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|- !colspan=2|Regular (3D) polyhedra |- !Convex !Star |- align=center |150px |150px |- !colspan=2|Regular 2D tessellations |- !Euclidean !Hyperbolic |- align=center |150px |150px |- !colspan=2|Regular 4D polytopes |- !Convex !Star |- align=center |150px |150px |- !colspan=2|Regular 3D tessellations |- !Euclidean !Hyperbolic |- align=center |150px |150px |} This page lists the regular polytopes and regular polytope compounds in Euclidean, spherical and 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 a (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, () or , is represented by Coxeter diagram . The regular polytopes are grouped by dimension and subgrouped by convex, nonconvex and infinite forms. Nonconvex forms use the same vertices as the convex forms, but have intersecting facets. Infinite forms tessellate a one-lower-dimensional Euclidean space. Infinite forms can be extended to tessellate a hyperbolic space. Hyperbolic space is like normal space at a small scale, but parallel lines diverge at a distance. This allows vertex figures to have negative angle defects, like making a vertex with seven equilateral triangles and allowing it to lie flat. It cannot be done in a regular plane, but can be at the right scale of a hyperbolic plane. A more general definition of regular polytopes which do not have simple Schläfli symbols includes regular skew polytopes and regular skew apeirotopes with nonplanar facets or vertex figures. ==Overview== This table shows a summary of regular polytope counts by dimension. * There are no Euclidean regular star tessellations in any number of dimensions. 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「List of regular polytopes and compounds」の詳細全文を読む スポンサード リンク
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