Hinton - The Fourth Dimension.pdf

A FOUR-DIMENSIONAL FIGURE 127
pick out by the rule the two points 201, 102—G, and K.
Here they occur in one plane and he can measure the
distance between them. In his first representation they
occur at G and K in separate figures.
Thus the plane being would find that the ends of each
of the lines was distant by the diagonal of a unit square
from the corresponding end of the last and he could then
place the three lines in their right relative position.
Joining them he would have the figure of a hexagon.
We may also notice that the plane being could make
a representation of the whole cube
simultaneously. The three squares,
shown in perspective in fig. 70, all
lie in one plane, and on these the
plane being could pick out any
selection of points just as well as on
three separate squares. He would
Fig. 70.
obtain a hexagon by joining the
points marked. This hexagon, as
drawn, is of the right shape, but it would not be so if
actual squares were used instead of perspective, because
the relation between the separate squares as they lie in
the plane figure is not their real relation. The figure,
however, as thus constructed, would give him an idea of
the correct figure, and he could determine it accurately
by remembering that distances in each square were
correct, but in passing from one square to another their
distance in the third dimension had to be taken into
account.
Coming now to the figure made by selecting according
to our rule from the whole mass of points given by four
axes and four positions in each, we must first draw a
catalogue figure in which the whole assemblage is shown.
We can represent this assemblage of points by four
solid figures. The first giving all those positions which