The fourth dimension

A FOUR-DIMENSIONAL FIGURE 131

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 2x, where the points in question are marked with a * in fig. 73. 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 draw-

ing will serve as a <a representation of the ly () whole series, if we { )

take care to remember where we are, whether feel

in a Oh, a Ih, a 2h, Fig. 74.

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.

In putting the sections together some of the outlines in them disappear. The line Tw for instance is not wanted.

We notice that paTw and tTwrs are each the half of a hexagon. Now Qv and vp lie in one straight line.