Hinton - The Fourth Dimension.pdf

A FOUR-DIMENSIONAL FIGURE 133
note when we look at it, whether we consider it as a 0h, a
1h, a 2h, etc., cube. Putting then the 0h, 1h, 2h, 3h, 4h
cubes of each row in one, we have five cubes with the sides
of each containing five positions, the first of these five
cubes represents the 0l points, and has in it the i points
from 0 to 4, the j points from 0 to 4, the k points from
0 to 4, while we have to specify with regard to any
selection we make from it, whether we regard it as a 0h,
a 1h, a 2h, a 3h, or a 4h figure. In fig. 76 each cube is
represented by two drawings, one of the front part, the
other of the rear part.
Let then our five cubes be arranged before us and our
selection be made according to the rule. Take the first
figure in which all points are 0l points. We cannot
have 0 with any other letter. Then, keeping in the first
figure, which is that of the 0l positions, take first of all
that selection which always contains 1h. We suppose,
therefore, that the cube is a 1h cube, and in it we take
i, j, k in combination with 4, 3, 2 according to the rule.
The figure we obtain is a hexagon, as shown, the one
in front. The points on the right hand have the same
figures as those on the left, with the first two numerals
interchanged. Next keeping still to the 0l figure let
us suppose that the cube before us represents a section
at a distance 2 in the h direction. Let all the points
in it be considered as 2h points. We then have a 0l, 2h
region, and have the sets ijk and 431 left over. We
must then pick out in accordance with our rule all such
points as 4i, 3j, 1k.
These are shown in the figure and we find that we can
draw them without confusion, forming the second hexagon
from the front. Going on in this way it will be seen
that in each of the five figures a set of hexagons is picked
out, which put together form a three-space figure something
like the tetrakaidekagon.