Page:Popular Science Monthly Volume 83.djvu/395
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THE FOURTH DIMENSION
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 posi- tion of the moving cube, and 13 described by the bounding lines of the moving cube.
Bounding Cubes. — Finally, of bounding cubes, 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 :
BOUNDAEIES
�� �Points
�Lines
�Squares
�Cubes
�One-dimensional unit
�2
4
8
16
�1
4 12 32
� 1
6 24
�
�Two-dimensional unit
�
�Three-dimensional unit
Four-dimensional unit
�1 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 hyper- space, 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.^ This may be made clear by analogy. Suppose we have in a plane (Fig. 12) a nearly closed polygon 'A right glove turned inside out in our space becomes a left glove.
��
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