Hinton - The Fourth Dimension.pdf

216
THE FOURTH DIMENSION
turn in parallel planes, keeping their relative positions.
z
The point B, for instance, will
describe a circle. At one time
it will be above the line A, at
D
another time below it. Hence
F
it rotates round A.
G H Not only two rods but any
x
number of rotes crossing the
C E plane will move round it har-
w
moniously. We can think of
Fig. 10 (138).
this rotation by supposing the
rods standing up from one line
to move round that line and remembering that it is
not inconsistent with this rotation for the rods standing
up along another line also to move round it, the relative
positions of all the rods being preserved. Now, if the
rods are thick together, they may represent a disk of
matter, and se see that a disk of matter can rotate
round a central plane.
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-dimensional
turning, all the contour must be free. The whole
contour 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 dis.
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.