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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 Irrational winding of a torus ： ウィキペディア英語版 Irrational winding of a torus In topology, a branch of mathematics, an irrational winding of a torus is a continuous injection of a line into a torus that is used to set up several counterexamples. A related notion is the Kronecker foliation of a torus, a foliation formed by the set of all translates of a given irrational winding. == Definition == One way of constructing a torus is as the quotient space $T^2\; =\; \backslash mathbb^2\; /\; \backslash mathbb^2$ of a two-dimensional real vector space by the additive subgroup of integer vectors, with the corresponding projection $\backslash pi:\; \backslash mathbb^2\; \backslash to\; T^2$. Each point in the torus has as its preimage one of the translates of the square lattice $\backslash mathbb^2$ in $\backslash mathbb^2$, and $\backslash pi$ factors through a map that takes any point in the plane to a point in the unit square $[0,\; 1)^2$ given by the fractional parts of the original point's Cartesian coordinates. Now consider a line in $\backslash mathbb^2$ given by the equation ''y = kx''. If the slope ''k'' of the line is rational, then it can be represented by a fraction and a corresponding lattice point of $\backslash mathbb^2$. It can be shown that then the projection of this line is a simple closed curve on a torus. If, however, ''k'' is irrational, then it will not cross any lattice points except 0, which means that its projection on the torus will not be a closed curve, and the restriction of $\backslash pi$ on this line is injective. Moreover, it can be shown that the image of this restricted projection as a subspace, called the irrational winding of a torus, is dense in the torus. 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「Irrational winding of a 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.