. . . . . . . . "Schwarz triangle"@en . . . "SchwarzTriangle"@en . . . . . . . . "\u5728\u5E7E\u4F55\u5B78\u4E2D\uFF0C\u65BD\u74E6\u8328\u4E09\u89D2\u5F62\uFF08\u82F1\u8A9E\uFF1ASchwarz triangle\uFF09\u662F\u4E00\u500B\u7403\u9762\u4E09\u89D2\u5F62\uFF0C\u53EF\u7528\u65BC\u7403\u9762\u9472\u5D4C\uFF0C\u900F\u904E\u5728\u5176\u908A\u7DE3\u53CD\u5C04\uFF0C\u4F46\u662F\u53EF\u80FD\u6703\u91CD\u758A\u3002\u4ED6\u5011\u88AB\u6B78\u985E\u65BC\u65BD\u74E6\u83281873\u3002 \u65BD\u74E6\u8328\u4E09\u89D2\u5F62\u9664\u4E86\u53EF\u4EE5\u5B9A\u7FA9\u5728\u7403\u9762\u4E4B\u5916\uFF0C\u4E5F\u53EF\u4EE5\u5B9A\u7FA9\u65BC\u6B50\u5E7E\u91CC\u5F97\u5E73\u9762\u6216\u96D9\u66F2\u9762\uFF0C\u800C\u505A\u6210\u4FBF\u9762\u9472\u5D4C\u6216\u96D9\u66F2\u9762\u9472\u5D4C\u3002\u5728\u7403\u9762\u4E0A\u7684\u6BCF\u500B\u65BD\u74E6\u8328\u4E09\u89D2\u5F62\u5B9A\u7FA9\u4E86\u4E00\u500B\u6709\u9650\u7FA4\uFF0C\u800C\u5728\u6B50\u6C0F\u6216\u96D9\u66F2\u5E73\u9762\uFF0C\u5247\u6703\u5B9A\u7FA9\u51FA\u4E00\u500B\u7121\u9650\u7FA4\u3002 \u65BD\u74E6\u8328\u4E09\u89D2\u5F62\u662F\u7531\u4E09\u500B\u6709\u7406\u6578(p q r)\u4F86\u4EE3\u8868\u6BCF\u500B\u9802\u9EDE\u7684\u89D2\u5EA6\u3002\u503Cn/d\u8868\u793A\u7684\u9802\u89D2\u70BA\u534A\u5713\u7684d/n\uFF0C\u201C2\u201D\u8868\u662F\u4E00\u500B\u76F4\u89D2\u3002\u82E5p\u3001q\u3001r\u7686\u70BA\u6574\u6578\uFF0C\u5247\u5C07\u5176\u7A31\u70BA\u83AB\u6BD4\u70CF\u65AF\u4E09\u89D2\u5F62\uFF08\u82F1\u8A9E\uFF1AM\u00F6bius triangle\uFF09\u4E26\u4E14\u5C0D\u61C9\u65BC\u4E00\u500B\u6C92\u6709\u91CD\u758A\u7684\u9472\u5D4C\uFF0C\u5176\u5C0D\u7A31\u7FA4\u7A31\u70BA\u4E00\u500B\u4E09\u89D2\u7FA4\u3002\u5728\u7403\u9762\u79FB\u5171\u67093\u500B\u83AB\u6BD4\u70CF\u65AF\u4E09\u89D2\u5F62\u52A0\u4E00\u500B\u55AE\u53C3\u6578\u65CF\uFF1B\u5728\u6B50\u6C0F\u5E73\u9762\u4E0A\u6709\u4E09\u500B\u83AB\u6BD4\u70CF\u65AF\u4E09\u89D2\u5F62\uFF1B\u800C\u5728\u7F85\u6C0F\u96D9\u66F2\u7A7A\u9593\u4E2D\u6709\u4E09\u500B\u53C3\u6578\u65CF\u7684\u83AB\u6BD4\u70CF\u65AF\u4E09\u89D2\u5F62\uFF0C\u4E26\u6C92\u6709\u7279\u4F8B\u3002"@zh . . . . "\u0645\u062B\u0644\u062B \u0634\u0641\u0627\u0631\u0632"@ar . . . . . . "In geometry, a Schwarz triangle, named after Hermann Schwarz, is a spherical triangle that can be used to tile a sphere (spherical tiling), possibly overlapping, through reflections in its edges. They were classified in . These can be defined more generally as tessellations of the sphere, the Euclidean plane, or the hyperbolic plane. Each Schwarz triangle on a sphere defines a finite group, while on the Euclidean or hyperbolic plane they define an infinite group. A Schwarz triangle is represented by three rational numbers (p q r), each representing the angle at a vertex. The value n\u2044d means the vertex angle is d\u2044n of the half-circle. \"2\" means a right triangle. When these are whole numbers, the triangle is called a M\u00F6bius triangle, and corresponds to a non-overlapping tiling, and the symmetry group is called a triangle group. In the sphere there are three M\u00F6bius triangles plus one one-parameter family; in the plane there are three M\u00F6bius triangles, while in hyperbolic space there is a three-parameter family of M\u00F6bius triangles, and no exceptional objects."