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Uniform tilings in hyperbolic plane Однородные мозаики на гиперболической плоскости
rdfs:comment
In hyperbolic geometry, a uniform hyperbolic tiling (or regular, quasiregular or semiregular hyperbolic tiling) is an edge-to-edge filling of the hyperbolic plane which has regular polygons as faces and is vertex-transitive (transitive on its vertices, isogonal, i.e. there is an isometry mapping any vertex onto any other). It follows that all vertices are congruent, and the tiling has a high degree of rotational and translational symmetry. В гиперболической геометрии однородная (правильная, квазиправильная или полуправильная) гиперболическая мозаика — это заполнение гиперболической плоскости правильными многоугольниками ребро-к-ребру со свойством вершинной транзитивности (это мозаика транзитивная относительно вершин, изогональная, т.е. существует движение, переводящее любую вершину в любую другую). Отсюда следует, что все вершины конгруэнтны и мозаика имеет высокую степень вращательной и трансляционной симметрии.
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В гиперболической геометрии однородная (правильная, квазиправильная или полуправильная) гиперболическая мозаика — это заполнение гиперболической плоскости правильными многоугольниками ребро-к-ребру со свойством вершинной транзитивности (это мозаика транзитивная относительно вершин, изогональная, т.е. существует движение, переводящее любую вершину в любую другую). Отсюда следует, что все вершины конгруэнтны и мозаика имеет высокую степень вращательной и трансляционной симметрии. Однородные мозаики однозначно определяются их вершинной конфигурацией, последовательностью чисел, представляющих число сторон многоугольников вокруг каждой вершины. Например, 7.7.7 представляет семиугольную мозаику, имеющую 3 семиугольника вокруг каждой вершины. Она правильна, поскольку все многоугольники имеют один размер. Таким образом, её можно задать символом Шлефли {7,3}. Однородные мозаики могут быть правильными (если они также транзитивны по граням и рёбрам), квазиправильными (если они рёберно транзитивны, но не транзитивны по граням) или полуправильными (если они не транзитивны ни по рёбрам, ни по граням). Для правильных треугольников (p q 2) существуют две правильные мозаики с символами Шлефли {p,q} и {q,p}. In hyperbolic geometry, a uniform hyperbolic tiling (or regular, quasiregular or semiregular hyperbolic tiling) is an edge-to-edge filling of the hyperbolic plane which has regular polygons as faces and is vertex-transitive (transitive on its vertices, isogonal, i.e. there is an isometry mapping any vertex onto any other). It follows that all vertices are congruent, and the tiling has a high degree of rotational and translational symmetry. Uniform tilings can be identified by their vertex configuration, a sequence of numbers representing the number of sides of the polygons around each vertex. For example, 7.7.7 represents the heptagonal tiling which has 3 heptagons around each vertex. It is also regular since all the polygons are the same size, so it can also be given the Schläfli symbol {7,3}. Uniform tilings may be regular (if also face- and edge-transitive), quasi-regular (if edge-transitive but not face-transitive) or semi-regular (if neither edge- nor face-transitive). For right triangles (p q 2), there are two regular tilings, represented by Schläfli symbol {p,q} and {q,p}.
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