \uADF8\uC744 \uC218 \uC788\uB294 \uACF5\uAC04\uC744 \uB300\uC0C1\uC73C\uB85C \uD558\uB294 \uAE30\uD558\uD559\uC774\uB2E4. \uC720\uD074\uB9AC\uB4DC \uAE30\uD558\uD559\uC758 \uC81C5\uACF5\uB9AC \"\uC9C1\uC120 \uBC16\uC758 \uD55C \uC810\uC744 \uC9C0\uB098\uBA74\uC11C \uADF8 \uC9C1\uC120\uC5D0 \uD3C9\uD589\uD55C \uC9C1\uC120\uC740 \uB2E8 \uD558\uB098 \uC874\uC7AC\uD55C\uB2E4\"\uAC00 \uC131\uB9BD\uD558\uC9C0 \uC54A\uB294 \uACF5\uAC04\uC744 \uB2E4\uB8E8\uB294 \uAE30\uD558\uD559\uC73C\uB85C, \uC30D\uACE1\uAE30\uD558\uD559, \uD0C0\uC6D0\uAE30\uD558\uD559, \uD0DD\uC2DC\uAE30\uD558\uD559 \uB4F1\uC774 \uC788\uB2E4. 19\uC138\uAE30\uC5D0 \uC81C5\uACF5\uB9AC\uB97C \uBD80\uC815\uD574\uB3C4 \uB2E4\uB978 \uACF5\uB9AC\uC640\uB294 \uC544\uBB34\uB7F0 \uBAA8\uC21C\uC774 \uC5C6\uC74C\uC774 \uBC1D\uD600\uC9C0\uBA74\uC11C \uB4F1\uC7A5\uD558\uC600\uB2E4. \uC5F0\uAD6C\uD55C \uC218\uD559\uC790\uB85C\uB294 \uB2C8\uCF5C\uB77C\uC774 \uB85C\uBC14\uCCC5\uC2A4\uD0A4 \u00B7 \uBCF4\uC5EC\uC774 \uC57C\uB178\uC2DC \u00B7 \uBCA0\uB978\uD558\uB974\uD2B8 \uB9AC\uB9CC\uC774 \uC720\uBA85\uD558\uB2E4. \uBE44\uC720\uD074\uB9AC\uB4DC \uAE30\uD558\uD559\uC740 \uC5ED\uC0AC\uC801\uC73C\uB85C\uB294 \uACF5\uB9AC\uB860\uC801\uC73C\uB85C \uAD6C\uC131\uB418\uC9C0\uB9CC \uD604\uB300\uC801\uC778 \uACAC\uD574\uB85C\uB294 \uBE44\uC720\uD074\uB9AC\uB4DC \uAE30\uD558\uD559\uC744 \uB9AC\uB9CC \uAE30\uD558\uD559\uC758 \uD2B9\uC218\uD55C \uC608 \uB610\uB294 \uACE0\uC804\uC801\uC778 \uBAA8\uB378\uB85C \uAC04\uC8FC\uD55C\uB2E4. \uADF8\uB9AC\uACE0 \uD604\uC7AC\uAE4C\uC9C0 13\uAC1C \uC774\uC0C1\uC758 \uAE30\uD558\uD559\uC774 \uD0C4\uC0DD\uB418\uACE0 \uCCB4\uACC4\uD654\uB418\uC5C8\uB2E4."@ko ,
"La nee\u016Dklidaj geometrioj estas du spacoj, geometrie studataj, kiuj malsamas je la pli vaste konata E\u016Dklida geometrio. Pli specife, ili malobeas la E\u016Dklidajn postulaton pri paralelon. Efektive, tiuj geometrioj uzas nerektajn kurbojn anstata\u016D la rektajn liniojn de la E\u016Dklida geometrio."@eo ,
"\u975E\u6B27\u51E0\u91CC\u5F97\u51E0\u4F55\uFF0C\u7B80\u79F0\u975E\u6B27\u51E0\u4F55\uFF0C\u662F\u591A\u4E2A\u51E0\u4F55\u5F62\u5F0F\u7CFB\u7EDF\u7684\u7EDF\u79F0\uFF0C\u4E0E\u6B27\u51E0\u91CC\u5F97\u51E0\u4F55\u7684\u5DEE\u522B\u5728\u4E8E\u7B2C\u4E94\u516C\u8BBE\u3002"@zh ,
"\u064A\u0639\u0628\u0631 \u0645\u0635\u0637\u0644\u062D \u0627\u0644\u0647\u0646\u062F\u0633\u0629 \u0627\u0644\u0644\u0627\u0625\u0642\u0644\u064A\u062F\u064A\u0629 \u0641\u064A \u0639\u0644\u0645 \u0627\u0644\u0631\u064A\u0627\u0636\u064A\u0627\u062A \u0639\u0646 \u0627\u0644\u0647\u0646\u062F\u0633\u0629 \u0627\u0644\u0625\u0647\u0644\u064A\u0644\u062C\u064A\u0629 \u0648\u0627\u0644\u0647\u0646\u062F\u0633\u0629 \u0627\u0644\u0632\u0627\u0626\u062F\u064A\u0629 \u0648\u0627\u0644\u062A\u064A \u0647\u064A \u0645\u0642\u0627\u0628\u0644 \u0627\u0644\u0647\u0646\u062F\u0633\u0629 \u0627\u0644\u0625\u0642\u0644\u064A\u062F\u064A\u0629. \u0627\u0644\u0641\u0631\u0642 \u0627\u0644\u0623\u0633\u0627\u0633\u064A \u0628\u064A\u0646 \u0627\u0644\u0647\u0646\u062F\u0633\u0629 \u0627\u0644\u0625\u0642\u0644\u064A\u062F\u064A\u0629 \u0648\u0627\u0644\u0647\u0646\u062F\u0633\u0629 \u0627\u0644\u0644\u0627\u0625\u0642\u0644\u064A\u062F\u064A\u0629 \u0647\u0648 \u0641\u064A \u0637\u0628\u064A\u0639\u0629 . \u062D\u064A\u062B \u062A\u0646\u0635 \u0645\u0633\u0644\u0645\u0629 \u0625\u0642\u0644\u064A\u062F\u0633 \u0627\u0644\u062E\u0627\u0645\u0633\u0629 \u0623\u0646 \u0641\u064A \u0627\u0644\u0645\u0633\u062A\u0648\u064A \u0627\u0644\u062B\u0646\u0627\u0626\u064A \u0627\u0644\u0623\u0628\u0639\u0627\u062F \u0645\u0646 \u0623\u062C\u0644 \u0623\u064A \u0645\u0633\u062A\u0642\u064A\u0645 l \u0648\u0646\u0642\u0637\u0629 A \u0644\u0627 \u062A\u0642\u0639 \u0639\u0644\u0649 \u0627\u0644\u0645\u0633\u062A\u0642\u064A\u0645 l \u064A\u0648\u062C\u062F \u0645\u0633\u062A\u0642\u064A\u0645 \u0648\u062D\u064A\u062F \u064A\u0645\u0631 \u0645\u0646 A \u0648\u0644\u0627 \u064A\u062A\u0642\u0627\u0637\u0639 \u0645\u0639 l. \u0641\u064A \u0627\u0644\u0647\u0646\u062F\u0633\u0629 \u0627\u0644\u0632\u0627\u0626\u062F\u064A\u0629 \u064A\u0648\u062C\u062F \u0639\u062F\u062F \u0644\u0627\u0646\u0647\u0627\u0626\u064A \u0645\u0646 \u0627\u0644\u0645\u0633\u062A\u0642\u064A\u0645\u0627\u062A \u0627\u0644\u062A\u064A \u062A\u0645\u0631 \u0628\u0640 A \u0628\u062F\u0648\u0646 \u0623\u0646 \u062A\u0642\u0637\u0639 l \u0628\u064A\u0646\u0645\u0627 \u0641\u064A \u0627\u0644\u0647\u0646\u062F\u0633\u0629 \u0627\u0644\u0625\u0647\u0644\u064A\u0644\u062C\u064A\u0629 \u0641\u0625\u0646 \u0627\u0644\u0645\u0633\u062A\u0642\u064A\u0645\u064A\u0646 \u0627\u0644\u0645\u062A\u0648\u0627\u0632\u064A\u064A\u0646 \u064A\u062A\u0642\u0627\u0631\u0628\u0627\u0646 \u0648\u0645\u0646 \u062B\u0645 \u064A\u062A\u0642\u0627\u0637\u0639\u0627\u0646."@ar ,
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"Geometria nieeuklidesowa \u2013 geometria, kt\u00F3ra nie spe\u0142nia co najmniej jednego z aksjomat\u00F3w geometrii euklidesowej. Mo\u017Ce ona spe\u0142nia\u0107 tylko cz\u0119\u015B\u0107 z nich, przy czym mog\u0105 r\u00F3wnie\u017C obowi\u0105zywa\u0107 w niej inne, sprzeczne z aksjomatami i twierdzeniami geometrii Euklidesa."@pl ,
"Dalam matematika, geometri non-Euklides (bahasa Inggris: non-Euclidean geometry) adalah himpunan kecil geometri berdasarkan aksioma yang berkaitan erat dengan geometri Euklides. Jika geometri Euklides terbentang antara geometri metrik dan , geometri non-Euklides muncul saat ruang metrik tidak ada, atau diabaikan. Perbedaan mendasar dari geometri metrik adalah keadaan garis . Cara lain untuk menggambarkan perbedaan antara geometri tersebut adalah dengan menggambarkan dua garis lurus dengan panjang tak hingga yang keduanya tegak lurus dengan sebuah garis ketiga."@in ,
"La geometria no euclidiana es diferencia de la geometria euclidiana perqu\u00E8, en aquesta mena de geometria, el cinqu\u00E8 postulat d'Euclides no \u00E9s v\u00E0lid. No fou desenvolupada amb la intenci\u00F3 de precisar la nostra experi\u00E8ncia espacial, sin\u00F3 com una teoria axiom\u00E0tica en conflicte amb el cinqu\u00E8 postulat d'Euclides. Segons el model de la geometria no euclidiana, es demostra que el cinqu\u00E8 postulat d'Euclides no es pot deduir dels altres axiomes i que n'\u00E9s independent."@ca ,
