Hinton - The Fourth Dimension.pdf

THE SIMPLEST FOUR-DIMENSIONAL SOLID 167
blocks of cubes, 64 in each block. He we see, com-
paring it with the figure of 81 tesseracts, that the number
of the different regions shows a different tendency of
increase. But 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 is 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
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 coloured in
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 there are sections between the