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In mathematics, a 3-sphere (also called a glome) is a higher-dimensional analogue of a sphere. It consists of the set of points equidistant from a fixed central point in 4-dimensional Euclidean space. Just as an ordinary sphere (or 2-sphere) is a two-dimensional surface that forms the boundary of a ball in three dimensions, a 3-sphere is an object with three dimensions that forms the boundary of a ball in four dimensions. A 3-sphere is an example of a 3-manifold. ==Definition== In coordinates, a 3-sphere with center (''C''0, ''C''1, ''C''2, ''C''3) and radius ''r'' is the set of all points (''x''0, ''x''1, ''x''2, ''x''3) in real, 4-dimensional space (R4) such that : The 3-sphere centered at the origin with radius 1 is called the unit 3-sphere and is usually denoted ''S''3: : It is often convenient to regard R4 as the space with 2 complex dimensions (C2) or the quaternions (H). The unit 3-sphere is then given by : or : This description as the quaternions of norm one, identifies the 3-sphere with the versors in the quaternion division ring. Just as the unit circle is important for planar polar coordinates, so the 3-sphere is important in the polar view of 4-space involved in quaternion multiplication. See polar decomposition of a quaternion for details of this development of the three-sphere. This view of the 3-sphere is the basis for the study of elliptic space as developed by Georges Lemaître〔Georges Lemaître (1948) "Quaternions et espace elliptique", ''Acta'' Pontifical Academy of Sciences 12:57–78〕 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「3-sphere」の詳細全文を読む スポンサード リンク
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