The fourth dimension

REMARKS ON THE FIGURES 187

As the red line now runs in the fourth dimension, the guecessive sections can be called 7, 71, 72, 73, 74, these letters indicating that at distances 0, 4, 7, 3, 1 inch along the red axis we take all of the tesseract that can be found in a three-dimensional space, this three-dimensional space extending not at all in the fourth dimension, but up and down, right and left, far and near.

We can see what should replace the light yellow face of 7, when the section 7; comes in, by looking at the cube bo, fig. 107. What is distant in it one-quarter of an inch from the light yellow face in the red direction? It is an ochre section with orange and pink lines and red points ; see also fig. 103.

This square then forms the top square of m.. Now we

can determine the nomenclature of all the regions of 7 by considering what would be formed by the motion of this square along a blue axis. » But we can adopt another plan. Let us take a horizontal section of 7, and finding that section in the figures, of fig. 107 or fig. 103, from them determine what will replace it, going on in the red direction.

A section of the rp cube has green, light blue, green, light blue sides and blue points.

Now this square occurs on the base of each of the section figures, bi, bo, etc. In them we see that 1 inch in the red direction from it lies a section with brown and light purple lines and purple corners, the interior being of light brown. Hence this is the nomenclature of the section which in 7; replaces the section of ry made from a point along the blue axis.

Hence the colouring as given can be derived.

We have thus obtained a perfectly named group of tesseracts. We can take a group of eighty-one of them 3x3%x3x3, in four dimensions, and each tesseract will have its name null, red, white, yellow, blue, ete, and