If the two rotations are equal in velocity, every point in the body describes a circle. All points equally distant from the stationary point describe circles of equal size.
We can represent a four-dimensional sphere by means of two diagrams, in one of which we take the three axes, x, y, z; in the other the axes x, w, and z. In [fig. 13] we have the view of a four-dimensional sphere in the space of xyz. Fig. 13 shows all that we can see of the four sphere in the space of xyz, for it represents all the points in that space, which are at an equal distance from the centre.
Let us now take the xz section, and let the axis of w take the place of the y axis. Here, in [fig. 14], we have the space of xzw. In this space we have to take all the points which are at the same distance from the centre, consequently we have another sphere. If we had a three-dimensional sphere, as has been shown before, we should have merely a circle in the xzw space, the xz circle seen in the space of xzw. But now, taking the view in the space of xzw, we have a sphere in that space also. In a similar manner, whichever set of three axes we take, we obtain a sphere.
Showing axes xyz
Fig. 13 (141).
Showing axes xwz
Fig. 14 (142).
In [fig. 13], let us imagine the rotation in the direction xy to be taking place. The point x will turn to y, and p to p´. The axis zz´ remains stationary, and this axis is all of the plane zw which we can see in the space section exhibited in the figure.
In [fig. 14], imagine the rotation from z to w to be taking place. The w axis now occupies the position previously occupied by the y axis. This does not mean that the w axis can coincide with the y axis. It indicates that we are looking at the four-dimensional sphere from a different point of view. Any three-space view will show us three axes, and in [fig. 14] we are looking at xzw.
The only part that is identical in the two diagrams is the circle of the x and z axes, which axes are contained in both diagrams. Thus the plane zxz´ is the same in both, and the point p represents the same point in both diagrams. Now, in [fig. 14] let the zw rotation take place, the z axis will turn toward the point w of the w axis, and the point p will move in a circle about the point x.