Hinton - The Fourth Dimension.pdf

72
THE FOURTH DIMENSION
the whole contour corresponds to the ends of an axis of
rotation in our space. He can impart the rotation at any
point and take it off at any other point on the contour,
just as rotation round a line can in three-space be imparted
at one end of a rod and taken off at the other end.
A four-dimensional wheel can easily be described from
the analogy of the representation which a plane being
would form from himself of one of our wheels.
Suppose a wheel to move transverse to a plane, so that
the whole disk, which I will consider to be solid and
without spokes, can at the same time into contact with
the plane. It would appear as a circular portion of plane
matter completely enclosing another and smaller portion—
the axle.
This appearance would last, supposing the motion of
the wheel to continue until it had traversed the plane by
the extent of its thickness, when there would remain in
the plane only the small disk which is the section of the
axle. There would be no means obvious in the plane
at first by which the axle could be reached, except by
going through the substance of the wheel. But the
possibility of reaching it without destroying the substance
of the wheel would be shown by the continued existence
of the axle section after that of the wheel had disappeared.
In a similar way a four-dimensional wheel moving
transverse to our space would appear first as a solid sphere,
completely surrounding a smaller solid sphere. The
outer sphere would represent the wheel, and would last
until the wheel had traversed our space by a distance
equal to its thickness. Then the small sphere alone
would remain, representing the section of the axle. The
large sphere could move round the small one quite freely.
Any line in space could be taken as an axis, and round
this line the outer sphere could rotate, while the inner
sphere remained still. But in all these directions of