We can represent this assemblage of points by four solid figures. The first giving all those positions which are at a distance O from our space in the fourth dimension, the second showing all those that are at a distance 1, and so on.
These figures will each be cubes. The first two are drawn showing the front faces, the second two the rear faces. We will mark the points 0, 1, 2, 3, putting points at those distances along each of these axes, and suppose all the points thus determined to be contained in solid models of which our drawings in [fig. 71] are representatives. Here we notice that as on the plane 0i meant the whole line from which the distances in the i direction was measured, and as in space 0i means the whole plane from which distances in the i direction are measured, so now 0h means the whole space in which the first cube stands—measuring away from that space by a distance of one we come to the second cube represented.
Fig. 71.
Now selecting according to the rule every one of one kind with every other of every other kind, we must take, for instance, 3i, 2j, 1k, 0h. This point is marked 3210 at the lower star in the figure. It is 3 in the i direction, 2 in the j direction, 1 in the k direction, 0 in the h direction.
With 3i we must also take 1j, 2k, 0h. This point is shown by the second star in the cube 0h.
Fig. 72.
In the first cube, since all the points are 0h points, we can only have varieties in which i, j, k, are accompanied by 3, 2, 1.
The points determined are marked off in the diagram fig. 72, and lines are drawn joining the adjacent pairs in each figure, the lines being dotted when they pass within the substance of the cube in the first two diagrams.