The fourth dimension

A FOUR-DIMENSIONAL FIGURE 133

note when we look at it, whether we consider it as a Oh, a 1h, a 2h, ete., cube. Putting then the Oh, 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 OJ points, and has in it the points from 0 to 4, the j points from 0 to 4, the & 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 Oh, a lh, 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 OJ points. We cannot have 0 with any cther letter. Then, keeping in the first figure, which is that of the OJ 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, & 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 Ol figure let us suppose that the cube before us represents a section at a distance of 2 in the A direction. Let all the points in it be considered as 2h points. We then have a Ol, 2h region, and have the sets 7k and 431 left over. We must then pick out in accordance with our rule all such points as 47, 37, 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 tetrakaidecagon.