The fourth dimension

REMARKS ON THE FIGURES 19].

The cube with the yellow, red, blue axes is shown in ‘ fig. 113. If the red line is moved equal'y in the yellow and in the blue direction by four equal motions of 3 inch each, it takes the positions 11, 22, 33,

4 Red and ends as a red line. Yellow [ds 2. Now, the whole of this red, Buen yellow, blue, or brown cube apFig. 1s. pears as a series of faces on the

successive sections of the tesseract starting from the ochre cube and letting the blue axis run in the fourth dimension. Hence the plane traced out by the red line appears as a series of lines in the successive sections, in our ordinary way of representing the tesseract ; these lines are in different places in each successive section.

4 ie Yellow, P) |] Nu ite bg by b, b3 by

Fig. 114,

Thus drawing our initial enbe and the successive sections, calling them dg, bi, bs, b3, b,, fig. 115, we have the red line subject to this movement appearing in the positions indicated,

We will now investigate what positions in the tesseract another line in the pink face assumes when it is moved in a similar manner.

Take a section of the original cube containing a vertical line, 4, in the pink plane, fig. 115. We have, in the section, the yellow direction, but not the blue,