"En icke-euklidisk geometri \u00E4r en geometrisk teori d\u00E4r Euklides femte axiom, parallellaxiomet, inte g\u00E4ller. B\u00E5de hyperbolisk och elliptisk geometri \u00E4r icke-euklidiska. Den v\u00E4sentliga skillnaden mellan euklidisk och icke-euklidisk geometri \u00E4r de parallella linjernas natur. Inom euklidisk geometri och med start i en punkt A och en linje l, g\u00E5r det att dra endast en linje genom A som \u00E4r parallell med l. Inom hyperbolisk geometri finns det o\u00E4ndligt m\u00E5nga linjer genom A parallella med l och inom elliptisk geometri existerar inte parallella linjer."@sv ,
"Die nichteuklidischen Geometrien sind Spezialisierungen der absoluten Geometrie. Sie unterscheiden sich von der euklidischen Geometrie, die ebenfalls als eine Spezialisierung der absoluten Geometrie formuliert werden kann, dadurch, dass in ihnen das Parallelenaxiom nicht gilt."@de ,
"\u975E\u30E6\u30FC\u30AF\u30EA\u30C3\u30C9\u5E7E\u4F55\u5B66\uFF08\u3072\u30E6\u30FC\u30AF\u30EA\u30C3\u30C9\u304D\u304B\u304C\u304F\u3001\u82F1\u8A9E: non-Euclidean geometry\uFF09\u306F\u3001\u30E6\u30FC\u30AF\u30EA\u30C3\u30C9\u5E7E\u4F55\u5B66\u306E\u5E73\u884C\u7DDA\u516C\u6E96\u304C\u6210\u308A\u7ACB\u305F\u306A\u3044\u3068\u3057\u3066\u6210\u7ACB\u3059\u308B\u5E7E\u4F55\u5B66\u306E\u7DCF\u79F0\u3002\u975E\u30E6\u30FC\u30AF\u30EA\u30C3\u30C9\u306A\u5E7E\u4F55\u5B66\u306E\u516C\u7406\u7CFB\u3092\u6E80\u305F\u3059\u30E2\u30C7\u30EB\u306F\u69D8\u3005\u306B\u69CB\u6210\u3055\u308C\u308B\u304C\u3001\u8A08\u91CF\u3092\u3082\u3064\u5E7E\u4F55\u5B66\u30E2\u30C7\u30EB\u306E\u66F2\u7387\u3092\u4E00\u3064\u306E\u76EE\u5B89\u3068\u3057\u305F\u3068\u304D\u306E\u4E21\u6975\u7AEF\u306E\u5834\u5408\u3068\u3057\u3066\u3001\u81F3\u308B\u6240\u3067\u8CA0\u306E\u66F2\u7387\u3092\u3082\u3064\u53CC\u66F2\u5E7E\u4F55\u5B66\u3068\u81F3\u308B\u6240\u3067\u6B63\u306E\u66F2\u7387\u3092\u6301\u3064\u6955\u5186\u5E7E\u4F55\u5B66\uFF08\u7279\u306B\u7403\u9762\u5E7E\u4F55\u5B66\u306F\u6955\u5186\u5E7E\u4F55\u5B66\u306E\u4EE3\u8868\u7684\u306A\u30E2\u30C7\u30EB\u3067\u3042\u308B\uFF09\u304C\u77E5\u3089\u308C\u3066\u3044\u308B\u3002 \u30E6\u30FC\u30AF\u30EA\u30C3\u30C9\u306E\u5E7E\u4F55\u5B66\u306F\u3001\u81F3\u308B\u6240\u66F2\u73870\u306E\u4E16\u754C\u306E\u5E7E\u4F55\u3067\u3042\u308B\u3053\u3068\u304B\u3089\u3001\u53CC\u66F2\u30FB\u6955\u5186\u306B\u5BFE\u3057\u3066\u653E\u7269\u5E7E\u4F55\u5B66\u3068\u547C\u3076\u3053\u3068\u304C\u3042\u308B\u3002\u5E73\u6613\u306A\u8A00\u8449\u3067\u8868\u73FE\u3059\u308B\u306A\u3089\u3070\u3001\u300C\u5E73\u9762\u4E0A\u306E\u5E7E\u4F55\u5B66\u300D\u3067\u3042\u308B\u30E6\u30FC\u30AF\u30EA\u30C3\u30C9\u5E7E\u4F55\u5B66\u306B\u5BFE\u3057\u3066\u3001\u300C\u66F2\u9762\u4E0A\u306E\u5E7E\u4F55\u5B66\u300D\u304C\u975E\u30E6\u30FC\u30AF\u30EA\u30C3\u30C9\u5E7E\u4F55\u5B66\u3067\u3042\u308B\u3002"@ja ,
"Geometria ez-euklidearra Euklidesek bere Elementuak tratatuan ezarritako eta proportzioak betetzen ez dituen geometriako edozein sistema formali deitzen zaio. Ez da geometria ez-euklidear bakarra existitzen; asko existitzen dira. Hauek guztiak kasu partikularrak dira. Hala ere, geometriaren kurbadura intrintsekoa puntu batetik bestera aldatzeko aukera onartzen bada, Riemannen geometriaren kasu orokor bat lortzen da, erlatibitate orokorraren teorian gertatzen den bezala."@eu ;
rdfs:seeAlso dbr:Hyperbolic_geometry .
@prefix foaf: .
dbr:Non-Euclidean_geometry foaf:depiction ,
,
,
.
@prefix dcterms: .
@prefix dbc: .
dbr:Non-Euclidean_geometry dcterms:subject dbc:Non-Euclidean_geometry ;
dbo:abstract "\u041D\u0435\u0435\u0432\u043A\u043B\u0456\u0434\u043E\u0432\u0430 \u0433\u0435\u043E\u043C\u0435\u0442\u0440\u0456\u044F \u2014 \u0443 \u0431\u0443\u043A\u0432\u0430\u043B\u044C\u043D\u043E\u043C\u0443 \u0440\u043E\u0437\u0443\u043C\u0456\u043D\u043D\u0456 \u2014 \u0431\u0443\u0434\u044C-\u044F\u043A\u0430 \u0433\u0435\u043E\u043C\u0435\u0442\u0440\u0438\u0447\u043D\u0430 \u0441\u0438\u0441\u0442\u0435\u043C\u0430, \u0432\u0456\u0434\u043C\u0456\u043D\u043D\u0430 \u0432\u0456\u0434 \u0433\u0435\u043E\u043C\u0435\u0442\u0440\u0456\u0457 \u0415\u0432\u043A\u043B\u0456\u0434\u0430; \u043F\u0440\u043E\u0442\u0435 \u0442\u0440\u0430\u0434\u0438\u0446\u0456\u0439\u043D\u043E \u0442\u0435\u0440\u043C\u0456\u043D \u00AB\u041D\u0435\u0435\u0432\u043A\u043B\u0456\u0434\u043E\u0432\u0430 \u0433\u0435\u043E\u043C\u0435\u0442\u0440\u0456\u044F\u00BB \u0437\u0430\u0441\u0442\u043E\u0441\u043E\u0432\u0443\u0454\u0442\u044C\u0441\u044F \u0443 \u0432\u0443\u0436\u0447\u043E\u043C\u0443 \u0441\u0435\u043D\u0441\u0456 \u0439 \u0441\u0442\u043E\u0441\u0443\u0454\u0442\u044C\u0441\u044F \u043B\u0438\u0448\u0435 \u0434\u0432\u043E\u0445 \u0433\u0435\u043E\u043C\u0435\u0442\u0440\u0438\u0447\u043D\u0438\u0445 \u0441\u0438\u0441\u0442\u0435\u043C: \u0433\u0456\u043F\u0435\u0440\u0431\u043E\u043B\u0456\u0447\u043D\u043E\u0457 \u0433\u0435\u043E\u043C\u0435\u0442\u0440\u0456\u0457 \u0439 \u0441\u0444\u0435\u0440\u0438\u0447\u043D\u043E\u0457 \u0433\u0435\u043E\u043C\u0435\u0442\u0440\u0456\u0457. \u042F\u043A \u0456 \u0435\u0432\u043A\u043B\u0456\u0434\u043E\u0432\u0430 \u0446\u0456 \u0433\u0435\u043E\u043C\u0435\u0442\u0440\u0456\u0457 \u043D\u0430\u043B\u0435\u0436\u0430\u0442\u044C \u0434\u043E \u043C\u0435\u0442\u0440\u0438\u0447\u043D\u0438\u0445 \u0433\u0435\u043E\u043C\u0435\u0442\u0440\u0456\u0439 \u0442\u0440\u0438\u0432\u0438\u043C\u0456\u0440\u043D\u043E\u0433\u043E \u043F\u0440\u043E\u0441\u0442\u043E\u0440\u0443 \u043F\u043E\u0441\u0442\u0456\u0439\u043D\u043E\u0457 \u0441\u0435\u043A\u0446\u0456\u0439\u043D\u043E\u0457 \u043A\u0440\u0438\u0432\u0438\u043D\u0438. \u041D\u0443\u043B\u044C\u043E\u0432\u0430 \u043A\u0440\u0438\u0432\u0438\u043D\u0430 \u0432\u0456\u0434\u043F\u043E\u0432\u0456\u0434\u0430\u0454 \u0435\u0432\u043A\u043B\u0456\u0434\u043E\u0432\u0456\u0439 \u0433\u0435\u043E\u043C\u0435\u0442\u0440\u0456\u0457, \u0434\u043E\u0434\u0430\u0442\u043D\u0430 \u2014 \u0441\u0444\u0435\u0440\u0438\u0447\u043D\u0456\u0439, \u0432\u0456\u0434'\u0454\u043C\u043D\u0430 \u2014 \u0433\u0456\u043F\u0435\u0440\u0431\u043E\u043B\u0456\u0447\u043D\u0456\u0439 \u0433\u0435\u043E\u043C\u0435\u0442\u0440\u0456\u0457. \u0421\u0443\u0442\u0442\u0454\u0432\u0430 \u0440\u0456\u0437\u043D\u0438\u0446\u044F \u043C\u0456\u0436 \u043C\u0435\u0442\u0440\u0438\u0447\u043D\u0438\u043C\u0438 \u0433\u0435\u043E\u043C\u0435\u0442\u0440\u0456\u044F\u043C\u0438 \u043E\u043F\u0438\u0441\u0443\u0454\u0442\u044C\u0441\u044F \u0456\u0441\u043D\u0443\u0432\u0430\u043D\u043D\u044F\u043C \u043F\u0430\u0440\u0430\u043B\u0435\u043B\u044C\u043D\u0438\u0445 \u043F\u0440\u044F\u043C\u0438\u0445. \u041F'\u044F\u0442\u0438\u0439 \u043F\u043E\u0441\u0442\u0443\u043B\u0430\u0442 \u0415\u0432\u043A\u043B\u0456\u0434\u0430 \u0430\u0431\u043E \u0430\u043A\u0441\u0456\u043E\u043C\u0430 \u043F\u0440\u043E \u043F\u0430\u0440\u0430\u043B\u0435\u043B\u044C\u043D\u0456 \u043F\u0440\u044F\u043C\u0456 \u0441\u0442\u0432\u0435\u0440\u0434\u0436\u0443\u0454, \u0449\u043E \u0443 \u0434\u0432\u043E\u0432\u0438\u043C\u0456\u0440\u043D\u0456\u0439 \u043F\u043B\u043E\u0449\u0438\u043D\u0456 \u0434\u043B\u044F \u0431\u0443\u0434\u044C-\u044F\u043A\u043E\u0457 \u0437\u0430\u0434\u0430\u043D\u043E\u0457 \u043F\u0440\u044F\u043C\u043E\u0457 \u2113 \u0442\u0430 \u0442\u043E\u0447\u043A\u0438 A, \u044F\u043A\u0430 \u043D\u0435 \u043D\u0430\u043B\u0435\u0436\u0438\u0442\u044C \u2113, \u0456\u0441\u043D\u0443\u0454 \u0440\u0456\u0432\u043D\u043E \u043E\u0434\u043D\u0430 \u043F\u0440\u044F\u043C\u0430, \u044F\u043A\u0430 \u043F\u0440\u043E\u0445\u043E\u0434\u0438\u0442\u044C \u0447\u0435\u0440\u0435\u0437 A \u0456 \u043D\u0435 \u043F\u0435\u0440\u0435\u0442\u0438\u043D\u0430\u0454 \u2113. \u0423 \u0433\u0456\u043F\u0435\u0440\u0431\u043E\u043B\u0456\u0447\u043D\u0456\u0439 \u0433\u0435\u043E\u043C\u0435\u0442\u0440\u0456\u0457, \u043D\u0430\u0432\u043F\u0430\u043A\u0438, \u0447\u0435\u0440\u0435\u0437 A \u043F\u0440\u043E\u0445\u043E\u0434\u0438\u0442\u044C \u043D\u0435\u0441\u043A\u0456\u043D\u0447\u0435\u043D\u043D\u043E \u0431\u0430\u0433\u0430\u0442\u043E \u043F\u0440\u044F\u043C\u0438\u0445, \u044F\u043A\u0456 \u043D\u0435 \u043F\u0435\u0440\u0435\u0442\u0438\u043D\u0430\u044E\u0442\u044C \u2113. \u0422\u043E\u0434\u0456 \u044F\u043A \u0432 \u0435\u043B\u0456\u043F\u0442\u0438\u0447\u043D\u0456\u0439 \u0433\u0435\u043E\u043C\u0435\u0442\u0440\u0456\u0457 \u0431\u0443\u0434\u044C-\u044F\u043A\u0430 \u043F\u0440\u044F\u043C\u0430, \u0449\u043E \u043F\u0440\u043E\u0445\u043E\u0434\u0438\u0442\u044C \u0447\u0435\u0440\u0435\u0437 A, \u043F\u0435\u0440\u0435\u0442\u0438\u043D\u0430\u0454 \u2113 (\u0442\u043E\u0431\u0442\u043E, \u043F\u0430\u0440\u0430\u043B\u0435\u043B\u044C\u043D\u0438\u0445 \u043F\u0440\u044F\u043C\u0438\u0445 \u0443 \u0446\u0456\u0439 \u0433\u0435\u043E\u043C\u0435\u0442\u0440\u0456\u0457 \u0432\u0437\u0430\u0433\u0430\u043B\u0456 \u043D\u0435 \u0456\u0441\u043D\u0443\u0454). \u0406\u043D\u0448\u0438\u0439 \u0441\u043F\u043E\u0441\u0456\u0431 \u043E\u043F\u0438\u0441\u0430\u0442\u0438 \u0440\u0456\u0437\u043D\u0438\u0446\u044E \u043C\u0456\u0436 \u0446\u0438\u043C\u0438 \u0433\u0435\u043E\u043C\u0435\u0442\u0440\u0456\u044F\u043C\u0438 \u043F\u043E\u043B\u044F\u0433\u0430\u0454 \u0432 \u0442\u043E\u043C\u0443, \u0449\u043E\u0431 \u0440\u043E\u0437\u0433\u043B\u044F\u043D\u0443\u0442\u0438 \u0434\u0432\u0456 \u043F\u0440\u044F\u043C\u0456, \u044F\u043A\u0456 \u043F\u0435\u0440\u043F\u0435\u043D\u0434\u0438\u043A\u0443\u043B\u044F\u0440\u043D\u0456 \u0434\u043E \u0442\u0440\u0435\u0442\u044C\u043E\u0457 \u043F\u0440\u044F\u043C\u043E\u0457: \n* \u0412 \u0435\u0432\u043A\u043B\u0456\u0434\u043E\u0432\u0456\u0439 \u0433\u0435\u043E\u043C\u0435\u0442\u0440\u0456\u0457 \u0434\u0432\u0456 \u043F\u0440\u044F\u043C\u0456 \u0437\u0430\u043B\u0438\u0448\u0430\u044E\u0442\u044C\u0441\u044F \u043D\u0430 \u043F\u043E\u0441\u0442\u0456\u0439\u043D\u0456\u0439 \u0432\u0456\u0434\u0441\u0442\u0430\u043D\u0456 \u043E\u0434\u043D\u0430 \u0432\u0456\u0434 \u043E\u0434\u043D\u043E\u0457 (\u043F\u0435\u0440\u043F\u0435\u043D\u0434\u0438\u043A\u0443\u043B\u044F\u0440, \u043F\u0440\u043E\u0432\u0435\u0434\u0435\u043D\u0438\u0439 \u0434\u043E \u043F\u0435\u0440\u0448\u043E\u0457 \u043F\u0440\u044F\u043C\u043E\u0457 \u0432 \u0431\u0443\u0434\u044C-\u044F\u043A\u0456\u0439 \u0457\u0457 \u0442\u043E\u0447\u0446\u0456, \u043F\u0435\u0440\u0435\u0442\u043D\u0435 \u0434\u0440\u0443\u0433\u0443 \u043F\u0440\u044F\u043C\u0443, \u0456 \u0434\u043E\u0432\u0436\u0438\u043D\u0430 \u0432\u0456\u0434\u0440\u0456\u0437\u043A\u0430, \u044F\u043A\u0438\u0439 \u0437'\u0454\u0434\u043D\u0443\u0454 \u0442\u043E\u0447\u043A\u0438 \u043F\u0435\u0440\u0435\u0442\u0438\u043D\u0443, \u0454 \u043F\u043E\u0441\u0442\u0456\u0439\u043D\u043E\u044E). \u0422\u0430\u043A\u0456 \u043F\u0440\u044F\u043C\u0456 \u0432\u0456\u0434\u043E\u043C\u0456 \u044F\u043A \u043F\u0430\u0440\u0430\u043B\u0435\u043B\u0456. \n* \u0423 \u0433\u0456\u043F\u0435\u0440\u0431\u043E\u043B\u0456\u0447\u043D\u0456\u0439 \u0433\u0435\u043E\u043C\u0435\u0442\u0440\u0456\u0457 \u0434\u0432\u0456 \u043F\u0440\u044F\u043C\u0456, \u043F\u0435\u0440\u043F\u0435\u043D\u0434\u0438\u043A\u0443\u043B\u044F\u0440\u043D\u0456 \u0434\u043E \u0442\u0440\u0435\u0442\u044C\u043E\u0457, \u00AB\u0440\u043E\u0437\u0431\u0456\u0433\u0430\u044E\u0442\u044C\u0441\u044F\u00BB \u043E\u0434\u043D\u0430 \u0432\u0456\u0434 \u043E\u0434\u043D\u043E\u0457, \u0432\u0456\u0434\u0434\u0430\u043B\u044F\u044E\u0447\u0438\u0441\u044C, \u044F\u043A\u0449\u043E \u0440\u0443\u0445\u0430\u0442\u0438\u0441\u044C \u0432\u0456\u0434 \u0442\u043E\u0447\u043E\u043A \u043F\u0435\u0440\u0435\u0442\u0438\u043D\u0443 \u0456\u0437 \u0437\u0430\u0433\u0430\u043B\u044C\u043D\u0438\u043C \u043F\u0435\u0440\u043F\u0435\u043D\u0434\u0438\u043A\u0443\u043B\u044F\u0440\u043E\u043C[\u0434\u0436\u0435\u0440\u0435\u043B\u043E?]. \n* \u0412 \u0435\u043B\u0456\u043F\u0442\u0438\u0447\u043D\u0456\u0439 (\u0441\u0444\u0435\u0440\u0438\u0447\u043D\u0456\u0439) \u0433\u0435\u043E\u043C\u0435\u0442\u0440\u0456\u0457 \u0442\u0430\u043A\u0456 \u043F\u0440\u044F\u043C\u0456 \u043F\u043E\u0441\u0442\u0443\u043F\u043E\u0432\u043E \u00AB\u043D\u0430\u0431\u043B\u0438\u0436\u0430\u044E\u0442\u044C\u0441\u044F\u00BB \u043E\u0434\u043D\u0430 \u0434\u043E \u043E\u0434\u043D\u043E\u0457 \u0456 \u0432\u0440\u0435\u0448\u0442\u0456-\u0440\u0435\u0448\u0442 \u2014 \u043F\u0435\u0440\u0435\u0442\u0438\u043D\u0430\u044E\u0442\u044C\u0441\u044F."@uk ,
