Ludwig Schläfli Euclidean space higher dimensions polytopes higher-dimensional analogues polygons polyhedra Platonic solids regular convex polytopes in dimension four Euclid geometry
At mid-century Ludwig Schläfli developed the general concept of Euclidean space, extending Euclidean geometry to higher dimensions. He defined polyschemes, later called polytopes, which are the higher-dimensional analogues of polygons and polyhedra. He developed their theory and discovered all the regular polytopes, i.e. the 



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-dimensional analogues of regular polygons and Platonic solids. He found there are six regular convex polytopes in dimension four, and three in all higher dimensions.
