Hinton - The Fourth Dimension.pdf

THE SIMPLEST FOUR-DIMENSIONAL SOLID 169
We get then altogether, as three-dimensional regions,
ochre, brown, light purple, light green.
Finally, there is the region which corresponds to a
mixture of all the colours; there is only one region such
as this. It is the one that springs from ochre by the
addition of blue—this colour we call light brown.
Looking at the light brown region we see that it
increases in four ways. Hence, the tesseracts of which it
is composed increase in number in each of four dimensions,
and the shape they form does not remain thin in
any of the four dimensions. Consequently this region
becomes the solid content of the block of tesseracts itself;
it is the real four-dimensional solid. All the other regions
are then boundaries of this light brown region. If we
suppose the process of increasing the number of tesseracts
and decreasing their size carried on indefinitely, then
the light brown coloured tesseracts become the whole
interior mass, the three-coloured tesseracts become threedimensional
boundaries, thin in one dimension, and form
the ochre, the brown, the light blue, the light green.
The two-coloured tesseracts become two-dimensional
boundaries, thin in two dimensions, e.g., the pink, the
green, the purple, the orange, the light blue, the light
yellow. The one-coloured tesseracts become bounding
lines, thin in three dimensions, and the null points become
bounding corners, thin in four dimensions. From these
thin real boundaries we can pass in thought to the
abstractions—points, lines, faces, solids—bounding the
four-dimensional solid, which in this case is light brown
coloured, and under this supposition the light brown
coloured region is the only real one, is the only one which
is not an abstraction.
It should be observed that, in taking a square as the
representation of a cube on a plane, we only represent
one face, or the section between two faces. The squares,