The fourth dimension

212 THE FOURTH DIMENSION

so we will let y go and draw win its place. What will be our view of the cube?

Evidently we shall have simply the square that is in

the plane of xz, the square AcDB.

The rest of the cube stretches in

D the y direction, and, as we have

w none of the space so determined,

we have only the face of the cube.

x This is represented in fig. 4.

A c Now, suppose the whole cube to

Fig. 4 (182). 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 aBop.

Let this face begin toturn. Fig. 5 represents one of the positions it will occupy; the line AB remains on the gaxis. The rest of the face extends between the xz and the w direction.

Now, since we can take any three A axes, let us look at what lies in Fig. 5 (133). the space of zyw, and examine the turning there. We must now let the 2 axis disappear and let the w axis run in the direction in which the z ran.

Making this representation, what do we see of the cube? Obviously we see only the lower face. The rest 6 of the cube lies in the space of xyz.

é In the space of xyz we have merely

A Cc the base of the cube lying in the Fig. 6 (134). plane of zy, as shown in fig. 6.

Now let the « to w turning take place. The square ACEG will turn about the line ar. This edge will remain along the y axis and will be stationary, however far the square turns.