186 
HINTON. 
Let us imagine the cube in our space, which is the base 
of a tesseract, to turn about one of its edges. The rotation 
will carry the whole body with it, and each of the cubic 
sections will rotate. The axis we see in our space will re¬ 
main unchanged, and likewise the series of axes parallel to 
it about which each of the parallel cubic sections rotates. 
The assemblage of all of these is a plane. 
Hence in four dimensions a body rotates about a plane. 
There is no such thing as rotation round an axis. 
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-dimen¬ 
sional 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 could appear as the production of a looking-glass 
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 appear¬ 
ances would be unimaginable in three-dimensional space. 
Let us now apply the method by which 
a plane being could examine the na¬ 
ture of rotation about a line in our 
examination of rotation about a plane. 
Fig. 3 represents a cube in our space, 
the three axes x, y, z denoting its three 
dimensions. Let w represent the fourth 
~ x dimension. Now, since in our space we 
can represent any three dimensions, we 
can, if we choose, make a representation 
of what is in the space determined by the three axes x, z, w. 
F/g.3. 
