We are not limited to any particular direction for the lines in the plane about which we suppose the rotation of the particular sections to take place. Let us draw the section of the cube, [fig. 36], through A, F, C, H, forming a sloping plane. Now since the fourth dimension is at right angles to every line in our space it is at right angles to this section also. We can represent our space by drawing an axis at right angles to the plane ACEG, our space is then determined by the plane ACEG, and the perpendicular axis. If we let this axis drop and suppose the fourth axis, w, to take its place, we have a representation of the space which runs off in the fourth dimension from the plane ACEG. In this space we shall see simply the section ACEG of the cube, and nothing else, for one cube does not extend to any distance in the fourth dimension.
Fig. 42.
If, keeping this plane, we bring in the fourth dimension, we shall have a space in which simply this section of the cube exists and nothing else. The section can turn about the line AF, and parallel sections can turn about parallel lines. Thus in considering the rotation about a plane we can draw any lines we like and consider the rotation as taking place in sections about them.
To bring out this point more clearly let us take two parallel lines, A and B, in the space of xyz, and let CD and EF be two rods running above and below the plane of xy, from these lines. If we turn these rods in our space about the lines A and B, as the upper end of one, F, is going down, the lower end of the other, C, will be coming up. They will meet and conflict. But it is quite possible for these two rods each of them to turn about the two lines without altering their relative distances.
To see this suppose the y axis to go, and let the w axis take its place. We shall see the lines A and B no longer, for they run in the y direction from the points G and H.
Fig. 43.
Fig. 43 is a picture of the two rods seen in the space of xzw. If they rotate in the direction shown by the arrows—in the z to w direction—they move parallel to one another, keeping their relative distances. Each will rotate about its own line, but their rotation will not be inconsistent with their forming part of a rigid body.
Now we have but to suppose a central plane with rods crossing it at every point, like CD and EF cross the plane of xy, to have an image of a mass of matter extending equal distances on each side of a diametral plane. As two of these rods can rotate round, so can all, and the whole mass of matter can rotate round its diametral plane.