Swinging the tesseract into our space about the pink face of the ochre cube we likewise find that the parallel to the white axis is cut at half its length by the sectional space. Hence in a section made when the tesseract had passed half across our space the parallels to the red, white, yellow axes, which are now in our space, are cut by the section space, each of them half way along, and for this stage of the traversing motion we should have fig. 126. The section made of this cube by the plane in which the sectional space cuts it, is an equilateral triangle with purple, l. blue, green points, and l. purple, brown, l. green lines. [Illustration: Fig. 126.] Thus the original ochre triangle, with null points and pink, orange, light yellow lines, would be succeeded by a triangle coloured in manner just described. This triangle would initially be only a very little smaller than the original triangle, it would gradually diminish, until it ended in a point, a null point. Each of its edges would be of the same length. Thus the successive sections of the successive planes into which we analyse the cutting space would be a tetrahedron of the description shown (fig. 123), and the whole interior of the tetrahedron would be light brown. [Illustration: Fig. 127. Front view. The rear faces.] In fig. 127 the tetrahedron is represented by means of its faces as two triangles which meet in the p. line, and two rear triangles which join on to them, the diagonal of the pink face being supposed to run vertically upward. We have now reached a natural termination. The reader may pursue the subject in further detail, but will find no essential novelty. I conclude with an indication as to the manner in which figures previously given may be used in determining sections by the method developed above. Applying this method to the tesseract, as represented in Chapter IX., sections made by a space cutting the axes equidistantly at any distance can be drawn, and also the sections of tesseracts arranged in a block. If we draw a plane, cutting all four axes at a point six units distance from null, we have a slanting space. This space cuts the red, white, yellow axes in the points LMN (fig. 128), and so in the region of our space before we go off into the fourth dimension, we have the plane represented by LMN extended. This is what is common to the slanting space and our space. [Illustration: Fig. 128.] This plane cuts the ochre cube in the triangle EFG. Comparing this with (fig. 72) _oh_, we see that the hexagon there drawn is part of the triangle EFG.