To show these figures we must suppose the ochre cube to be on a movable stand. When the red line swings out into the unknown dimension, and the blue line comes in downwards, a cube appears below the place occupied by the ochre cube. The dotted cube shows where the ochre cube was. That cube has gone and a different cube runs downwards from its base. This cube has white, yellow, and blue axes. Its top is a light yellow square, and hence its interior is light yellow + blue or light green. Its front face is formed by the white line moving along the blue axis, and is therefore light blue, the left-hand side is formed by the yellow line moving along the blue axis, and therefore green. As the red line now runs in the fourth dimension, the successive sections can he called _r__{0}, _r__{1}, _r__{2}, _r__{3}, _r__{4}, these letters indicating that at distances 0, 1/4, 2/4, 3/4, 1 inch along the red axis we take all of the tesseract that can be found in a three-dimensional space, this three-dimensional space extending not at all in the fourth dimension, but up and down, right and left, far and near. We can see what should replace the light yellow face of _r__{0}, when the section _r__{1} comes in, by looking at the cube _b__{0}, fig. 107. What is distant in it one-quarter of an inch from the light yellow face in the red direction? It is an ochre section with orange and pink lines and red points; see also fig. 103. This square then forms the top square of _r__{1}. Now we can determine the nomenclature of all the regions of _r__{1} by considering what would be formed by the motion of this square along a blue axis. But we can adopt another plan. Let us take a horizontal section of _r__{0}, and finding that section in the figures, of fig. 107 or fig. 103, from them determine what will replace it, going on in the red direction. A section of the _r__{0} cube has green, light blue, green, light blue sides and blue points. Now this square occurs on the base of each of the section figures, _b__{1}, _b__{2}, 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 _r__{1} replaces the section of _r__{0} 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.