The fourth dimension

A FOUR-DIMENSIONAL FIGURE 127

pick out by the rule the two points 201, 102—c, and K. Here they occur in one plane and he can measure the distance between them. In his first representation they occur at G and K in separate figures.

Thus the plane being would find that the ends of each of the lines was distant by the diagonal of a unit square from the corresponding end of the last and he could then place the three lines in their right relative position. Joining them he would have the figure of a hexagon.

We may also notice that the plane being could make a representation of the whole cube simultaneously. The three squares, shown in perspective in fig. 70, all lie in one plane, and on these the plane being could pick out any selection of points just as well as on three separate squares. He would obtain a hexagon by joining the points marked. This hexagon, as drawn, is of the right shape, but it would not be so if actual squares were used instead of perspective, because the relation between the separate squares as they lie in the plane figure is not their real relation. The figure, however, as thus constructed, would give him an idea of the correct figure, and he could determine it accurately by remembering that distances in each square were correct, but in passing from one square to another their distance in the third dimension had to be taken into account.

Coming now to the figure made by selecting according to our rule from the whole mass of points given by four axes and four positions in each, we must first draw a catalogue figure in which the whole assemblage is shown.

We can represent this assemblage of points by four solid figures. The first giving all those positions which

Fig. 70,