"Niet-euclidische meetkunde is meetkunde waarbij het vijfde postulaat van Euclides (het parallellenpostulaat) niet wordt aangenomen. Euclides ging bij zijn meetkunde uit van een aantal postulaten (axioma's). De meeste daarvan zijn eenvoudig, maar het vijfde vormt een uitzondering. Het postulaat heeft diverse vormen, maar de bekendste is waarschijnlijk \"Gegeven een rechte l en een punt P dat niet op l ligt, dan is er in het vlak door l en P maar \u00E9\u00E9n rechte door P die l niet snijdt.\" (Euclides' oorspronkelijke vorm was gecompliceerder.) Er zijn twee typen niet-euclidische meetkunde: \n* In hyperbolische meetkunde gaan er door P oneindig veel lijnen die l niet snijden. \n* In elliptische meetkunde gaat er door P geen lijn die l niet snijdt: alle lijnen snijden elkaar. Overigens is het voor elliptische meetkunde nodig ook andere postulaten van Euclides aan te passen. Lange tijd heeft men geprobeerd het parallellenpostulaat te bewijzen uit de andere axioma's, maar achteraf bleken alle bewijzen fout, doordat er ergens toch een 'evident' feit was gebruikt dat echter niet uit de overblijvende axioma's volgt, en dus equivalent was aan het parallellenpostulaat. In de 19e eeuw werd de stap genomen het parallellenpostulaat te laten vallen. Drie wiskundigen: de Rus Nikolaj Ivanovitsj Lobatsjevski (publicatie in 1829), de Hongaar J\u00E1nos Bolyai (publicatie in 1832) en de Duitser Carl Friedrich Gauss (ongepubliceerd, maar voor 1832) ontdekten ieder voor zich de principes van de hyperbolische meetkunde. In 1733 had overigens Giovanni Saccheri al een flink aantal stellingen afgeleid, in een poging het parallellenpostulaat door middel van reductio ad absurdum te bewijzen. De elliptische meetkunde werd ge\u00EFntroduceerd door Bernhard Riemann in 1854, als onderdeel van een veel grotere klasse van meetkunden (zie de riemann-meetkunde)."@nl ,
"Geometria nieeuklidesowa \u2013 geometria, kt\u00F3ra nie spe\u0142nia co najmniej jednego z aksjomat\u00F3w geometrii euklidesowej. Mo\u017Ce ona spe\u0142nia\u0107 tylko cz\u0119\u015B\u0107 z nich, przy czym mog\u0105 r\u00F3wnie\u017C obowi\u0105zywa\u0107 w niej inne, sprzeczne z aksjomatami i twierdzeniami geometrii Euklidesa."@pl ,
"Dalam matematika, geometri non-Euklides (bahasa Inggris: non-Euclidean geometry) adalah himpunan kecil geometri berdasarkan aksioma yang berkaitan erat dengan geometri Euklides. Jika geometri Euklides terbentang antara geometri metrik dan , geometri non-Euklides muncul saat ruang metrik tidak ada, atau diabaikan. Perbedaan mendasar dari geometri metrik adalah keadaan garis . Cara lain untuk menggambarkan perbedaan antara geometri tersebut adalah dengan menggambarkan dua garis lurus dengan panjang tak hingga yang keduanya tegak lurus dengan sebuah garis ketiga."@in ,
"\u975E\u6B27\u51E0\u91CC\u5F97\u51E0\u4F55\uFF0C\u7B80\u79F0\u975E\u6B27\u51E0\u4F55\uFF0C\u662F\u591A\u4E2A\u51E0\u4F55\u5F62\u5F0F\u7CFB\u7EDF\u7684\u7EDF\u79F0\uFF0C\u4E0E\u6B27\u51E0\u91CC\u5F97\u51E0\u4F55\u7684\u5DEE\u522B\u5728\u4E8E\u7B2C\u4E94\u516C\u8BBE\u3002"@zh ,
"La g\u00E9om\u00E9trie non euclidienne (GNE) est, en math\u00E9matiques, une th\u00E9orie g\u00E9om\u00E9trique ayant recours aux axiomes et postulats pos\u00E9s par Euclide dans les \u00C9l\u00E9ments, sauf le postulat des parall\u00E8les. Les diff\u00E9rentes g\u00E9om\u00E9tries non euclidiennes sont issues de la volont\u00E9 de d\u00E9montrer le cinqui\u00E8me postulat (le postulat d'Euclide, qui semble peu satisfaisant car trop complexe \u00E9tait peut-\u00EAtre redondant)."@fr ,
"Na matem\u00E1tica, uma geometria n\u00E3o euclidiana \u00E9 uma geometria baseada num sistema axiom\u00E1tico distinto da geometria euclidiana. Modificando o axioma das paralelas, que postula que por um ponto exterior a uma reta passa exatamente uma reta paralela \u00E0 inicial, obt\u00EAm-se as geometrias el\u00EDptica e hiperb\u00F3lica (geometria de Lobachevsky). Na geometria el\u00EDptica n\u00E3o h\u00E1 nenhuma reta paralela \u00E0 inicial, enquanto que na geometria hiperb\u00F3lica existe uma infinidade de rectas paralelas \u00E0 inicial que passam no mesmo ponto. Na geometria el\u00EDptica a soma dos \u00E2ngulos internos de um triangulo \u00E9 maior que dois \u00E2ngulos retos, enquanto na geometria hiperb\u00F3lica esta soma \u00E9 menor que dois \u00E2ngulos retos. Na el\u00EDptica, temos que a circunfer\u00EAncia de um c\u00EDrculo \u00E9 menor do que PI vezes o seu di\u00E2metro, enquanto na hiperb\u00F3lica esta circunfer\u00EAncia \u00E9 maior que PI vezes o di\u00E2metro. O cr\u00E9dito pela descoberta das geometrias n\u00E3o euclidianas geralmente \u00E9 atrelado \u00E0s figuras dos matem\u00E1ticos Carl Friedrich Gauss, e Bernhard Riemann."@pt ,
"\u064A\u0639\u0628\u0631 \u0645\u0635\u0637\u0644\u062D \u0627\u0644\u0647\u0646\u062F\u0633\u0629 \u0627\u0644\u0644\u0627\u0625\u0642\u0644\u064A\u062F\u064A\u0629 \u0641\u064A \u0639\u0644\u0645 \u0627\u0644\u0631\u064A\u0627\u0636\u064A\u0627\u062A \u0639\u0646 \u0627\u0644\u0647\u0646\u062F\u0633\u0629 \u0627\u0644\u0625\u0647\u0644\u064A\u0644\u062C\u064A\u0629 \u0648\u0627\u0644\u0647\u0646\u062F\u0633\u0629 \u0627\u0644\u0632\u0627\u0626\u062F\u064A\u0629 \u0648\u0627\u0644\u062A\u064A \u0647\u064A \u0645\u0642\u0627\u0628\u0644 \u0627\u0644\u0647\u0646\u062F\u0633\u0629 \u0627\u0644\u0625\u0642\u0644\u064A\u062F\u064A\u0629. \u0627\u0644\u0641\u0631\u0642 \u0627\u0644\u0623\u0633\u0627\u0633\u064A \u0628\u064A\u0646 \u0627\u0644\u0647\u0646\u062F\u0633\u0629 \u0627\u0644\u0625\u0642\u0644\u064A\u062F\u064A\u0629 \u0648\u0627\u0644\u0647\u0646\u062F\u0633\u0629 \u0627\u0644\u0644\u0627\u0625\u0642\u0644\u064A\u062F\u064A\u0629 \u0647\u0648 \u0641\u064A \u0637\u0628\u064A\u0639\u0629 . \u062D\u064A\u062B \u062A\u0646\u0635 \u0645\u0633\u0644\u0645\u0629 \u0625\u0642\u0644\u064A\u062F\u0633 \u0627\u0644\u062E\u0627\u0645\u0633\u0629 \u0623\u0646 \u0641\u064A \u0627\u0644\u0645\u0633\u062A\u0648\u064A \u0627\u0644\u062B\u0646\u0627\u0626\u064A \u0627\u0644\u0623\u0628\u0639\u0627\u062F \u0645\u0646 \u0623\u062C\u0644 \u0623\u064A \u0645\u0633\u062A\u0642\u064A\u0645 l \u0648\u0646\u0642\u0637\u0629 A \u0644\u0627 \u062A\u0642\u0639 \u0639\u0644\u0649 \u0627\u0644\u0645\u0633\u062A\u0642\u064A\u0645 l \u064A\u0648\u062C\u062F \u0645\u0633\u062A\u0642\u064A\u0645 \u0648\u062D\u064A\u062F \u064A\u0645\u0631 \u0645\u0646 A \u0648\u0644\u0627 \u064A\u062A\u0642\u0627\u0637\u0639 \u0645\u0639 l. \u0641\u064A \u0627\u0644\u0647\u0646\u062F\u0633\u0629 \u0627\u0644\u0632\u0627\u0626\u062F\u064A\u0629 \u064A\u0648\u062C\u062F \u0639\u062F\u062F \u0644\u0627\u0646\u0647\u0627\u0626\u064A \u0645\u0646 \u0627\u0644\u0645\u0633\u062A\u0642\u064A\u0645\u0627\u062A \u0627\u0644\u062A\u064A \u062A\u0645\u0631 \u0628\u0640 A \u0628\u062F\u0648\u0646 \u0623\u0646 \u062A\u0642\u0637\u0639 l \u0628\u064A\u0646\u0645\u0627 \u0641\u064A \u0627\u0644\u0647\u0646\u062F\u0633\u0629 \u0627\u0644\u0625\u0647\u0644\u064A\u0644\u062C\u064A\u0629 \u0641\u0625\u0646 \u0627\u0644\u0645\u0633\u062A\u0642\u064A\u0645\u064A\u0646 \u0627\u0644\u0645\u062A\u0648\u0627\u0632\u064A\u064A\u0646 \u064A\u062A\u0642\u0627\u0631\u0628\u0627\u0646 \u0648\u0645\u0646 \u062B\u0645 \u064A\u062A\u0642\u0627\u0637\u0639\u0627\u0646."@ar ,
"En icke-euklidisk geometri \u00E4r en geometrisk teori d\u00E4r Euklides femte axiom, parallellaxiomet, inte g\u00E4ller. B\u00E5de hyperbolisk och elliptisk geometri \u00E4r icke-euklidiska. Den v\u00E4sentliga skillnaden mellan euklidisk och icke-euklidisk geometri \u00E4r de parallella linjernas natur. Inom euklidisk geometri och med start i en punkt A och en linje l, g\u00E5r det att dra endast en linje genom A som \u00E4r parallell med l. Inom hyperbolisk geometri finns det o\u00E4ndligt m\u00E5nga linjer genom A parallella med l och inom elliptisk geometri existerar inte parallella linjer. Ett annat s\u00E4tt att beskriva skillnaderna mellan geometrierna: betrakta tv\u00E5 linjer i ett plan som b\u00E5da \u00E4r vinkelr\u00E4ta mot en tredje linje. Inom euklidisk och hyperbolisk geometri \u00E4r de tv\u00E5 linjerna parallella. Inom euklidisk geometri f\u00F6rblir avst\u00E5ndet mellan de tv\u00E5 linjerna konstant, medan inom hyperbolisk geometri \u00F6kar avst\u00E5ndet mellan linjerna med \u00F6kande avst\u00E5nd fr\u00E5n sk\u00E4rningspunkterna med den gemensamma vinkelr\u00E4ta linjen. Inom elliptisk geometri minskar avst\u00E5ndet mellan linjerna tills linjerna sk\u00E4r varandra; s\u00E5ledes existerar inga parallella linjer inom elliptisk geometri. Beteende hos linjer med gemensam ortogonal linje i vardera av de tre sorternas geometri"@sv ,
"In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those that specify Euclidean geometry. As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean geometry arises by either replacing the parallel postulate with an alternative, or relaxing the metric requirement. In the former case, one obtains hyperbolic geometry and elliptic geometry, the traditional non-Euclidean geometries. When the metric requirement is relaxed, then there are affine planes associated with the , which give rise to that have also been called non-Euclidean geometry. The essential difference between the metric geometries is the nature of parallel lines. Euclid's fifth postulate, the parallel postulate, is equivalent to Playfair's postulate, which states that, within a two-dimensional plane, for any given line l and a point A, which is not on l, there is exactly one line through A that does not intersect l. In hyperbolic geometry, by contrast, there are infinitely many lines through A not intersecting l, while in elliptic geometry, any line through A intersects l. Another way to describe the differences between these geometries is to consider two straight lines indefinitely extended in a two-dimensional plane that are both perpendicular to a third line (in the same plane): \n* In Euclidean geometry, the lines remain at a constant distance from each other (meaning that a line drawn perpendicular to one line at any point will intersect the other line and the length of the line segment joining the points of intersection remains constant) and are known as parallels. \n* In hyperbolic geometry, they \"curve away\" from each other, increasing in distance as one moves further from the points of intersection with the common perpendicular; these lines are often called ultraparallels. \n* In elliptic geometry, the lines \"curve toward\" each other and intersect."@en ,
"Una geometria non euclidea \u00E8 una geometria costruita negando o non accettando alcuni postulati euclidei.Viene detta anche metageometria."@it ,
"Neeukleidovsk\u00E1 geometrie je obecn\u00E9 ozna\u010Den\u00ED pro takov\u00E9 geometrie (tj. syst\u00E9my spl\u0148uj\u00EDc\u00ED prvn\u00ED \u010Dty\u0159i Eukleidovy postul\u00E1ty), kter\u00E9 nespl\u0148uj\u00ED p\u00E1t\u00FD Eukleid\u016Fv postul\u00E1t. Jej\u00EDmi nejd\u016Fle\u017Eit\u011Bj\u0161\u00EDmi p\u0159\u00EDpady jsou hyperbolick\u00E1 geometrie, (a jej\u00ED zvl\u00E1\u0161tn\u00ED p\u0159\u00EDpad sf\u00E9rick\u00E1 geometrie, tedy geometrie kuloplochy), Riemannova geometrie a . Geometrie spl\u0148uj\u00EDc\u00ED i p\u00E1t\u00FD postul\u00E1t se naz\u00FDv\u00E1 eukleidovsk\u00E1."@cs ,
