Let him suppose the cube, in the position of the drawing, [fig. 124], turned so that the pink face lies against his plane. He can see the line from the null r point to the null wh point, and can see (compare [fig. 119]) that it cuts AB a parallel to his red axis, drawn at a point half way along the white line, in a point B, half way up. I shall speak of the axis as having the length of an edge of the cube. Similarly, by letting the cube turn so that the light yellow square swings against his plane, he can see (compare [fig. 119]) that a parallel to his yellow axis drawn from a point half-way along the white axis, is cut at half its length by the trace of the section plane in the light yellow face.
Hence when the cube had passed half-way through he would have—instead of the orange line with null points, which he had at first—an ochre line of half its length, with pink and light yellow points. Thus, as the cube passed slowly through his plane, he would have a succession of lines gradually diminishing in length and forming an equilateral triangle. The whole interior would be ochre, the line from which it started would be orange. The succession of points at the ends of the succeeding lines would form pink and light yellow lines and the final point would be null. Thus looking at the successive lines in the section plane as it and the cube passed across his plane he would determine the figure cut out bit by bit.
Coming now to the section of the tesseract, let us imagine that the tesseract and its cutting space pass slowly across our space; we can examine portions of it, and their relation to portions of the cutting space. Take the section space which passes through the four points, null r, wh, y, b; we can see in the ochre cube ([fig. 119]) the plane belonging to this section space, which passes through the three extremities of the red, white, yellow axes.
Now let the tesseract pass half way through our space. Instead of our original axes we have parallels to them, purple, light blue, and green, each of the same length as the first axes, for the section of the tesseract is of exactly the same shape as its ochre cube.
But the sectional space seen at this stage of the transference would not cut the section of the tesseract in a plane disposed as at first.
To see where the sectional space would cut these parallels to the original axes let the tesseract swing so that, the orange face remaining stationary, the blue line comes in to the left.
Fig. 125.
Here ([fig. 125]) we have the null r, y, b points, and of the sectional space all we see is the plane through these three points in it.
In this figure we can draw the parallels to the red and yellow axes and see that, if they started at a point half way along the blue axis, they would each be cut at a point so as to be half of their previous length.