Front view.The rear faces.Fig. 127.
Front view.The rear faces.Fig. 127.
Infig. 127the 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 thepointsLMN(fig. 128), and so in the region of our space before we go off into the fourth dimension, we have the plane represented byLMNextended. This is what is common to the slanting space and our space.
Fig. 128.
Fig. 128.
This plane cuts the ochre cube in the triangleEFG.
Comparing this with (fig. 72)oh, we see that the hexagon there drawn is part of the triangleEFG.
Let us now imagine the tesseract and the slanting space both together to pass transverse to our space, a distance of one unit, we have in 1ha section of the tesseract, whose axes are parallels to the previous axes. The slanting space cuts them at a distance of five units along each. Drawing the plane through these points in 1hit will be found to cut the cubical section of the tesseract in the hexagonal figure drawn. In 2h(fig. 72) the slanting space cuts the parallels to the axes at a distance of four along each, and the hexagonal figure is the section of this section of the tesseract by it. Finally when 3hcomes in the slanting space cuts the axes at a distance of three along each, and the section is a triangle, of which the hexagon drawn is a truncated portion. After this the tesseract, which extends only three units in each of the four dimensions, has completely passed transverse of our space, and there is no more of it to be cut. Hence, putting the plane sections together in the right relations, we have the section determined by the particular slanting space: namely an octahedron.