Now this square occurs on the base of each of the section figures, b1, b2, etc. In them we see that 1/4 inch in the red direction from it lies a section with brown and light purple lines and purple corners, the interior being of light brown. Hence this is the nomenclature of the section which in r1 replaces the section of r0 made from a point along the blue axis.
Hence the colouring as given can be derived.
We have thus obtained a perfectly named group of tesseracts. We can take a group of eighty-one of them 3 × 3 × 3 × 3, in four dimensions, and each tesseract will have its name null, red, white, yellow, blue, etc., and whatever cubic view we take of them we can say exactly what sides of the tesseracts we are handling, and how they touch each other.[5]
[5] At this point the reader will find it advantageous, if he has the models, to go through the manipulations described in the appendix.
Thus, for instance, if we have the sixteen tesseracts shown below, we can ask how does null touch blue.
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.
