The fourth dimension

A FOUR-DIMENSIONAL FIGURE 125

new dimension the signification of a compound symbol, such as “oi,” alters. In the plane it meant the line ac, In space it means the whole plane acp.

Now, it is evident that we have twenty-seven positions, each of them named. If the reader will follow this nomenclature in respect of the positions marked in the figures he will have no difficulty in assigning names to each one of the twenty-seven positions. A is 0%, oj, ok. It is at the distance 0 along i, 0 along J, 0 along k, and io can be written in short 000, where the ijk symbols are omitted.

The point immediately above is 001, for it is no distance in the @ direction, and a distance of 1 in the & direction. Again, looking at B, it is at a distance of 2 from A, or from the plane abc, in the @ direction, 0 in the j direction from the plane ap, and 0 in the & direction, measured from the plane asc. Hence it is 200 written for 22, 0), Ok.

Now, out of these twenty-seven ‘things ” or compounds of position and dimension, select those which are given by the rule, every one of one kind with every other of every other kind.

k Take 2 of thez kind. With this

- we must have a 1 of the g kind,

and then by the rule we can only have a O of the & kind, for if we had any other of the & kind we should repeat one of the kinds we already had. In 27, lj, 1h, for instance, 1 is repeated. The point Fig. 66. we obtain is that marked 210, fig. 66.

Proceeding in this way, we pick out the following cluster of points, fig. 67. They are joined by lines, dotted where they are hidden by the body of the cube, and we see that they form a figure—a hexagon which

+210 4