"\u03A3\u03C4\u03B1 \u03BC\u03B1\u03B8\u03B7\u03BC\u03B1\u03C4\u03B9\u03BA\u03AC, \u03BC\u03B9\u03B1 \u03BC\u03B7-\u0395\u03C5\u03BA\u03BB\u03B5\u03AF\u03B4\u03B5\u03B9\u03B1 \u03B3\u03B5\u03C9\u03BC\u03B5\u03C4\u03C1\u03AF\u03B1 \u03C3\u03C5\u03BD\u03AF\u03C3\u03C4\u03B1\u03C4\u03B1\u03B9 \u03B1\u03C0\u03CC \u03B4\u03CD\u03BF \u03B3\u03B5\u03C9\u03BC\u03B5\u03C4\u03C1\u03AF\u03B5\u03C2 \u03B2\u03B1\u03C3\u03B9\u03C3\u03BC\u03AD\u03BD\u03B5\u03C2 \u03C3\u03B5 \u03B1\u03BE\u03B9\u03CE\u03BC\u03B1\u03C4\u03B1 \u03C3\u03C4\u03B5\u03BD\u03AC \u03C3\u03C5\u03BD\u03B4\u03B5\u03B4\u03B5\u03BC\u03AD\u03BD\u03B1 \u03BC\u03B5 \u03B1\u03C5\u03C4\u03AC \u03C0\u03BF\u03C5 \u03C0\u03C1\u03BF\u03C3\u03B4\u03B9\u03BF\u03C1\u03AF\u03B6\u03BF\u03C5\u03BD \u03C4\u03B7\u03BD \u0395\u03C5\u03BA\u03BB\u03B5\u03AF\u03B4\u03B5\u03B9\u03B1 \u03B3\u03B5\u03C9\u03BC\u03B5\u03C4\u03C1\u03AF\u03B1. \u039A\u03B1\u03B8\u03CE\u03C2 \u03B7 \u0395\u03C5\u03BA\u03BB\u03B5\u03AF\u03B4\u03B5\u03B9\u03B1 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\u03B7 \u03BC\u03AF\u03B1 \u03B1\u03C0\u03CC \u03C4\u03B7\u03BD \u03AC\u03BB\u03BB\u03B7, \u03B1\u03C5\u03BE\u03AC\u03BD\u03BF\u03BD\u03C4\u03B1\u03C2 \u03C4\u03B7\u03BD \u03BC\u03B5\u03C4\u03B1\u03BE\u03CD \u03C4\u03BF\u03C5\u03C2 \u03B1\u03C0\u03CC\u03C3\u03C4\u03B1\u03C3\u03B7 \u03BA\u03B1\u03B8\u03CE\u03C2 \u03B7 \u03BC\u03AF\u03B1 \u03B1\u03C0\u03BF\u03BC\u03B1\u03BA\u03C1\u03CD\u03BD\u03B5\u03C4\u03B1\u03B9 \u03B1\u03C0\u03CC \u03C4\u03B1 \u03C3\u03B7\u03BC\u03B5\u03AF\u03B1 \u03C4\u03BF\u03BC\u03AE\u03C2 \u03BC\u03B5 \u03C4\u03B7\u03BD \u03BA\u03BF\u03B9\u03BD\u03AE \u03BA\u03AC\u03B8\u03B5\u03C4\u03B7; \u03C4\u03AD\u03C4\u03BF\u03B9\u03B5\u03C2 \u03B5\u03C5\u03B8\u03B5\u03AF\u03B5\u03C2 \u03C3\u03C5\u03C7\u03BD\u03AC \u03B1\u03C0\u03BF\u03BA\u03B1\u03BB\u03BF\u03CD\u03BD\u03C4\u03B1\u03B9 \u03C5\u03C0\u03B5\u03C1\u03C0\u03B1\u03C1\u03AC\u03BB\u03BB\u03B7\u03BB\u03B5\u03C2. \n* \u03A3\u03C4\u03B7\u03BD \u03B5\u03BB\u03BB\u03B5\u03B9\u03C0\u03C4\u03B9\u03BA\u03AE \u03B3\u03B5\u03C9\u03BC\u03B5\u03C4\u03C1\u03AF\u03B1 \u03BA\u03B1\u03BC\u03C0\u03C5\u03BB\u03CE\u03BD\u03BF\u03C5\u03BD \u03B7 \u03BC\u03AF\u03B1 \u03C0\u03C1\u03BF\u03C2 \u03C4\u03B7\u03BD \u03AC\u03BB\u03BB\u03B7 \u03BA\u03B1\u03B9 \u03C4\u03AD\u03BC\u03BD\u03BF\u03BD\u03C4\u03B1\u03B9."@el ,
"\uBE44\uC720\uD074\uB9AC\uB4DC \uAE30\uD558\uD559(non-Euclidean geometry)\uC740 \uC9C1\uC120 \uBC16\uC758 \uD55C \uC810\uC5D0\uC11C \uC9C1\uC120\uC5D0 \uD3C9\uD589\uD55C \uC9C1\uC120\uC744 \uB450 \uAC1C \uC774\uC0C1 \uADF8\uC744 \uC218 \uC788\uB294 \uACF5\uAC04\uC744 \uB300\uC0C1\uC73C\uB85C \uD558\uB294 \uAE30\uD558\uD559\uC774\uB2E4. \uC720\uD074\uB9AC\uB4DC \uAE30\uD558\uD559\uC758 \uC81C5\uACF5\uB9AC \"\uC9C1\uC120 \uBC16\uC758 \uD55C \uC810\uC744 \uC9C0\uB098\uBA74\uC11C \uADF8 \uC9C1\uC120\uC5D0 \uD3C9\uD589\uD55C \uC9C1\uC120\uC740 \uB2E8 \uD558\uB098 \uC874\uC7AC\uD55C\uB2E4\"\uAC00 \uC131\uB9BD\uD558\uC9C0 \uC54A\uB294 \uACF5\uAC04\uC744 \uB2E4\uB8E8\uB294 \uAE30\uD558\uD559\uC73C\uB85C, \uC30D\uACE1\uAE30\uD558\uD559, \uD0C0\uC6D0\uAE30\uD558\uD559, \uD0DD\uC2DC\uAE30\uD558\uD559 \uB4F1\uC774 \uC788\uB2E4. 19\uC138\uAE30\uC5D0 \uC81C5\uACF5\uB9AC\uB97C \uBD80\uC815\uD574\uB3C4 \uB2E4\uB978 \uACF5\uB9AC\uC640\uB294 \uC544\uBB34\uB7F0 \uBAA8\uC21C\uC774 \uC5C6\uC74C\uC774 \uBC1D\uD600\uC9C0\uBA74\uC11C \uB4F1\uC7A5\uD558\uC600\uB2E4. \uC5F0\uAD6C\uD55C \uC218\uD559\uC790\uB85C\uB294 \uB2C8\uCF5C\uB77C\uC774 \uB85C\uBC14\uCCC5\uC2A4\uD0A4 \u00B7 \uBCF4\uC5EC\uC774 \uC57C\uB178\uC2DC \u00B7 \uBCA0\uB978\uD558\uB974\uD2B8 \uB9AC\uB9CC\uC774 \uC720\uBA85\uD558\uB2E4. \uBE44\uC720\uD074\uB9AC\uB4DC \uAE30\uD558\uD559\uC740 \uD0C0\uC6D0\uAE30\uD558\uD559(elliptic geometry)\uACFC \uC30D\uACE1\uAE30\uD558\uD559(hyperbolic geometry)\uC758 \uCD1D\uCE6D\uC774\uAE30\uB3C4 \uD558\uB2E4. \uB300\uD45C\uC801\uC778 \uD559\uC790\uB85C\uB294 \uCE74\uB97C \uD504\uB9AC\uB4DC\uB9AC\uD788 \uAC00\uC6B0\uC2A4, \uBCA0\uB978\uD558\uB974\uD2B8 \uB9AC\uB9CC \uB4F1\uC774 \uC788\uB2E4. \uB9AC\uB9CC\uC740 \u201C\uAD6C \uC704\uC5D0\uC11C\uB294 \uD55C \uC9C1\uC120\uACFC \uADF8 \uC9C1\uC120 \uC704\uC5D0 \uC788\uC9C0 \uC54A\uC740 \uC810\uC774 \uC8FC\uC5B4\uC84C\uC744 \uB54C, \uADF8 \uC9C1\uC120\uACFC \uD3C9\uD589\uD558\uACE0 \uADF8 \uC810\uC744 \uC9C0\uB098\uB294 \uC9C1\uC120\uC740 \uC5C6\uB2E4.\u201D\uACE0 \uB9D0\uD588\uC73C\uBA70, \uAC00\uC6B0\uC2A4\uB294 \uBC18\uB300\uB85C \u201C\uC9C0\uAD6C \uC704\uC5D0\uC11C\uB294 \uD55C \uC9C1\uC120\uACFC \uADF8 \uC9C1\uC120 \uC704\uC5D0 \uC788\uC9C0 \uC54A\uC740 \uC810\uC774 \uC8FC\uC5B4\uC84C\uC744 \uB54C, \uADF8 \uC9C1\uC120\uACFC \uD3C9\uD589\uD558\uACE0 \uADF8 \uC810\uC744 \uC9C0\uB098\uB294 \uC9C1\uC120\uC740 \uB458 \uC774\uC0C1\uC774\uB2E4.\u201D\uACE0 \uB9D0\uD588\uB2E4. \uC774\uB294 \uAC01\uAC01 \uD0C0\uC6D0 \uAE30\uD558\uD559\uACFC \uC30D\uACE1 \uAE30\uD558\uD559\uC758 \uAE30\uCD08\uAC00 \uB418\uC5C8\uB2E4. \uC0BC\uAC01\uD615\uC758 \uB0B4\uAC01\uC758 \uD569\uC774 180\uB3C4\uC778 \uC720\uD074\uB9AC\uB4DC \uAE30\uD558\uD559\uACFC\uB294 \uB2EC\uB9AC \uBE44\uC720\uD074\uB9AC\uB4DC \uAE30\uD558\uD559\uC5D0\uC11C\uB294 \uC0BC\uAC01\uD615\uC758 \uB0B4\uAC01\uC758 \uD569\uC774 180\uB3C4\uAC00 \uC544\uB2C8\uB77C \uC774\uBCF4\uB2E4 \uD06C\uAC70\uB098(\uD0C0\uC6D0 \uAE30\uD558\uD559) \uC791\uB2E4(\uC30D\uACE1 \uAE30\uD558\uD559). \uBE44\uC720\uD074\uB9AC\uB4DC \uAE30\uD558\uD559\uC740 \uC5ED\uC0AC\uC801\uC73C\uB85C\uB294 \uACF5\uB9AC\uB860\uC801\uC73C\uB85C \uAD6C\uC131\uB418\uC9C0\uB9CC \uD604\uB300\uC801\uC778 \uACAC\uD574\uB85C\uB294 \uBE44\uC720\uD074\uB9AC\uB4DC \uAE30\uD558\uD559\uC744 \uB9AC\uB9CC \uAE30\uD558\uD559\uC758 \uD2B9\uC218\uD55C \uC608 \uB610\uB294 \uACE0\uC804\uC801\uC778 \uBAA8\uB378\uB85C \uAC04\uC8FC\uD55C\uB2E4. \uADF8\uB9AC\uACE0 \uD604\uC7AC\uAE4C\uC9C0 13\uAC1C \uC774\uC0C1\uC758 \uAE30\uD558\uD559\uC774 \uD0C4\uC0DD\uB418\uACE0 \uCCB4\uACC4\uD654\uB418\uC5C8\uB2E4."@ko ,
