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In elementary geometry, a polytope is a geometric object with flat sides, and may exist in any general number of dimensions ''n'' as an ''n''-dimensional polytope or ''n''-polytope. For example a two-dimensional polygon is a 2-polytope and a three-dimensional polyhedron is a 3-polytope. Some theories further generalize the idea to include such objects as unbounded (apeirotopes and tessellations), decompositions or tilings of curved manifolds such as spherical polyhedra, and set-theoretic abstract polytopes. Polytopes in more than three dimensions were first discovered by Ludwig Schläfli. The term "polytop" was coined by the mathematician Reinhold Hoppe, writing in German, and was introduced to English mathematicians in its present form by Alicia Boole Stott. ==Approaches to definition== The term ''polytope'' is nowadays a broad term that covers a wide class of objects, and different definitions are attested in mathematical literature. Many of these definitions are not equivalent, resulting in different sets of objects being called ''polytopes''. They represent different approaches to generalizing the convex polytopes to include other objects with similar properties. The original approach broadly followed by Ludwig Schläfli, Thorold Gosset and others begins with the extension by analogy into four or more dimensions, of the idea of a polygon and polyhedron respectively in two and three dimensions.〔Coxeter (1973)〕 Attempts to generalise the Euler characteristic of polyhedra to higher-dimensional polytopes led to the development of topology and the treatment of a decomposition or CW-complex as analogous to a polytope.〔Richeson, S.; ''Euler's Gem: The Polyhedron Formula and the Birth of Topology'', Princeton University, 2008.〕 In this approach, a polytope may be regarded as a tessellation or decomposition of some given manifold. An example of this approach defines a polytope as a set of points that admits a simplicial decomposition. In this definition, a polytope is the union of finitely many simplices, with the additional property that, for any two simplices that have a nonempty intersection, their intersection is a vertex, edge, or higher dimensional face of the two.〔Grünbaum (2003)〕 However this definition does not allow star polytopes with interior structures, and so is restricted to certain areas of mathematics. The discovery of star polyhedra and other unusual constructions led to the idea of a polyhedron as a bounding surface, ignoring its interior.〔Cromwell, P.; ''Polyhedra'', CUP (ppbk 1999) pp 205 ff.〕 In this light convex polytopes in ''p''-space are equivalent to tilings of the (''p''−1)-sphere, while others may be tilings of other elliptic, flat or toroidal (''p''−1)-surfaces – see elliptic tiling and toroidal polyhedron. A polyhedron is understood as a surface whose faces are polygons, a 4-polytope as a hypersurface whose facets (cells) are polyhedra, and so forth. The idea of constructing a higher polytope from those of lower dimension is also sometimes extended downwards in dimension, with an (edge) seen as a 1-polytope bounded by a point pair, and a point or vertex as a 0-polytope. This approach is used for example in the theory of abstract polytopes. In certain fields of mathematics, the terms "polytope" and "polyhedron" are used in a different sense: a ''polyhedron'' is the generic object in any dimension (referred to as ''polytope'' in this Wikipedia article) and ''polytope'' means a bounded polyhedron.〔Nemhauser and Wolsey, "Integer and Combinatorial Optimization," 1999, ISBN 978-0471359432, Definition 2.2.〕 This terminology is typically confined to polytopes and polyhedra that are convex. With this terminology, a convex polyhedron is the intersection of a finite number of halfspaces and is defined by its sides while a convex polytope is the convex hull of a finite number of points and is defined by its vertices. 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「Polytope」の詳細全文を読む スポンサード リンク
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