The fourth dimension

RECAPITULATION AND EXTENSION 221

take the place of the y axis. Here, in fig. 14, we have the space of zw. 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 threedimensional sphere, as has been shown before, we should have merely a circle in the xzw space, the xz circle seen in the space of wzw. But now, taking the view in the space of xzw, we have a sphere in that space also. Ina similar manner, whichever set of three axes we take, we obtain a sphere.

~ z ay a fa Showing axes ee ty xwz Ray )

2’ 2 Fig. 13 (141). Fig. 14 (142),

In fig. 13, let us imagine the rotation in the direction wy to be taking place. The point x will turn to y, and p top’. The axis 22’ 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 zg axes, which axes are contained in both diagrams. Thus the plane zaz’ is the same in both, and the point p represents the same point in both