@en . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . "Triangolo di Schwarz"@it . . . "In geometria, un triangolo di Schwarz \u00E8 un triangolo sferico che pu\u00F2 essere utilizzato per tassellare una sfera (creando una cosiddetta tassellatura sferica), eventualmente sovrapponendosi, attraverso continue riflessioni sui suoi bordi. Tali triangoli sono stati classificati dal matematico tedesco Hermann Schwarz nel 1873 e sono stati cos\u00EC chiamati proprio in suo onore. I triangoli di Schwarz possono essere pi\u00F9 generalmente definiti come una tassellatura della sfera, del piano euclideo o di un piano iperbolico, tuttavia, ogni triangolo di Schwarz su una sfera definisce un gruppo finito, mentre su un piano euclideo o iperbolico essi definiscono un gruppo infinito. Un triangolo di Schwarz \u00E8 rappresentato da tre numeri razionali (p q r) ciascuno dei quali rappresenta uno dei suoi angoli al vertice, essendo in particolare il quoziente dell'angolo piatto diviso per il valore dell'angolo al vertice. Ci\u00F2 implica ad esempio che se uno dei tre numeri \u00E8 un 2, il triangolo in questione sar\u00E0 un triangolo rettangolo . Quando tutti e tre i numeri sono interi, il triangolo \u00E8 chiamato triangolo di M\u00F6bius e corrisponde a una tassellatura non sovrapposta, inoltre il gruppo di simmetria \u00E8 chiamato gruppo triangolare. Su una sfera \u00E8 possibile individuare tre triangoli di M\u00F6bius pi\u00F9 una famiglia a un solo parametro; nel piano euclideo ci sono tre triangoli di M\u00F6bius, mentre nello spazio iperbolico c'\u00E8 una sola famiglia di triangoli di M\u00F6bius a tre parametri."@it . . . . . . "1104450002"^^ . . "Schwarz triangle"@en . . "\u0422\u0440\u0435\u0443\u0433\u043E\u043B\u044C\u043D\u0438\u043A \u0428\u0432\u0430\u0440\u0446\u0430"@ru . . . . . . . . . . . . "\u65BD\u74E6\u8328\u4E09\u89D2\u5F62"@zh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . "80013"^^ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . "In geometry, a Schwarz triangle, named after Hermann Schwarz, is a spherical triangle that can be used to tile a sphere (spherical tiling), possibly overlapping, through reflections in its edges. They were classified in . These can be defined more generally as tessellations of the sphere, the Euclidean plane, or the hyperbolic plane. Each Schwarz triangle on a sphere defines a finite group, while on the Euclidean or hyperbolic plane they define an infinite group."