# Page:Popular Science Monthly Volume 83.djvu/395

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THE FOURTH DIMENSION

${\displaystyle ABCD}$, and 4 described by the bounding lines of the moving square; and the hypercube has 24,—6 each from the initial and the final position of the moving cube, and 12 described by the bounding lines of the moving cube.

Bounding Cubes.—Finally, of bounding cubes, ${\displaystyle ABCD-G}$ has one (itself); and the hypercube has 8,—one each from the initial and the final position of the moving cube, and 6 described by the bounding squares of the moving cube.

The results obtained for the boundaries may be conveniently exhibited by the following table:

Boundaries

 Points Lines Squares Cubes One-dimensional unit 2 1 0 0 Two-dimensional unit 4 4 1 0 Three-dimensional unit 8 12 6 1 Four-dimensional unit 16 32 24 8

Freedom of movement is greater in hyperspace than in our space. The degrees of freedom of a rigid body in our space are 6, namely, 3 translations along and 3 rotations about 3 axes, while the fixing of 3 of its points, not in a straight line, prevents all movement. In hyperspace, however, with 3 of its points fixed, it could still rotate about the plane of those 3 points. A rigid body has 10 possible different movements in hyperspace, namely, 4 translations along 4 axes, and 6 rotations about 6 planes, while at least 4 of its points must be fixed to prevent all movement.

In hyperspace, a sphere of flexible material could without stretching or tearing be turned inside out. Two links of a chain could be separated without breaking them. Our knots would be useless. In hyperspace, as we have seen, it would be entirely possible to pass in and out of a sphere

Fig. 12.

(or other enclosed space). A right glove turned over through space of four dimensions becomes a left glove, but notice that when the glove is turned over, it is not turned inside out.[1] This may be made clear by analogy. Suppose we have in a plane (Fig. 12) a nearly closed polygon

1. A right glove turned inside out in our space becomes a left glove.