The fourth dimension

THE SIMPLEST FOUR-DIMENSIONAL SOLID 167

blocks of cubes, 64 in each block. Here we see, comparing it with the figure of 81 tesseracts, that the number of the different regions show a different tendency of increase. By taking five blocks of five divisions each way this would become even more clear.

We see, fig. 102, that starting from the point at any corner, the white coloured regions only extend out in a line. The same is true for the yellow, red, and blue. With regard to the latter it should be noticed that the line of blues does not consist in regions next to each other in the drawing, but in portions which come in in different cubes. The portions which lie next to one another in the fourth dimension must always be represented so, when we have a three-dimensional representation. Again, those regions such as the pink one, go on increasing in two dimensions, About the pink region this is seen without going out of the cube itself, the pink regions increase in length and height, but in no other dimension. In examining these regions it is sufficient to take one as a sample.

The purple increases in the same manner, for it comes in in a succession from below to above in block 2, and in a succession from block to block in 2 and 3. Now, a succession from below to above represents a continuous extension upwards, and a succession from block to block represents a continuous extension in the fourth dimension. Thus the purple regions increase in two dimensions, the upward and the fourth, so when we take a very great many divisions, and let each become very small, the purple region forms a two-dimensional extension.

In the same way, looking at the regions marked 1. b. or light blue, which starts nearest a corner, we see that the tesseracts occupying it increase in length from left to right, forming a line, and that there are as many lines of light blue tesseracts as thep> are sections between the