"Geometria ez-euklidearra Euklidesek bere Elementuak tratatuan ezarritako eta proportzioak betetzen ez dituen geometriako edozein sistema formali deitzen zaio. Ez da geometria ez-euklidear bakarra existitzen; asko existitzen dira. Espazio homogeneoetara mugatzen bagara, hau da, espazioko puntu bakoitzean kurbadura berdina duten espazioez ari bagara, hiru geometria bereizten dira: geometria euklidearra, geometria eliptikoa eta geometria hiperbolikoa. Geometria horien arteko desberdintasunak deskribatzeko modu bat ondokoak kontsideratuz lortzen da: plano bidimentsional batean bi lerro zuzen hartu; eta erreferentzia bezala, horiekiko \u201Cperpendikularra\u201D den beste lerro zuzen bat irudikatu. \n* Geometria euklidearrak Euklidesen bost postulatuak betetzen ditu eta bere kurbadura zero da. Geometria honetan bi lerroak distantzia berera mantentzen dira beti, eta paralelo izena hartzen dute.Triangelu baten barneko hiru angeluen baturak 180\u00B0 ematen du beti. \n* Geometria eliptikoak Euklidesen lehenengo lau postulatuak betetzen ditu eta kurbadura negatiboa du. Geometria honetan bi lerroak ez dira distantzia berera mantentzen. Erreferentziatzat hartutako lerroarengandik urrundu ahala lerroen arteko distantzia handitu egiten da. Ondorioz, triangelu baten barneko hiru angeluen batura beti 180\u00B0 baino txikiagoa da. \n* Geometria hiperbolikoak Euklidesen lehenengo lau postulatuak betetzen ditu eta kurbadura positiboa du. Geometria honetan ere lerroak ez dira distantzia berera mantentzen. Erreferentziatzat hartutako lerroarengandik urrundu ahala lerroen arteko distantzia txikiagotu egiten da. Ondorioz, triangelu baten barneko hiru angeluen batura beti 180\u00B0 baino handiagoa da. Hauek guztiak kasu partikularrak dira. Hala ere, geometriaren kurbadura intrintsekoa puntu batetik bestera aldatzeko aukera onartzen bada, Riemannen geometriaren kasu orokor bat lortzen da, erlatibitate orokorraren teorian gertatzen den bezala."@eu ,
"La geometria no euclidiana es diferencia de la geometria euclidiana perqu\u00E8, en aquesta mena de geometria, el cinqu\u00E8 postulat d'Euclides no \u00E9s v\u00E0lid. No fou desenvolupada amb la intenci\u00F3 de precisar la nostra experi\u00E8ncia espacial, sin\u00F3 com una teoria axiom\u00E0tica en conflicte amb el cinqu\u00E8 postulat d'Euclides. Segons el model de la geometria no euclidiana, es demostra que el cinqu\u00E8 postulat d'Euclides no es pot deduir dels altres axiomes i que n'\u00E9s independent. La geometria no euclidiana s'obt\u00E9 a mesura que s'omet o es modifica el cinqu\u00E8 postulat d'Euclides. Les possibilitats fonamentals de modificaci\u00F3 s\u00F3n: \n* Entre una recta i un punt situat fora de la recta no hi ha cap paral\u00B7lela. Per tant, dues rectes diferents en un mateix nivell es toquen sempre. Aquesta hip\u00F2tesi no \u00E9s compatible amb la resta d'axiomes de la geometria euclidiana. S'arriba, per tant, a la conclusi\u00F3 que entre dos punts nom\u00E9s hi pot haver una recta d'uni\u00F3. Aquest fet condueix a la geometria el\u00B7l\u00EDptica. Un model il\u00B7lustratiu de la geometria el\u00B7l\u00EDptica bidimensional \u00E9s la geometria d'una superf\u00EDcie esf\u00E8rica, en qu\u00E8 la suma d'angles d'un triangle \u00E9s superior a 180\u00B0. \n* Entre una recta i un punt situat fora de la recta hi ha, com a m\u00EDnim, dues paral\u00B7leles. Amb la qual cosa la resta d'axiomes euclidians es mantenen. D'aix\u00F2, se n'obt\u00E9 la geometria hiperb\u00F2lica. Un exemple bidimensional d'aquesta geometria \u00E9s una superf\u00EDcie amb forma de sell\u00F3, en la qual la suma dels angles d'un triangle situat damunt d'aquesta superf\u00EDcie \u00E9s menor a 180\u00B0. Actualment, la geometria no euclidiana t\u00E9 un paper molt important en la f\u00EDsica te\u00F2rica i en la cosmologia. Segons la teoria de la relativitat, difereix de la geometria del cosmos perqu\u00E8 la gravitaci\u00F3 \"plega\" l'espai. Un dels misteris m\u00E9s importants de la f\u00EDsica actual \u00E9s saber si la geometria de l'univers \u00E9s, en l\u00EDnies generals, esf\u00E8rica (el\u00B7l\u00EDptica), plana (\u00E9s a dir, euclidiana) o hiperb\u00F2lica."@ca ,
"Se denomina geometr\u00EDa no euclidiana o no eucl\u00EDdea, a cualquier sistema formal de geometr\u00EDa cuyos postulados y proposiciones difieren en alg\u00FAn asunto de los establecidos por Euclides en su tratado Elementos. No existe un solo sistema de geometr\u00EDa no eucl\u00EDdea, sino muchos, aunque si se restringe la discusi\u00F3n a espacios homog\u00E9neos, en los que la curvatura del espacio es la misma en cada punto, en los que los puntos del espacio son indistinguibles, pueden distinguirse tres formulaciones\u200B de geometr\u00EDas: \n* La geometr\u00EDa euclidiana satisface los cinco postulados de Euclides y tiene curvatura cero (es decir se supone en un espacio plano por lo que la suma de los tres \u00E1ngulos interiores de un tri\u00E1ngulo da siempre 180\u00B0). \n* La geometr\u00EDa hiperb\u00F3lica satisface solo los cuatro primeros postulados de Euclides y tiene curvatura negativa (en esta geometr\u00EDa, por ejemplo, la suma de los tres \u00E1ngulos interiores de un tri\u00E1ngulo es inferior a 180\u00B0). \n* La geometr\u00EDa el\u00EDptica satisface solo los cuatro primeros postulados de Euclides y tiene curvatura positiva (en esta geometr\u00EDa, por ejemplo, la suma de los tres \u00E1ngulos interiores de un tri\u00E1ngulo es mayor a 180\u00B0). Todos estos son casos particulares de geometr\u00EDas riemannianas, en los que la curvatura es constante, si se admite la posibilidad de que la curvatura intr\u00EDnseca de la geometr\u00EDa var\u00EDe de un punto a otro se tiene un caso de geometr\u00EDa riemanniana general, como sucede en la teor\u00EDa de la relatividad general donde la gravedad causa una curvatura no homog\u00E9nea en el espacio-tiempo, siendo mayor la curvatura cerca de las concentraciones de masa, lo cual es percibido como un campo gravitatorio atractivo."@es ,
"Geoim\u00E9adrachta\u00ED a saothra\u00EDodh \u00F3n 18\u00FA c\u00E9ad anuas, bunaithe ar 5\u00FA aics\u00EDm Eoicl\u00EDd\u00E9is a thr\u00E9igean ar bheala\u00ED \u00E9ags\u00FAla. Ba \u00ED an aics\u00EDm sin nach f\u00E9idir ach l\u00EDne dh\u00EDreach amh\u00E1in a tharraingt tr\u00ED phointe ar leith at\u00E1 comhthreomhar le l\u00EDne dh\u00EDreach ar leith. Mar shampla, m\u00E1 ghlactar mar aics\u00EDm nua ina hionad sin gur f\u00E9idir n\u00EDos m\u00F3 n\u00E1 l\u00EDne dh\u00EDreach amh\u00E1in a tharraingt tr\u00ED phointe ar leith at\u00E1 comhthreomhar le l\u00EDne dh\u00EDreach ar leith, forbra\u00EDtear geoim\u00E9adrachta\u00ED hipearb\u00F3ileacha, mar at\u00E1 d\u00E9anta ag Gau\u00DF is Lobachevsky. Ba \u00ED seo an ch\u00E9ad gheoim\u00E9adracht fhisici\u00FAil shochreidte mar mhalairt ar gheoim\u00E9adracht Eoicl\u00EDd\u00E9is. Mar shampla eile, m\u00E1 ghlactar mar aisc\u00EDm eile nach f\u00E9idir aon l\u00EDne dh\u00EDreach a tharraingt tr\u00ED phointe ar leith at\u00E1 comhthreomhar le l\u00EDne dh\u00EDreach ar leith, forbra\u00EDtear geoim\u00E9adracht \u00E9ilipseach, agus rinne Riemann obair bhun\u00FAsach ar an gcoincheap teib\u00ED seo. T\u00E1 geoim\u00E9adrachta\u00ED neamh-Eoicl\u00EDd\u00E9acha teib\u00ED eile \u00E1 saothr\u00FA freisin."@ga ,
