Hinton - The Fourth Dimension.pdf

THE SIMPLEST FOUR-DIMENSIONAL SOLID 161
but the thought of an abstract boundary, the face of
a cube.
Let us now take our eight coloured cubes, which form
a cube in space, and ask what additions we must make
to them to represent the simplest collection of four-dimensional
bodies—namely, a group of them of the same extent
in every direction. In plane space we have four squares.
In solid space we have eight cubes. So we should expect
in four-dimensional space to have sixteen four-dimensional
bodies—bodies which in four-dimensional space
correspond to cubes in three-dimensional space, and these
bodies we call tesseracts.
Given then the null, white, red, yellow cubes, and
those which make up the block, we
notice that we represent perfectly
well the extension in three directions
(fig. 98). From the null point of the
x
(Orange hidden)
Fig. 98.
null cube, travelling one inch, we
come to the white cube; travelling
one inch away we come to the yellow
cube; travelling one inch up we come
to the red cube. Now, if there is a
fourth dimension, then travelling
from the same null point for one
inch in that direction, we must come to the body lying
beyond the null region.
I say null region, not cube; for with the introduction
of the fourth dimension each of our cubes must become
something different from cubes. If they are to have
existence in the fourth dimension, they must be “filled
up from” in this fourth dimension.
Now we will assume that as we get a transference from
null to white going in one way, from null to yellow going
in another, so going from null in the fourth direction we
have a transference from null to blue, using thus the