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uniform polytope
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, although even-sided polygons can be seen as uniform by alternating two colors of edges, represented by a 2-ring Coxeter diagram .
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.
== Operations ==
Nearly every uniform polytope can be generated by a Wythoff construction, and represented by a Coxeter diagram. Notable exceptions include the grand antiprism in four dimensions. The terminology for the convex uniform polytopes used in uniform polyhedron, uniform 4-polytope, uniform 5-polytope, uniform 6-polytope, uniform tiling, and convex uniform honeycomb articles were coined by Norman Johnson.
Equivalently, the Wythoffian polytopes can be generated by applying basic operations to the regular polytopes in that dimension. This approach was first used by Johannes Kepler, and is the basis of the Conway polyhedron notation.
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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