Take models 4, 5, 6. Place 4, or suppose No. 4 of the tesseract views placed, with its orange face coincident with the orange face of 1, red line to red line, and yellow line to yellow line, with the blue line pointing to the left. Then remove cube 1 and we have the tesseract face which comes in when the white axis runs in the positive unknown, and the blue axis comes into our space.
Now place catalogue cube 5 in some position, it does not matter which, say to the left; and place it so that there is a correspondence of colour corresponding to the colour of the line that runs out of space. The line that runs out of space is white, hence, every part of this cube 5 should differ from the corresponding part of 4 by an alteration in the direction of white.
Thus we have white points in 5 corresponding to the null points in 4. We have a pink line corresponding to a red line, a light yellow line corresponding to a yellow line, an ochre face corresponding to an orange face. This cube section is completely named in Chapter XI. Finally cube 6 is a replica of 1.
These catalogue cubes will enable us to set up our models of the block of tesseracts.
First of all for the set of tesseracts, which beginning in our space reach out one inch in the unknown, we have the pattern of catalogue cube 4.
We see that we can build up a block of twenty-seven tesseract faces after the colour scheme of cube 4, by taking the left-hand wall of block 1, then the left-hand wall of block 2, and finally that of block 3. We take, that is, the three first walls of our previous arrangement to form the first cubic block of this new one.
This will represent the cubic faces by which the group of tesseracts in its new position touches our space. We have running up, null f., red f., null f. In the next vertical line, on the side remote from us, we have yellow f., orange f., yellow f., and then the first colours over again. Then the three following columns are, blue f., purple f., blue f.; green f., brown f., green f.; blue f., purple f., blue f. The last three columns are like the first.
These tesseracts touch our space, and none of them are by any part of them distant more than an inch from it. What lies beyond them in the unknown?
This can be told by looking at catalogue cube 5. According to its scheme of colour we see that the second wall of each of our old arrangements must be taken. Putting them together we have, as the corner, white f. above it, pink f. above it, white f. The column next to this remote from us is as follows:—light yellow f., ochre f., light yellow f., and beyond this a column like the first. Then for the middle of the block, light blue f., above it light purple, then light blue. The centre column has, at the bottom, light green f., light brown f. in the centre and at the top light green f. The last wall is like the first.
The third block is made by taking the third walls of our previous arrangement, which we called the normal one.