Fig. 112.
If we ask again how red touches light blue tesseract, let us rearrange our group, [fig. 112], or rather turn it about so that we have a different space view of it; let the red axis and the white axis run up and right, and let the blue axis come in space towards us, then the yellow axis runs in the fourth dimension. We have then two blocks in which the bounding cubes of the tesseracts are given, differently arranged with regard to us—the arrangement is really the same, but it appears different to us. Starting from the plane of the red and white axes we have the four squares of the null, white, red, pink tesseracts as shown in A, on the red, white plane, unaltered, only from them now comes out towards us the blue axis. Hence we have null, white, red, pink tesseracts in contact with our space by their cubes which have the red, white, blue axis in them, that is by the light purple cubes. Following on these four tesseracts we have that which comes next to them in the blue direction, that is the four blue, light blue, purple, light purple. These are likewise in contact with our space by their light purple cubes, so we see a block as named in the figure, of which each cube is the one determined by the red, white, blue, axes.
The yellow line now runs out of space; accordingly one inch on in the fourth dimension we come to the tesseracts which follow on the eight named in C, [fig. 112], in the yellow direction.
These are shown in C.y1, [fig. 112]. Between figure C and C.y1 is that four-dimensional mass which is formed by moving each of the cubes in C one inch in the fourth dimension—that is, along a yellow axis; for the yellow axis now runs in the fourth dimension.
In the block C we observe that red (light purple cube) touches light blue (light purple cube) by a point. Now these two cubes moving together remain in contact during the period in which they trace out the tesseracts red and light blue. This motion is along the yellow axis, consequently red and light blue touch by a yellow line.
We have seen that the pink face moved in a yellow direction traces out a cube; moved in the blue direction it also traces out a cube. Let us ask what the pink face will trace out if it is moved in a direction within the tesseract lying equally between the yellow and blue directions. What section of the tesseract will it make?
We will first consider the red line alone. Let us take a cube with the red line in it and the yellow and blue axes.
Fig. 113.
The cube with the yellow, red, blue axes is shown in [fig. 113]. If the red line is moved equally in the yellow and in the blue direction by four equal motions of ¼ inch each, it takes the positions 11, 22, 33, and ends as a red line.