Hinton - The Fourth Dimension.pdf

A FOUR-DIMENSIONAL FIGURE 123
The rule will be made more evident by a simple
application.
Let us consider one dimension and one position. I will
call the axis i, and the position o.
o
Here the figure is the position o on the line i. Take
now two dimensions and two positions on each.
We have the two positions o; 1 on i, and the two
o o 1 j positions o, 1 on j, fig. 63. These give
1
rise to a certain complexity. I will let
the two lines i and j meet in the
i
Fig. 63.
position I call o on each, and I will
consider i as a direction starting equally
from every position on j, and j as
starting equally from every position on i. We thus
obtain the following figure:--A is both oi and oj, B is 1 i
A C and oj, and so on as shown in fig. 63b.
oi, oj oi, 1j
The positions on AC are all oi positions.
They are, if we like to consider them
1, i oj
B
1, i 1j
D
that way, points at no distance in the i
direction from the line AC. We can
call the line AC the oi line. Similarly
i
Fig 63b.
the points on AB are those no distance
from AB in the j direction, and we can
call them oj points and the line AB the oj line. Again,
the line CD can be called the 1j line, because the points
on it are at a distance 1 in the j direction.
We have then four positions or points named as shown,
and, considering directions and positions as “kinds,” we
have the combination of two kinds with two kinds. Now,
selecting every one of one kind with every other of every
other kind will mean that we take 1 of the kind i and
i