"\u041D\u0435\u0435\u0432\u043A\u043B\u0438\u0434\u043E\u0432\u0430 \u0433\u0435\u043E\u043C\u0435\u0442\u0440\u0438\u044F \u2014 \u0432 \u0431\u0443\u043A\u0432\u0430\u043B\u044C\u043D\u043E\u043C \u043F\u043E\u043D\u0438\u043C\u0430\u043D\u0438\u0438 \u2014 \u043B\u044E\u0431\u0430\u044F \u0433\u0435\u043E\u043C\u0435\u0442\u0440\u0438\u0447\u0435\u0441\u043A\u0430\u044F \u0441\u0438\u0441\u0442\u0435\u043C\u0430, \u043A\u043E\u0442\u043E\u0440\u0430\u044F \u043E\u0442\u043B\u0438\u0447\u0430\u0435\u0442\u0441\u044F \u043E\u0442 \u0433\u0435\u043E\u043C\u0435\u0442\u0440\u0438\u0438 \u0415\u0432\u043A\u043B\u0438\u0434\u0430; \u043E\u0434\u043D\u0430\u043A\u043E \u0442\u0440\u0430\u0434\u0438\u0446\u0438\u043E\u043D\u043D\u043E \u0442\u0435\u0440\u043C\u0438\u043D \u00AB\u043D\u0435\u0435\u0432\u043A\u043B\u0438\u0434\u043E\u0432\u0430 \u0433\u0435\u043E\u043C\u0435\u0442\u0440\u0438\u044F\u00BB \u043F\u0440\u0438\u043C\u0435\u043D\u044F\u0435\u0442\u0441\u044F \u0432 \u0431\u043E\u043B\u0435\u0435 \u0443\u0437\u043A\u043E\u043C \u0441\u043C\u044B\u0441\u043B\u0435 \u0438 \u043E\u0442\u043D\u043E\u0441\u0438\u0442\u0441\u044F \u0442\u043E\u043B\u044C\u043A\u043E \u043A \u0434\u0432\u0443\u043C \u0433\u0435\u043E\u043C\u0435\u0442\u0440\u0438\u0447\u0435\u0441\u043A\u0438\u043C \u0441\u0438\u0441\u0442\u0435\u043C\u0430\u043C: \u0433\u0435\u043E\u043C\u0435\u0442\u0440\u0438\u0438 \u041B\u043E\u0431\u0430\u0447\u0435\u0432\u0441\u043A\u043E\u0433\u043E \u0438 \u0441\u0444\u0435\u0440\u0438\u0447\u0435\u0441\u043A\u043E\u0439 \u0433\u0435\u043E\u043C\u0435\u0442\u0440\u0438\u0438 (\u0438\u043B\u0438 \u0441\u0445\u043E\u0436\u0435\u0439 \u0441 \u043D\u0435\u0439 \u0433\u0435\u043E\u043C\u0435\u0442\u0440\u0438\u0438 \u0420\u0438\u043C\u0430\u043D\u0430). \u041A\u0430\u043A \u0438 \u0435\u0432\u043A\u043B\u0438\u0434\u043E\u0432\u0430, \u044D\u0442\u0438 \u0433\u0435\u043E\u043C\u0435\u0442\u0440\u0438\u0438 \u043E\u0442\u043D\u043E\u0441\u044F\u0442\u0441\u044F \u043A \u043C\u0435\u0442\u0440\u0438\u0447\u0435\u0441\u043A\u0438\u043C \u0433\u0435\u043E\u043C\u0435\u0442\u0440\u0438\u044F\u043C \u043F\u0440\u043E\u0441\u0442\u0440\u0430\u043D\u0441\u0442\u0432\u0430 \u043F\u043E\u0441\u0442\u043E\u044F\u043D\u043D\u043E\u0439 \u043A\u0440\u0438\u0432\u0438\u0437\u043D\u044B.\u041D\u0443\u043B\u0435\u0432\u0430\u044F \u043A\u0440\u0438\u0432\u0438\u0437\u043D\u0430 \u0441\u043E\u043E\u0442\u0432\u0435\u0442\u0441\u0442\u0432\u0443\u0435\u0442 \u0435\u0432\u043A\u043B\u0438\u0434\u043E\u0432\u043E\u0439 \u0433\u0435\u043E\u043C\u0435\u0442\u0440\u0438\u0438, \u043F\u043E\u043B\u043E\u0436\u0438\u0442\u0435\u043B\u044C\u043D\u0430\u044F \u2014 \u0441\u043E\u0432\u043F\u0430\u0434\u0430\u044E\u0449\u0438\u043C \u043F\u043E \u043B\u043E\u043A\u0430\u043B\u044C\u043D\u044B\u043C \u0441\u0432\u043E\u0439\u0441\u0442\u0432\u0430\u043C \u0441\u0444\u0435\u0440\u0438\u0447\u0435\u0441\u043A\u043E\u0439 \u0438\u043B\u0438 \u0433\u0435\u043E\u043C\u0435\u0442\u0440\u0438\u0438 \u0420\u0438\u043C\u0430\u043D\u0430, \u043E\u0442\u0440\u0438\u0446\u0430\u0442\u0435\u043B\u044C\u043D\u0430\u044F \u2014 \u0433\u0435\u043E\u043C\u0435\u0442\u0440\u0438\u0438 \u041B\u043E\u0431\u0430\u0447\u0435\u0432\u0441\u043A\u043E\u0433\u043E."@ru ,
"La nee\u016Dklidaj geometrioj estas du spacoj, geometrie studataj, kiuj malsamas je la pli vaste konata E\u016Dklida geometrio. Pli specife, ili malobeas la E\u016Dklidajn postulaton pri paralelon. Efektive, tiuj geometrioj uzas nerektajn kurbojn anstata\u016D la rektajn liniojn de la E\u016Dklida geometrio."@eo ,
"Die nichteuklidischen Geometrien sind Spezialisierungen der absoluten Geometrie. Sie unterscheiden sich von der euklidischen Geometrie, die ebenfalls als eine Spezialisierung der absoluten Geometrie formuliert werden kann, dadurch, dass in ihnen das Parallelenaxiom nicht gilt."@de ,
"\u975E\u30E6\u30FC\u30AF\u30EA\u30C3\u30C9\u5E7E\u4F55\u5B66\uFF08\u3072\u30E6\u30FC\u30AF\u30EA\u30C3\u30C9\u304D\u304B\u304C\u304F\u3001\u82F1\u8A9E: non-Euclidean geometry\uFF09\u306F\u3001\u30E6\u30FC\u30AF\u30EA\u30C3\u30C9\u5E7E\u4F55\u5B66\u306E\u5E73\u884C\u7DDA\u516C\u6E96\u304C\u6210\u308A\u7ACB\u305F\u306A\u3044\u3068\u3057\u3066\u6210\u7ACB\u3059\u308B\u5E7E\u4F55\u5B66\u306E\u7DCF\u79F0\u3002\u975E\u30E6\u30FC\u30AF\u30EA\u30C3\u30C9\u306A\u5E7E\u4F55\u5B66\u306E\u516C\u7406\u7CFB\u3092\u6E80\u305F\u3059\u30E2\u30C7\u30EB\u306F\u69D8\u3005\u306B\u69CB\u6210\u3055\u308C\u308B\u304C\u3001\u8A08\u91CF\u3092\u3082\u3064\u5E7E\u4F55\u5B66\u30E2\u30C7\u30EB\u306E\u66F2\u7387\u3092\u4E00\u3064\u306E\u76EE\u5B89\u3068\u3057\u305F\u3068\u304D\u306E\u4E21\u6975\u7AEF\u306E\u5834\u5408\u3068\u3057\u3066\u3001\u81F3\u308B\u6240\u3067\u8CA0\u306E\u66F2\u7387\u3092\u3082\u3064\u53CC\u66F2\u5E7E\u4F55\u5B66\u3068\u81F3\u308B\u6240\u3067\u6B63\u306E\u66F2\u7387\u3092\u6301\u3064\u6955\u5186\u5E7E\u4F55\u5B66\uFF08\u7279\u306B\u7403\u9762\u5E7E\u4F55\u5B66\u306F\u6955\u5186\u5E7E\u4F55\u5B66\u306E\u4EE3\u8868\u7684\u306A\u30E2\u30C7\u30EB\u3067\u3042\u308B\uFF09\u304C\u77E5\u3089\u308C\u3066\u3044\u308B\u3002 \u30E6\u30FC\u30AF\u30EA\u30C3\u30C9\u306E\u5E7E\u4F55\u5B66\u306F\u3001\u81F3\u308B\u6240\u66F2\u73870\u306E\u4E16\u754C\u306E\u5E7E\u4F55\u3067\u3042\u308B\u3053\u3068\u304B\u3089\u3001\u53CC\u66F2\u30FB\u6955\u5186\u306B\u5BFE\u3057\u3066\u653E\u7269\u5E7E\u4F55\u5B66\u3068\u547C\u3076\u3053\u3068\u304C\u3042\u308B\u3002\u5E73\u6613\u306A\u8A00\u8449\u3067\u8868\u73FE\u3059\u308B\u306A\u3089\u3070\u3001\u300C\u5E73\u9762\u4E0A\u306E\u5E7E\u4F55\u5B66\u300D\u3067\u3042\u308B\u30E6\u30FC\u30AF\u30EA\u30C3\u30C9\u5E7E\u4F55\u5B66\u306B\u5BFE\u3057\u3066\u3001\u300C\u66F2\u9762\u4E0A\u306E\u5E7E\u4F55\u5B66\u300D\u304C\u975E\u30E6\u30FC\u30AF\u30EA\u30C3\u30C9\u5E7E\u4F55\u5B66\u3067\u3042\u308B\u3002"@ja ;
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