@en . . . . . . . . . . "\u0641\u064A \u0627\u0644\u0647\u0646\u062F\u0633\u0629 \u0627\u0644\u0631\u064A\u0627\u0636\u064A\u0629\u060C \u0645\u062B\u0644\u062B \u0634\u0641\u0627\u0631\u0632 (\u0628\u0627\u0644\u0625\u0646\u062C\u0644\u064A\u0632\u064A\u0629: Schwarz triangle)\u200F\u060C \u0647\u0648 \u0645\u062B\u0644\u062B \u0643\u0631\u0648\u064A \u064A\u0645\u0643\u0646 \u0627\u0633\u062A\u0639\u0645\u0627\u0644\u0647 \u0645\u0646 \u0623\u062C\u0644 \u062A\u063A\u0637\u064A\u0629 \u0627\u0644\u0643\u0631\u0629 \u0643\u0627\u0645\u0644\u0629\u064B.\u0633\u0645\u064A \u0647\u0630\u0627 \u0627\u0644\u0645\u062B\u0644\u062B \u0647\u0643\u0630\u0627 \u0646\u0633\u0628\u0629 \u0625\u0644\u0649 \u0639\u0627\u0644\u0645 \u0627\u0644\u0631\u064A\u0627\u0636\u064A\u0627\u062A \u0647\u064A\u0631\u0645\u0627\u0646 \u0634\u0641\u0627\u0631\u0632."@ar . . . . . . . . . . . . . . . . . . . . . . . . . . . "En matematiko, triangulo de Schwarz estas sfera triangulo kiu povas esti uzata por kaheli sferon. \u0108iu triangulo de Schwarz difinas finian grupon - \u011Dian triangulan grupon. Pri \u011Di unuafoje en 1873 en Zuriko publikigis la germana matematikisto Hermann Amandus Schwarz (1843-1921), tial la nomo. Triangulo de Schwarz estas prezentata per tri racionalaj nombroj (a b c), \u0109iu el ili prezentas la angulo je vertico. Valoro n/d de la nombro signifas ke la vertica angulo estas d/n de la duoncirklo. Se \u011Di estas 2 tio estas orta triangulo. Estas nur 4 grupoj, anka\u016D nomataj kiel trianguloj de M\u00F6bius: 1. \n* (2 2 p) - duedra simetrio 2. \n* (2 3 3) - kvaredra simetrio 3. \n* (2 3 4) - okedra simetrio 4. \n* (2 3 5) - dudekedra simetrio Do estas kvar familioj de trianguloj de Schwarz surbaze de ilia simetrio."@eo . . . . . . . . . . . . . . . . . "\u5728\u5E7E\u4F55\u5B78\u4E2D\uFF0C\u65BD\u74E6\u8328\u4E09\u89D2\u5F62\uFF08\u82F1\u8A9E\uFF1ASchwarz triangle\uFF09\u662F\u4E00\u500B\u7403\u9762\u4E09\u89D2\u5F62\uFF0C\u53EF\u7528\u65BC\u7403\u9762\u9472\u5D4C\uFF0C\u900F\u904E\u5728\u5176\u908A\u7DE3\u53CD\u5C04\uFF0C\u4F46\u662F\u53EF\u80FD\u6703\u91CD\u758A\u3002\u4ED6\u5011\u88AB\u6B78\u985E\u65BC\u65BD\u74E6\u83281873\u3002 \u65BD\u74E6\u8328\u4E09\u89D2\u5F62\u9664\u4E86\u53EF\u4EE5\u5B9A\u7FA9\u5728\u7403\u9762\u4E4B\u5916\uFF0C\u4E5F\u53EF\u4EE5\u5B9A\u7FA9\u65BC\u6B50\u5E7E\u91CC\u5F97\u5E73\u9762\u6216\u96D9\u66F2\u9762\uFF0C\u800C\u505A\u6210\u4FBF\u9762\u9472\u5D4C\u6216\u96D9\u66F2\u9762\u9472\u5D4C\u3002\u5728\u7403\u9762\u4E0A\u7684\u6BCF\u500B\u65BD\u74E6\u8328\u4E09\u89D2\u5F62\u5B9A\u7FA9\u4E86\u4E00\u500B\u6709\u9650\u7FA4\uFF0C\u800C\u5728\u6B50\u6C0F\u6216\u96D9\u66F2\u5E73\u9762\uFF0C\u5247\u6703\u5B9A\u7FA9\u51FA\u4E00\u500B\u7121\u9650\u7FA4\u3002 \u65BD\u74E6\u8328\u4E09\u89D2\u5F62\u662F\u7531\u4E09\u500B\u6709\u7406\u6578(p q