Regular polytope

MarinOsion45 0 275 01.26 05:35
This article includes a list of references, but its sources remain unclear because it has insufficient inline citations. Please help to improve this article by introducing more precise citations. (July 2014) (Learn how and when to remove this template message) Regular polytope examples A regular pentagon is a polygon, a two-dimensional polytope with 5 edges, represented by Schläfli symbol {5}. A regular dodecahedron is a polyhedron, a three-dimensional polytope, with 12 pentagonal faces, represented by Schläfli symbol {5,3}. A regular dodecaplex is a 4-polytope, a four-dimensional polytope, with 120 dodecahedral cells, represented by Schläfli symbol {5,3,3}. (shown here as a Schlegel diagram) A regular cubic honeycomb is a tessellation, an infinite three-dimensional polytope, represented by Schläfli symbol {4,3,4}. The 256 vertices and 1024 edges of an 8-cube can be shown in this orthogonal projection (Petrie polygon) In mathematics, a regular polytope is a polytope whose symmetry group acts transitively on its flags, thus giving it the highest degree of symmetry. All its elements or j-faces (for all 0 ≤ j ≤ n, where n is the dimension of the polytope) — cells, faces and so on — are also transitive on the symmetries of the polytope, and are regular polytopes of dimension ≤ n. Regular polytopes are the generalized analog in any number of dimensions of regular polygons (for example, the square or the regular pentagon) and regular polyhedra (for example, the cube). The strong symmetry of the regular polytopes gives them an aesthetic quality that interests both non-mathematicians and mathematicians. Classically, a regular polytope in n dimensions may be defined as having regular facets [(n − 1)-faces] and regular vertex figures. These two conditions are sufficient to ensure that all faces are alike and all vertices are alike. Note, however, that this definition does not work for abstract polytopes. A regular polytope can be represented by a Schläfli symbol of the form {a, b, c, ...., y, z}, with regular facets as {a, b, c, ..., y}, and regular vertex figures as {b, c, ..., y, z}. Contents 1 Classification and description 1.1 Schläfli symbols 1.2 Duality of the regular polytopes 1.3 Regular simplices 1.4 Measure polytopes (hypercubes) 1.5 Cross polytopes (orthoplexes) 2 History of discovery 2.1 Convex polygons and po 밤헌터


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