Uniform polytope
Isogonal polytope with uniform facets / From Wikipedia, the free encyclopedia
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In geometry, a uniform polytope of dimension three or higher is a vertex-transitive polytope bounded by uniform facets. The uniform polytopes in two dimensions are the regular polygons (the definition is different in 2 dimensions to exclude vertex-transitive even-sided polygons that alternate two different lengths of edges).
This article may be confusing or unclear to readers. (September 2008) |
2D | 3D |
---|---|
Truncated triangle or uniform hexagon, with Coxeter diagram . |
Truncated octahedron, |
4D | 5D |
Truncated 16-cell, |
Truncated 5-orthoplex, |
This is a generalization of the older category of semiregular polytopes, but also includes the regular polytopes. Further, star regular faces and vertex figures (star polygons) are allowed, which greatly expand the possible solutions. A strict definition requires uniform polytopes to be finite, while a more expansive definition allows uniform honeycombs (2-dimensional tilings and higher dimensional honeycombs) of Euclidean and hyperbolic space to be considered polytopes as well.