The fourth dimension

68 THE FOURTH DIMENSION

Let. us suppose that we let the y axis drop, and that Zz we represent the w axis as occupying its direction. We have in fig. 37 a drawing of what we should then see of the cube. The square ABCD, remains unchanged, for that 7 nn is in the plane of zz, and we still have that plane. But from this plane the cube stretches out in the direction of the y axis. Now the y axis is gone, and so we have no more of the cube than the face ABCD. Considering now this face ABCD, we see that it is free to turn about the line AB. It can rotate in the z tow direction about this line. In fig. 38 it is shown on its way, and it can _ evidently continue this rotation till * it lies on the other side of the z axis in the plane of zz.

We can also take a section parallel to the face ABCD, and then letting drop all of our space except the plane of that section, introduce the w axis, running in the old y direction. This section can be represented by the same drawing, fig. 38, and we see that it can rotate about the line on its left until it swings half way round and runs in the opposite direction to that which it ran in before. These turnings of the different sections are not inconsistent, and taken all together they will bring the cube from the position shown in fig. 36 to that shown in fig. 41.

Since we have three axes at our disposal in our space, we are not obliged to represent the w axis by any particular one. We may let any axis we like disappear, and let the fourth axis take its place.

In fig. 36 suppose the ¢ axis to go. We have then

D

Fig. 37.

Fig. 38.