The fourth dimension

REMARKS ON THE FIGURES 197

space, and the triangle they determine is common to the tesseract and the cutting space. Hence this boundary is a triangle haying a light yellow line, which is the same as the light yellow line of the first figure, a light blue line and a green line.

We have now traced the cutting space between every set of three that can be made out of the four points in which it cuts the tesseract, and have got four faces which all join on to each other by lines.

The triangles are shown in fig. 123 as they join on to the triangle in the ochre cube. But ' they join on each to the other in an exactly similar manner; their edges are all identical two and two. They form a closed figure, a tetrahedron, enclosing a light brown portion which is the portion of the cutting space which lies inside the tesseract.

We cannot expect to see this light brown portion, any more than a plane being could expect to see the inside of a cube if an angle of it were pushed through his plane. ¢ All he can do is to come upon the boundaries of it in a different way to that in which he would if it passed straight through his plane.

Thus in this solid section ; the whole interior lies perfectly open in the fourth dimension. Go round it as we may we are simply looking at the boundaries of the tesseract which penetrates through our solid sheet. If the tesseract were not to pass across so far, the triangle

Null-b. Fig. 122.