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In mathematics, the ''n''-sphere is the generalization of the ordinary sphere to spaces of arbitrary dimension. For any natural number ''n'', an ''n''-sphere of radius ''r'' is defined as the set of points in (''n'' + 1)-dimensional Euclidean space which are at distance ''r'' from a central point, where the radius ''r'' may be any positive real number. Thus, the ''n''-sphere centred at the origin is defined by: : It is an ''n''-dimensional manifold in Euclidean (''n'' + 1)-space. In particular: :a 0-sphere is the pair of points at the ends of a (one-dimensional) line segment, :a 1-sphere is the circle, which is the one-dimensional circumference of a (two-dimensional) disk in the plane, :a 2-sphere is the two-dimensional surface of a (three-dimensional) ball in three-dimensional space. Spheres of dimension are sometimes called hyperspheres, with a 3-sphere sometimes known as a glome. The ''n''-sphere of unit radius centered at the origin is called the unit ''n''-sphere, denoted ''S''''n''. The unit ''n''-sphere is often referred to as ''the'' ''n''-sphere. An ''n''-sphere is the surface or boundary of an -dimensional ball, and is an ''n''-dimensional manifold. For , the ''n''-spheres are the simply connected ''n''-dimensional manifolds of constant, positive curvature. The ''n''-spheres admit several other topological descriptions: for example, they can be constructed by gluing two ''n''-dimensional Euclidean spaces together, by identifying the boundary of an ''n''-cube with a point, or (inductively) by forming the suspension of an -sphere. ==Description== For any natural number ''n'', an ''n''-sphere of radius ''r'' is defined as the set of points in (''n'' + 1)-dimensional Euclidean space that are at distance ''r'' from some fixed point c, where ''r'' may be any positive real number and where c may be any point in (''n'' + 1)-dimensional space. In particular: * a 0-sphere is a pair of points , and is the boundary of a line segment (1-ball). * a 1-sphere is a circle of radius ''r'' centered at c, and is the boundary of a disk (2-ball). * a 2-sphere is an ordinary 2-dimensional sphere in 3-dimensional Euclidean space, and is the boundary of an ordinary ball (3-ball). * a 3-sphere is a sphere in 4-dimensional Euclidean space. 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「N-sphere」の詳細全文を読む スポンサード リンク
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