The fourth dimension

RECAPITULATION AND EXTENSION 211

We may regard the rotation from a different point of view. Consider four independent axes each at right angles to all the others, drawn in a four-dimensional body. Of these four axes we can see any three. The fourth extends normal to our space.

Rotation is the turning of one axis into a second, and the second turning to take the place of the negative of the first. It involves two axes. Thus, in this rotation of a four-dimensional body, two axes change and two remain at rest. Four-dimensional rotation is therefore a turning about a plane.

As in the case of a plane being, the result of rotation about a line would appear as the production of a lookingglass image of the original object on the other side of the line, so to us the result of a four-dimensional rotation would appear like the production of a looking-glass image of a body on the other side of a plane. The plane would be-the axis of the rotation, and the path of the body between its two appearances would be unimaginable in three-dimensional space.

Let us now apply the method by which a plane being could examine the nature of rotation about a line in our examination of rotation about a plane. Fig. 3 represents a cube in our space, the three axes 2, y, 2 denoting its three dimensions. Let w represent the fourth dimension. Now, since in our space we can represent any

Fig. 3 (131). three dimensions, we can, if we choose, make a representation of what is in the space determined by the three axes x, 2, w. ‘This is a threedimensional space determined by two of the axes we have drawn, x and 2, and in place of y the fourth axis, w. We cannot, keeping a and z, have both y and w in our space;