Hinton - The Fourth Dimension.pdf

A FOUR-DIMENSIONAL FIGURE 135
These separate figures are the successive stages in
which the four-dimensional figure in which they
cohere can be apprehended.
The first figure and the last are tetrakaidekagons.
These are two of the solid boundaries of the figure. The
other solid boundaries can be traced easily. Some of
them are complete from one face in the figure to the
corresponding face in the next, as for instance the solid
which extends from the hexagonal base of the first figure
to the equal hexagonal base of the second figure. This
kind of boundary is a hexagonal prism. The hexagonal
prism also occurs in another sectional series, as for
instance, in the square at the bottom of the first figure,
the oblong at the base of the second and the square at
the base of the third figure.
Other solid boundaries can be traced through four of
the five sectional figures. Thus taking the hexagon at
the top of the first figure we find in the next a hexagon
also, of which some alternate sides are elongated. The
top of the third figure is also a hexagon with the other
set of alternate rules elongated, and finally we come in
the fourth figure to a regular hexagon.
These four sections are the sections of a tetrakaidekagon
as can be recognised from the sections of this figure
which we have had previously. Hence the boundaries
are of two kinds, hexagonal prisms and tetrakaidekagons.
These four-dimensional figures exactly fill four-dimensional
space by equal repetitions of themselves.