As in the case of a plane being, the result of rotation about a line would appear as the production of a looking-glass image of the original object on the other side of the line, so to us the result of a four-dimensional rotation would appear like the production of a looking-glass image of a body on the other side of a plane. The plane would be the axis of the rotation, and the path of the body between its two appearances would be unimaginable in three-dimensional space.
Fig. 3 (131).
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?
Fig. 4 (132).
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].
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.
Fig. 5 (133).