THE RECOGNITION OF THE FOURTH DIMENSION. 191 
Rotation round a plane is exactly analogous to rotation 
round an axis in three dimensions. If we want a rod to 
turn round, the ends must be free; so if we want a disk 
of matter to turn round its central plane by a four-dimen¬ 
sional turning, all the contour must be free. The whole con¬ 
tour corresponds to the ends of the rod. Each point of the 
contour can be looked on as the extremity of an axis in the 
body, round each point of which there is a rotation of the 
matter in the disk. 
If the one end of a rod be clamped, we can twist the rod, 
but not turn it round ; so if any part of the contour of a disk 
is clamped we can impart a twist to the disk, but not turn it 
round its central plane. In the case of extensible materials 
a long, thin rod will twist round its axis, even when the axis 
is curved, as, for instance, in the case of a ring of India 
rubber. 
In an analogous manner, in four dimensions we can have 
rotation round a curved plane, if I may use the expression. 
A sphere can be turned inside out in four dimensions. 
Let Fig. 11 represent a 
spherical surface on each 
side of which a layer of mat¬ 
ter exists. The thickness of 
the matter is represented by 
the rods C D and E F, ex¬ 
tending equally without and 
within. 
Now, take the section of 
the sphere by the y z plane 
we have a circle—Fig. 12. 
Now, let the w axis be drawn 
in place of the x axis so that 
we have the space of y z w 
represented. In this space all that there will be seen of the 
sphere is the circle drawn. 
Here we see that there is no obstacle to prevent the rods 
turning round. If the matter is so elastic that it will give 
Axis ofx running towards 
the observer. 
