TEE FOURTH DIMENSION 



39 1 



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 12 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 





2 



4 



8 



16 



1 



4 

 12 

 32 





 1 



6 



24 















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 



4 & 



IS- 



Fig. 12. 



f. 



i e 



(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. 5 This may be made clear by 

 analogy. Suppose we have in a plane (Fig. 12) a nearly closed polygon 



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



