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.
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.
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 1h a 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 1h it 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 3h comes 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.
CHAPTER XIV.[6]
A RECAPITULATION AND EXTENSION OF THE PHYSICAL ARGUMENT
[6] The contents of this chapter are taken from a paper read before the Philosophical Society of Washington. The mathematical portion of the paper has appeared in part in the Proceedings of the Royal Irish Academy under the title, “Cayley’s formulæ of orthogonal transformation,” Nov. 29th, 1903.
There are two directions of inquiry in which the research for the physical reality of a fourth dimension can be prosecuted. One is the investigation of the infinitely great, the other is the investigation of the infinitely small.