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In geometry, the Schläfli symbol is a notation of the form that defines regular polytopes and tessellations. The Schläfli symbol is named after the 19th-century mathematician Ludwig Schläfli who made important contributions in geometry and other areas. == Description == The Schläfli symbol is a recursive description, starting with a ''p''-sided regular polygon as '. For example, is an equilateral triangle, is a square and so on. A regular polyhedron which has ''q'' regular p-sided polygon faces around each vertex is represented by . For example, the cube has 3 squares around each vertex and is represented by . A regular 4-dimensional polytope, with ''r'' regular polyhedral cells around each edge is represented by , and so on. Regular polytopes can have star polygon elements, like the pentagram, with symbol , represented by the vertices of a pentagon but connected alternately. A facet of a regular polytope is . A regular polytope has a regular vertex figure. The vertex figure of a regular polytope is . The Schläfli symbol can represent a finite convex polyhedron, an infinite tessellation of Euclidean space, or an infinite tessellation of hyperbolic space, depending on the angle defect of the construction. A positive angle defect allows the vertex figure to ''fold'' into a higher dimension and loops back into itself as a polytope. A zero angle defect will tessellate space of the same dimension as the facets. A negative angle defect can't exist in ordinary space, but can be constructed in hyperbolic space. Usually, a facet or a vertex figure is assumed to be a finite polytope, but can sometimes be considered a tessellation itself. A regular polytope also has a dual polytope, represented by the ''Schläfli symbol'' elements in reverse order. A self-dual regular polytope will have a symmetric Schläfli symbol. 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「Schläfli symbol」の詳細全文を読む スポンサード リンク
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