Mathematical Thought from Ancient to Modern Times by Kline Morris;

Author:Kline, Morris;
Language: eng
Format: epub
Publisher: Oxford University Press USA - OSO
Published: 1972-08-11T16:00:00+00:00

Travel through each subsquare so that the path shown corresponds to the unit segment. Now divide the unit square into 16 subsquares numbered as shown in Figure 42.5 and join the centers of the 16 subsquares as shown.

We continue the process of dividing each subsquare into four parts, numbering them so that we can traverse the entire set by a continuous path. The desired curve is the limit of the successive polygonal curves formed at each stage. Since the subsquares and the parts of the unit segment both contract to a point as the subdivision continues, we can see intuitively that each point on the unit segment maps into one point on the square. In fact, if we fix on one point in the unit segment, say t = 2/3, then the image of this point is the limit of the successive images of t = 2/3 which appear in the successive polygons.

These examples show that the definition of a curve Jordan suggested is not satisfactory because a curve, according to this definition, can fill out a square. The question of what is meant by a curve remained open. Felix Klein remarked in 189830 that nothing was more obscure than the notion of a curve. This question was taken up by the topologists (Chap. 50, sec. 2).

Figure 42.5

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