The fourth dimension

RECAPITULATION AND EXTENSION 213

Thus, if the cube be turned by an x to w turning, both the edge aB and the edge ac remain stationary; hence the whole face ABEF in the yz plane remains fixed. The turning has taken place about the face ABEF.

Suppose this turning to continue

A. *till ac runs to the left from a.

Hig. 7 (155). The cube will occupy the position

shown in fig. 8. This is the looking-glass image of the

cube in fig. 3. By no rotation in three-dimensional space

ean the cube be brought from

the position in fig. 3 to that shown in fig. 8.

We can think of this turning as a turning of the face ABCD about AB, and a turning of each

C Ax» Section parallel to aBcp round

2“position _ {*"positian the vertical line in which it

Fig. 8 (136). intersects the face ABEF, the

space in which the turning takes place being a different one from that in which the cube lies.

One of the conditions, then, of our inquiry in the direction of the infinitely small is that we form the conception of a rotation about a plane. The production of a body in a state in which it presents the appearance of a looking-glass image of its former state is the criterion for a four-dimensional rotation.

There is some evidence for the occurrence of such transformations of bodies in the change of bodies from those which produce a right-handed polarisation of light to those which produce a left-handed polarisation; but this is not a point to which any very great importance can be attached.

Still, in this connection, let me quote a remark from