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In elementary geometry, a polyhedron (plural polyhedra or polyhedrons) is a solid in three dimensions with flat polygonal faces, straight edges and sharp corners or vertices. The word polyhedron comes from the Classical Greek πολύεδρον, as ''poly-'' (stem of πολύς, "many") + ''-hedron'' (form of ἕδρα, "base" or "seat"). Cubes and pyramids are examples of polyhedra. A polyhedron is said to be convex if its surface (comprising its faces, edges and vertices) does not intersect itself and the line segment joining any two points of the polyhedron is contained in the interior or surface. A polyhedron is a 3-dimensional example of the more general polytope in any number of dimensions. ==Basis for definition== In elementary geometry, the faces are polygons – regions of planes – meeting in pairs along their edges which are straight-line segments, and with the edges meeting in vertex points. Treating a polyhedron as a solid bounded by flat faces and straight edges is not very precise, for example it is difficult to reconcile with star polyhedra. Grünbaum (1994, p. 43) observed, "The Original Sin in the theory of polyhedra goes back to Euclid, and through Kepler, Poinsot, Cauchy and many others ... (that ) at each stage ... the writers failed to define what are the 'polyhedra' ...." Many definitions of "polyhedron" have been given within particular contexts, some more rigorous than others.〔Lakatos, I.; ''Proofs and refutations: The logic of mathematical discovery'' (2nd Ed.), CUP, 1977.〕 For example definitions based on the idea of a bounding surface rather than a solid are common.〔Cromwell (1997).〕 However such definitions are not always compatible in other mathematical contexts. One modern approach treats a geometric polyhedron as an injection into real space, a ''realisation'', of some abstract polyhedron.〔Grünbaum 2003〕 Any such polyhedron can be built up from different kinds of element or entity, each associated with a different number of dimensions: *3 dimensions: The interior is the volume bounded by the faces. It might or might not be realised as a solid body. *2 dimensions: A face is a ''polygon'' bounded by a circuit of edges, and usually also realises the flat (plane) region inside the boundary. These polygonal faces together make up the polyhedral surface. *1 dimension: An edge joins one vertex to another and one face to another, and is usually a line segment. The edges together make up the polyhedral skeleton. *0 dimensions: A vertex (plural vertices) is a corner point. Different approaches - and definitions - may require different realisations. Sometimes the interior volume is considered to be part of the polyhedron, sometimes only the surface is considered, and occasionally only the skeleton of edges or even just the set of vertices.〔 In such elementary geometric and set-based definitions, a polyhedron is typically understood as a three-dimensional example of the more general polytope in any number of dimensions. For example a polygon has a two-dimensional body and no faces, while a 4-polytope has a four-dimensional body and an additional set of three-dimensional "cells". In other mathematical disciplines, the term "polyhedron" may be used to refer to a variety of specialised constructs, some geometric and others purely algebraic or abstract. In such contexts definition of the term "polyhedron" may not be consistent with a polytope but rather in contrast to it.〔Grünbaum, B.; "Convex polytopes," 2nd Edition, Springer (2003).〕 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「Polyhedron」の詳細全文を読む スポンサード リンク
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