In this representation the square would not be shown, for in the plane of xz simply the line AB of the square is contained.
The plane being then would have before him, in [fig. 35], the representation of one line AB of his square and two axes, x and z, at right angles. Now it would be obvious to him that, by a turning such as he knows, by a rotation about a point, the line AB can turn round A, and occupying all the intermediate positions, such as AB1, come after half a revolution to lie as Ax produced through A.
Again, just as he can represent the vertical plane through AB, so he can represent the vertical plane through A´B´, [fig. 34], and in a like manner can see that the line A´B´ can turn about the point A´ till it lies in the opposite direction from that which it ran in at first.
Now these two turnings are not inconsistent. In his plane, if AB turned about A, and A´B´ about A´, the consistency of the square would be destroyed, it would be an impossible motion for a rigid body to perform. But in the turning which he studies portion by portion there is nothing inconsistent. Each line in the square can turn in this way, hence he would realise the turning of the whole square as the sum of a number of turnings of isolated parts. Such turnings, if they took place in his plane, would be inconsistent, but by virtue of a third dimension they are consistent, and the result of them all is that the square turns about the line AC and lies in a position in which it is the mirror image of what it was in its first position. Thus he can realise a turning about a line by relinquishing one of his axes, and representing his body part by part.
Let us apply this method to the turning of a cube so as to become the mirror image of itself. In our space we can construct three independent axes, x, y, z, shown in [fig. 36]. Suppose that there is a fourth axis, w, at right angles to each and every one of them. We cannot, keeping all three axes, x, y, z, represent w in our space; but if we relinquish one of our three axes we can let the fourth axis take its place, and we can represent what lies in the space, determined by the two axes we retain and the fourth axis.
Fig. 37.
Let us suppose that we let the y axis drop, and that 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 is in the plane of xz, 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 x to w 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 xz.
Fig. 38.