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5-cell Prism
In geometry, a uniform 5-polytope is a five-dimensional uniform polytope. By definition, a uniform 5-polytope is vertex-transitive and constructed from uniform 4-polytope Facet (geometry), facets. The complete set of convex uniform 5-polytopes has not been determined, but many can be made as Wythoff constructions from a small set of Coxeter groups, symmetry groups. These construction operations are represented by the permutations of rings of the Coxeter diagrams. History of discovery *Regular polytopes: (convex faces) **1852: Ludwig Schläfli proved in his manuscript ''Theorie der vielfachen Kontinuität'' that there are exactly 3 regular polytopes in 5 or more dimensions. *Convex semiregular polytopes: (Various definitions before Coxeter's uniform category) **1900: Thorold Gosset enumerated the list of nonprismatic semiregular convex polytopes with regular facets (convex regular 4-polytopes) in his publication ''On the Regular and Semi-Regular Figures in Space of n Dimension ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
List Of Regular Polytopes
This article lists the regular polytopes and regular polytope compounds in Euclidean geometry, Euclidean, spherical geometry, spherical and hyperbolic geometry, hyperbolic spaces. The Schläfli symbol describes every regular tessellation of an ''n''-sphere, Euclidean and hyperbolic spaces. A Schläfli symbol describing an ''n''-polytope equivalently describes a tessellation of an (''n'' − 1)-sphere. In addition, the symmetry of a regular polytope or tessellation is expressed as a Coxeter group, which Coxeter expressed identically to the Schläfli symbol, except delimiting by square brackets, a notation that is called Coxeter notation. Another related symbol is the Coxeter-Dynkin diagram which represents a symmetry group with no rings, and the represents regular polytope or tessellation with a ring on the first node. For example, the cube has Schläfli symbol , and with its octahedral symmetry, [4,3] or , it is represented by Coxeter diagram . The regular polytopes are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
