Opposite each point, on one side or the other of each cube, is written its name. It will be noticed that the figures are symmetrical right and left; and right and left the first two numbers are simply interchanged.
Now this being our selection of points, what figure do they make when all are put together in their proper relative positions?
To determine this we must find the distance between corresponding corners of the separate hexagons.
Fig. 73.
To do this let us keep the axes i, j, in our space, and draw h instead of k, letting k run out in the fourth dimension, [fig. 73].
Fig. 74.
Here we have four cubes again, in the first of which all the points are 0k points; that is, points at a distance zero in the k direction from the space of the three dimensions ijh. We have all the points selected before, and some of the distances, which in the last diagram led from figure to figure are shown here in the same figure, and so capable of measurement. Take for instance the points 3120 to 3021, which in the first diagram ([fig. 72]) lie in the first and second figures. Their actual relation is shown in fig. 73 in the cube marked 2K, where the points in question are marked with a *. We see that the distance in question is the diagonal of a unit square. In like manner we find that the distance between corresponding points of any two hexagonal figures is the diagonal of a unit square. The total figure is now easily constructed. An idea of it may be gained by drawing all the four cubes in the catalogue figure in one (fig. 74). These cubes are exact repetitions of one another, so one drawing will serve as a representation of the whole series, if we take care to remember where we are, whether in a 0h, a 1h, a 2h, or a 3h figure, when we pick out the points required. Fig. 74 is a representation of all the catalogue cubes put in one. For the sake of clearness the front faces and the back faces of this cube are represented separately.
The figure determined by the selected points is shown below.