Hinton - The Fourth Dimension.pdf

68
THE FOURTH DIMENSION
Let us suppose that we let the y axis drop, and that
z
we represent the w axis as occupying
its direction. We have in fig.
B D 37 a drawing of what we should
w
then see of the cube. The square
ABCD remains unchanged, for
x
that is in the play of xz, and we
A C
still have that plane. But from
Fig. 37.
this plane the cube stretches out
in the direction of the y axis. Now the y axis is gone, and
so we have no more of the cube than the face ABCD.
z
Considering now this face ABCD, we
see that it is free to turn about the
D line AB. It can rotate in the x to w
w B
direction about this line. In fig. 38
it is shown on its way, and it can
C
evidently continue this rotation till
A
x it lies on the other side of the z axis
Fig. 38.
in the plane of xz.
We can also take a section parallel to the face ABCD,
and then letting drop all our space except the plane of
that section, introduce the w axis, running in the old y
direction. This section can be represented by the same
drawing, fig. 38, and we see that it can rotate about the
line on its left until it swings half way round and runs in
the opposite direction to that which it ran in before.
These turnings of the different sections are not inconsistent,
and taken all together they will bring the cube
from the position shown in fig. 36 to that shown in
fig. 41.
Since we have three axes at our disposal in our space,
we are not obliged to represent the w axis by any particular
one. We may let any axis we like disappear, and let the
fourth axis take its place.
In fig. 36 suppose the z axis to go. We have then