[Illustration: Fig. 111.] In the arrangement given in fig. 111 we have the axes white, red, yellow, in space, blue running in the fourth dimension. Hence we have the ochre cubes as bases. Imagine now the tesseractic group to pass transverse to our space—we have first of all null ochre cube, white ochre cube, etc.; these instantly vanish, and we get the section shown in the middle cube in fig. 103, and finally, just when the tesseract block has moved one inch transverse to our space, we have null ochre cube, and then immediately afterwards the ochre cube of blue comes in. Hence the tesseract null touches the tesseract blue by its ochre cube, which is in contact, each and every point of it, with the ochre cube of blue. How does null touch white, we may ask? Looking at the beginning A, fig. 111, where we have the ochre cubes, we see that null ochre touches white ochre by an orange face. Now let us generate the null and white tesseracts by a motion in the blue direction of each of these cubes. Each of them generates the corresponding tesseract, and the plane of contact of the cubes generates the cube by which the tesseracts are in contact. Now an orange plane carried along a blue axis generates a brown cube. Hence null touches white by a brown cube. [Illustration: 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.y_{1}, fig. 112. Between figure C and C.y_{1} 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.