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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Clifford torus ： ウィキペディア英語版 Clifford torus In geometric topology, the Clifford torus is a special kind of torus sitting inside the unit 3-sphere S3 in R4, the Euclidean space of four dimensions. Or equivalently, it can be seen as a torus sitting inside C2 since C2 is topologically equivalent to R4. It is specifically the torus in S3 that is geometrically the cartesian product of two circles, each of radius sqrt(1/2). The Clifford torus is an example of a square torus, because it is isometric to a square with opposite sides identified. It is further known as a Euclidean 2-torus (the "2" is its topological dimension); figures drawn on it obey Euclidean geometry as if it were flat, whereas the surface of a common "doughnut"-shaped torus is positively curved on the outer rim and negatively curved on the inner. Although having a different geometry than the standard embedding of a torus in three-dimensional Euclidean space, the square torus can also be embedded into three-dimensional space, by the Nash embedding theorem; one possible embedding modifies the standard torus by a fractal set of ripples running in two perpendicular directions along the surface.〔.〕 == Formal definition == The unit circle S1 in R2 can be parameterized by an angle coordinate: : In another copy of R2, take another copy of the unit circle : Then the Clifford torus is : Since each copy of S1 is an embedded submanifold of R2, the Clifford torus is an embedded torus in R2 × R2 = R4. If R4 is given by coordinates (''x''1, ''y''1, ''x''2, ''y''2), then the Clifford torus is given by : $x\_1^2\; +\; y\_1^2\; =\; 1/2\; =\; x\_2^2\; +\; y\_2^2.\; \backslash ,$. This shows that in R4 the Clifford torus is a submanifold of the unit 3-sphere S3. It is easy to verify that the Clifford torus is in fact a minimal surface in S3. 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「Clifford torus」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.