Let us now apply the method by which a plane being could examine the nature of rotation about a line in our examination of rotation about a plane. Fig. 3 represents a cube in our space, the three axes _x_, _y_, _z_ denoting its three dimensions. Let _w_ represent the fourth dimension. Now, since in our space we can represent any three dimensions, we can, if we choose, make a representation of what is in the space determined by the three axes _x_, _z_, _w_. This is a three-dimensional space determined by two of the axes we have drawn, _x_ and _z_, and in place of _y_ the fourth axis, _w_. We cannot, keeping _x_ and _z_, have both _y_ and _w_ in our space; so we will let _y_ go and draw _w_ in its place. What will be our view of the cube? Evidently we shall have simply the square that is in the plane of _xz_, the square ACDB. The rest of the cube stretches in the _y_ direction, and, as we have none of the space so determined, we have only the face of the cube. This is represented in fig. 4. [Illustration: Fig. 4 (132).] Now, suppose the whole cube to be turned from the _x_ to the _w_ direction. Conformably with our method, we will not take the whole of the cube into consideration at once, but will begin with the face ABCD. 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. [Illustration: Fig. 5 (133).] 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. 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. [Illustration: Fig. 6 (134).] 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. 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. [Illustration: Fig. 7 (135).] 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. [Illustration: Fig. 8 (136).]