Hinton - The Fourth Dimension.pdf

THE HIGHER WORLD 69
simply the plane of xy and the square base of the
w
cube ACEG, fig. 39, is all that could
be seen of it. Let now the w axis
y
take the place of the z axis and
E
G
x
we have, in fig. 39 again, a representation
of the space of xyw, in
A C which all that exists of the cube is
Fig. 39. its square base. Now, by a turning
of x to w, this base can rotate around the line AE, it is
w
shown on its way in fig. 40, and
finally it will, after half a revolution,
y
G
lie on the other side of the y axis. In
a similar way we may rotate
E
C sections parallel to the base of the
A
Fig. 40.
x
xw rotation, and each of them comes
to run in the opposite direction from
that which they occupied at first.
Thus again the cube comes from the position of fig. 36
H F z
to that of fig. 41. In this x to
w turning, we see that it
takes place by the rotations of
D
y
B
sections parallel to the front
G
E
face about lines parallel to AB,
or else we may consider it as
C A x x
nd st
2 position 1 position
Fig. 41.
consisting of the rotations of
sections parallel to the base
about lines parallel to AE. It
is a rotation of the whole cube about the plane ABEF.
Two separate sections could not rotate about two separate
lines in our space without conflicting, but their motion is
consistent when we consider another dimension. Just,
then, as a plane being can think of rotation about a line as
a rotation about a number of points, these rotations not
interfering as they would if they took place in his twodimensional
space, so we can think of a rotation about a