r)\u4F86\u4EE3\u8868\u6BCF\u500B\u9802\u9EDE\u7684\u89D2\u5EA6\u3002\u503Cn/d\u8868\u793A\u7684\u9802\u89D2\u70BA\u534A\u5713\u7684d/n\uFF0C\u201C2\u201D\u8868\u662F\u4E00\u500B\u76F4\u89D2\u3002\u82E5p\u3001q\u3001r\u7686\u70BA\u6574\u6578\uFF0C\u5247\u5C07\u5176\u7A31\u70BA\u83AB\u6BD4\u70CF\u65AF\u4E09\u89D2\u5F62\uFF08\u82F1\u8A9E\uFF1AM\u00F6bius triangle\uFF09\u4E26\u4E14\u5C0D\u61C9\u65BC\u4E00\u500B\u6C92\u6709\u91CD\u758A\u7684\u9472\u5D4C\uFF0C\u5176\u5C0D\u7A31\u7FA4\u7A31\u70BA\u4E00\u500B\u4E09\u89D2\u7FA4\u3002\u5728\u7403\u9762\u79FB\u5171\u67093\u500B\u83AB\u6BD4\u70CF\u65AF\u4E09\u89D2\u5F62\u52A0\u4E00\u500B\u55AE\u53C3\u6578\u65CF\uFF1B\u5728\u6B50\u6C0F\u5E73\u9762\u4E0A\u6709\u4E09\u500B\u83AB\u6BD4\u70CF\u65AF\u4E09\u89D2\u5F62\uFF1B\u800C\u5728\u7F85\u6C0F\u96D9\u66F2\u7A7A\u9593\u4E2D\u6709\u4E09\u500B\u53C3\u6578\u65CF\u7684\u83AB\u6BD4\u70CF\u65AF\u4E09\u89D2\u5F62\uFF0C\u4E26\u6C92\u6709\u7279\u4F8B\u3002"@zh . . . . . . . . "\u0641\u064A \u0627\u0644\u0647\u0646\u062F\u0633\u0629 \u0627\u0644\u0631\u064A\u0627\u0636\u064A\u0629\u060C \u0645\u062B\u0644\u062B \u0634\u0641\u0627\u0631\u0632 (\u0628\u0627\u0644\u0625\u0646\u062C\u0644\u064A\u0632\u064A\u0629: Schwarz triangle)\u200F\u060C \u0647\u0648 \u0645\u062B\u0644\u062B \u0643\u0631\u0648\u064A \u064A\u0645\u0643\u0646 \u0627\u0633\u062A\u0639\u0645\u0627\u0644\u0647 \u0645\u0646 \u0623\u062C\u0644 \u062A\u063A\u0637\u064A\u0629 \u0627\u0644\u0643\u0631\u0629 \u0643\u0627\u0645\u0644\u0629\u064B.\u0633\u0645\u064A \u0647\u0630\u0627 \u0627\u0644\u0645\u062B\u0644\u062B \u0647\u0643\u0630\u0627 \u0646\u0633\u0628\u0629 \u0625\u0644\u0649 \u0639\u0627\u0644\u0645 \u0627\u0644\u0631\u064A\u0627\u0636\u064A\u0627\u062A \u0647\u064A\u0631\u0645\u0627\u0646 \u0634\u0641\u0627\u0631\u0632."@ar . . . . . . . . . . "\u0422\u0440\u0435\u0443\u0433\u043E\u043B\u044C\u043D\u0438\u043A \u0428\u0432\u0430\u0440\u0446\u0430 \u2014 \u0441\u0444\u0435\u0440\u0438\u0447\u0435\u0441\u043A\u0438\u0439 \u0442\u0440\u0435\u0443\u0433\u043E\u043B\u044C\u043D\u0438\u043A, \u043A\u043E\u0442\u043E\u0440\u044B\u0439 \u043C\u043E\u0436\u043D\u043E \u0438\u0441\u043F\u043E\u043B\u044C\u0437\u043E\u0432\u0430\u0442\u044C \u0434\u043B\u044F \u0441\u043E\u0437\u0434\u0430\u043D\u0438\u044F \u043C\u043E\u0437\u0430\u0438\u043A\u0438 \u043D\u0430 \u0441\u0444\u0435\u0440\u0435, \u0432\u043E\u0437\u043C\u043E\u0436\u043D\u043E \u0441 \u043D\u0430\u043B\u043E\u0436\u0435\u043D\u0438\u0435\u043C, \u043F\u0443\u0442\u0451\u043C \u043E\u0442\u0440\u0430\u0436\u0435\u043D\u0438\u0439 \u0442\u0440\u0435\u0443\u0433\u043E\u043B\u044C\u043D\u0438\u043A\u0430 \u043E\u0442\u043D\u043E\u0441\u0438\u0442\u0435\u043B\u044C\u043D\u043E \u0441\u0442\u043E\u0440\u043E\u043D. \u0422\u0440\u0435\u0443\u0433\u043E\u043B\u044C\u043D\u0438\u043A\u0438 \u043A\u043B\u0430\u0441\u0441\u0438\u0444\u0438\u0446\u0438\u0440\u043E\u0432\u0430\u043D\u044B \u0432 \u0440\u0430\u0431\u043E\u0442\u0435 \u043D\u0435\u043C\u0435\u0446\u043A\u043E\u0433\u043E \u043C\u0430\u0442\u0435\u043C\u0430\u0442\u0438\u043A\u0430 \u041A\u0430\u0440\u043B\u0430 \u0428\u0432\u0430\u0440\u0446\u0430 1873 \u0433\u043E\u0434\u0430. \u0422\u0440\u0435\u0443\u0433\u043E\u043B\u044C\u043D\u0438\u043A\u0438 \u0428\u0432\u0430\u0440\u0446\u0430 \u043C\u043E\u0436\u043D\u043E \u043E\u043F\u0440\u0435\u0434\u0435\u043B\u0438\u0442\u044C \u0432 \u0431\u043E\u043B\u0435\u0435 \u043E\u0431\u0449\u0435\u043C \u0432\u0438\u0434\u0435 \u043A\u0430\u043A \u043C\u043E\u0437\u0430\u0438\u043A\u0438 \u043D\u0430 \u0441\u0444\u0435\u0440\u0435, \u0435\u0432\u043A\u043B\u0438\u0434\u043E\u0432\u043E\u0439 \u0438\u043B\u0438 \u0433\u0438\u043F\u0435\u0440\u0431\u043E\u043B\u0438\u0447\u0435\u0441\u043A\u043E\u0439 \u043F\u043B\u043E\u0441\u043A\u043E\u0441\u0442\u0438. \u041A\u0430\u0436\u0434\u044B\u0439 \u0442\u0440\u0435\u0443\u0433\u043E\u043B\u044C\u043D\u0438\u043A \u0428\u0432\u0430\u0440\u0446\u0430 \u043D\u0430 \u0441\u0444\u0435\u0440\u0435 \u043E\u043F\u0440\u0435\u0434\u0435\u043B\u044F\u0435\u0442 \u043A\u043E\u043D\u0435\u0447\u043D\u0443\u044E \u0433\u0440\u0443\u043F\u043F\u0443, \u0432 \u0442\u043E \u0432\u0440\u0435\u043C\u044F \u043A\u0430\u043A \u043D\u0430 \u0435\u0432\u043A\u043B\u0438\u0434\u043E\u0432\u043E\u0439 \u043F\u043B\u043E\u0441\u043A\u043E\u0441\u0442\u0438 \u043E\u043D\u0438 \u043E\u043F\u0440\u0435\u0434\u0435\u043B\u044F\u044E\u0442 \u0431\u0435\u0441\u043A\u043E\u043D\u0435\u0447\u043D\u044B\u0435 \u0433\u0440\u0443\u043F\u043F\u044B."@ru . . "En matematiko, triangulo de Schwarz estas sfera triangulo kiu povas esti uzata por kaheli sferon. \u0108iu triangulo de Schwarz difinas finian grupon - \u011Dian triangulan grupon. Pri \u011Di unuafoje en 1873 en Zuriko publikigis la germana matematikisto Hermann Amandus Schwarz (1843-1921), tial la nomo. Triangulo de Schwarz estas prezentata per tri racionalaj nombroj (a b c), \u0109iu el ili prezentas la angulo je vertico. Valoro n/d de la nombro signifas ke la vertica angulo estas d/n de la duoncirklo. Se \u011Di estas 2 tio estas orta triangulo. Estas nur 4 grupoj, anka\u016D nomataj kiel trianguloj de M\u00F6bius:"@eo . . . . . . "2069325"^^ . . . . . . "Triangulo de Schwarz"@eo . . 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\u041C\u0451\u0431\u0438\u0443\u0441\u0430 \u0441 \u0442\u0440\u0435\u043C\u044F \u043F\u0430\u0440\u0430\u043C\u0435\u0442\u0440\u0430\u043C\u0438 \u0438 \u043D\u0435\u0442 ."@ru . . . . . "In geometria, un triangolo di Schwarz \u00E8 un triangolo sferico che pu\u00F2 essere utilizzato per tassellare una sfera (creando una cosiddetta tassellatura sferica), eventualmente sovrapponendosi, attraverso continue riflessioni sui suoi bordi. Tali triangoli sono stati classificati dal matematico tedesco Hermann Schwarz nel 1873 e sono stati cos\u00EC chiamati proprio in suo onore."@it . . . .