The successive sections of the series, y0, y1, y2, etc., can be considered as derived from sections of the b0 cube made at distances along the yellow axis. What is distant a quarter inch from the pink face in the yellow direction? This question is answered by taking a section from a point a quarter inch along the yellow axis in the cube b0, [fig. 107]. It is an ochre section with lines orange and light yellow. This section will therefore take the place of the pink face in y1 when we go on in the yellow direction. Thus, the first section, y1, will begin from an ochre face with light yellow and orange lines. The colour of the axis which lies in space towards us is blue, hence the regions of this section-cube are determined in nomenclature, they will be found in full in [fig. 105].
There remains only one figure to be drawn, and that is the one in which the red axis is replaced by the blue. Here, as before, if the red axis goes out into the positive sense of the fourth dimension, the blue line must come into our space in the negative sense of the direction which the red line has left. Accordingly, the first cube will come in beneath the position of our ochre cube, the one we have been in the habit of starting with.
Fig. 110.
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 r0, r1, r2, r3, r4, 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 r0, when the section r1 comes in, by looking at the cube b0, [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 r1. Now we can determine the nomenclature of all the regions of r1 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 r0, 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 r0 cube has green, light blue, green, light blue sides and blue points.