Hinton - The Fourth Dimension.pdf

A FOUR-DIMENSIONAL FIGURE 125
new dimension the signification of a compound symbol,
such as “oi,” alters. In the plane it meant the line AC.
In space it means the whole plane ACD.
Now, it is evident that we have twenty-seven positions,
each of them named. If the reader will follow this
nomenclature in respect of the positions marked in the
figures he will have no difficulty in assigning names to
each one of the twenty-seven positions. A is oi, oj, ok.
It is at the distance 0 along i, 0 along j, 0 along k, and
it can be written in short 000, where the ijk symbols
are omitted.
The point immediately above is 001, for it is no distance
in the i direction, and a distance of 1 in the k
direction. Again, looking at B, it is at a distance of 2
from A, or from the place ADC, in the i direction, 0 in the
j direction from the plane ABD, and 0 in the k direction,
measured from the plane ABC. Hence it is 200 written
for 2i, 0j, 0k.
Now, out of these twenty-seven “things” or compounds
of position and dimension, select those which are given by
the rule, every one of one kind with every other of every
other kind.
k
i
210
j
Fig. 66.
Take 2 of the i kind. With this
we must have a 1 of the j kind,
and then by the rule we can only
have a 0 of the k kind, for if we
had any other of the k kind we
should repeat one of the kinds we
already had. In 2i, 1j, 1k, for
instance, 1 is repeated. The point
we obtain is that marked 210, fig. 66.
Proceeding in this way, we pick out the following
cluster of points, fig. 67. They are joined by lines,
dotted where they are hidden by the body of the cube,
and we see that they form a figure—a hexagon which