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In mathematics, the are three disjoint connected open sets of the plane or open unit square with the counterintuitive property that they all have the same boundary. More than two sets with the same boundary are said to have the Wada property; examples include Wada basins in dynamical systems. The lakes of Wada were introduced by , who credited the discovery to Takeo Wada. His construction is similar to the construction by of an indecomposable continuum, and in fact it is possible for the common boundary of the three sets to be an indecomposable continuum. ==Construction of the lakes of Wada== The Lakes of Wada are formed by starting with a open unit square of dry land, and then digging 3 lakes according to the following rule: *On day ''n'' = 1, 2, 3,... extend lake ''n'' mod 3 (=0, 1, 2) so that it is open and connected and passes within a distance 1/''n'' of all remaining dry land. This should be done so that the remaining dry land remains homeomorphic to a open unit square. After an infinite number of days, the three lakes are still disjoint connected open sets, and the remaining dry land is the boundary of each of the 3 lakes. For example, the first five days might be (see the image on the right): # Dig a blue lake of width 1/3 passing within √2/3 of all dry land. # Dig a red lake of width 1/32 passing within √2/32 of all dry land. # Dig a green lake of width 1/33 passing within √2/33 of all dry land. # Extend the blue lake by a channel of width 1/34 passing within √2/34 of all dry land. (The small channel connects the thin blue lake to the thick one, near the middle of the image.) # Extend the red lake by a channel of width 1/35 passing within √2/35 of all dry land. (The tiny channel connects the thin red lake to the thick one, near the top left of the image.) A variation of this construction can produce a countable infinite number of connected lakes with the same boundary: instead of extending the lakes in the order 1, 2, 0, 1, 2, 0, 1, 2, 0, ...., extend them in the order 0, 0, 1, 0, 1, 2, 0, 1, 2, 3, 0, 1, 2, 3, 4, ...and so on. 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「Lakes of Wada」の詳細全文を読む スポンサード リンク
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