In the same way, measuring the distances from the face ADC, we see that every position in the face ADC is an oi position, and the whole plane of the face may be called an oi plane. Thus we see that with the introduction of a 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 ACD.
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 oi, 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 i direction, and a distance of 1 in the k direction. Again, looking at B, it is at a distance of 2 from A, or from the plane ADC, in the i direction, 0 in the j direction from the plane ABD, and 0 in the k direction, measured from the plane ABC. Hence it is 200 written for 2i, 0j, 0k.
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.
Fig. 66.
Take 2 of the i kind. With this we must have a 1 of the j kind, and then by the rule we can only have a 0 of the k kind, for if we had any other of the k kind we should repeat one of the kinds we already had. In 2i, 1j, 1k, for instance, 1 is repeated. The point we obtain is that marked 210, [fig. 66].
Fig. 67.
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 could be taken out of the cube and placed on a plane. It is a figure which will fill a plane by equal repetitions of itself. The plane being representing this construction in his plane would take three squares to represent the cube. Let us suppose that he takes the ij axes in his space and k represents the axis running out of his space, [fig. 68]. In each of the three squares shown here as drawn separately he could select the points given by the rule, and he would then have to try to discover the figure determined by the three lines drawn. The line from 210 to 120 is given in the figure, but the line from 201 to 102 or GK is not given. He can determine GK by making another set of drawings and discovering in them what the relation between these two extremities is.