Hinton - The Fourth Dimension.pdf

THE SIMPLEST FOUR-DIMENSIONAL SOLID 171
position. The “higher region” means a region with
another dimension in it.
Now the square can move and generate a cube. The
square light yellow moves and traces out the mass of the
cube. Letting the addition of red denote the region
made by the motion in the upward direction we get an
ochre solid. The light yellow face in its initial and
terminal positions give the two square boundaries of the
cube above and below. Then each of the four lines of the
light yellow square—white, yellow, and the white, yellow
opposite them—trace out a bounding square. So there
are in all six bounding squares, four of these squares being
designated in colour by adding red to the colour of the
generating lines. Finally, each point moving in the up
direction gives rise to a line coloured null + red, or red,
and then there are the initial and terminal positions of the
points giving eight points. The number of the lines is
evidently twelve, for the four lines of the light yellow
square give four lines in their initial, four lines in their
final position, while the four points trace out four lines,
that is altogether twelve lines.
Now the squares are each of them separate boundaries
of the cube, while the lines belong, each of them, to two
squares, thus the red line is that which is common to the
orange and pink squares.
Now suppose that there is a direction, the fourth
dimension, which is perpendicular alike to every one
of the space dimensions already used—a dimension
perpendicular, for instance, to up and to right hand,
so that the pink square moving in this direction traces
out a cube.
A dimension, moreover, perpendicular to the up and
away directions, so that the orange square moving in this
direction also traces out a cube, and the light yellow
square, too, moving in this direction traces out a cube.