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. |- valign=top |160px A regular dodecaplex is a 4-polytope, a four-dimensional polytope, with 120 dodecahedral cells, represented by Schläfli symbol . (shown here as a Schlegel diagram) |160px A regular cubic honeycomb is a tessellation, an infinite three-dimensional polytope, represented by Schläfli symbol . |- |colspan=2|320px 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 is transitive 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 () 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 , with regular facets as , and regular vertex figures as . ==Classification and description== Regular polytopes are classified primarily according to their dimensionality. They can be further classified according to symmetry. For example the cube and the regular octahedron share the same symmetry, as do the regular dodecahedron and icosahedron. Indeed, symmetry groups are sometimes named after regular polytopes, for example the tetrahedral and icosahedral symmetries. Three special classes of regular polytope exist in every dimensionality: *Regular simplex *Measure polytope (Hypercube) *Cross polytope (Orthoplex) In two dimensions there are infinitely many regular polygons. In three and four dimensions there are several more regular polyhedra and 4-polytopes besides these three. In five dimensions and above, these are the only ones. See also the list of regular polytopes. The idea of a polytope is sometimes generalised to include related kinds of geometrical object. Some of these have regular examples, as discussed in the section on historical discovery below. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Regular polytope」の詳細全文を読む スポンサード リンク
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