Let this face begin to turn. Fig. 5 represents one of the positions it will occupy; the line AB remains on the z axis. The rest of the face extends between the x and the w direction.
Now, since we can take any three axes, let us look at what lies in the space of zyw, and examine the turning there. We must now let the z axis disappear and let the w axis run in the direction in which the z ran.
Fig. 6 (134).
Making this representation, what do we see of the cube? Obviously we see only the lower face. The rest of the cube lies in the space of xyz. In the space of xyz we have merely the base of the cube lying in the plane of xy, as shown in [fig. 6].
Now let the x to w turning take place. The square ACEG will turn about the line AE. This edge will remain along the y axis and will be stationary, however far the square turns.
Fig. 7 (135).
Thus, if the cube be turned by an x to w turning, both the edge AB and the edge AC remain stationary; hence the whole face ABEF in the yz plane remains fixed. The turning has taken place about the face ABEF.
Suppose this turning to continue till AC runs to the left from A. The cube will occupy the position shown in [fig. 8]. This is the looking-glass image of the cube in [fig. 3]. By no rotation in three-dimensional space can the cube be brought from the position in [fig. 3] to that shown in [fig. 8].