Hinton - The Fourth Dimension.pdf

132
THE FOURTH DIMENSION
Hence these two hexagons fit together, forming one
hexagon, and the line TW is only wanted when we consider
a section of the whole figure, we thus obtain the
solid represented in the lower part of fig. 74. Equal
repetitions of this figure, called as tetrakaidekagon, will
fill up three-dimensional space.
To make the corresponding four-dimensional figure we
have to take five axes mutually at right angles with five
points on each. A catalogue of the positions determined
in five-dimensional space can be found thus.
Take a cube with five points on each of its axes, the
fifth point is at a distance of four units of length from the
first on any one of the axes. And since the fourth dimension
also stretches to a distance of four we shall need to
represent the succes-
4L
3L
2L
1L
0L
0H 1H 2H 3H 4H
Fig. 75.
sive sets of points at
distances 0, 1, 2, 3, 4,
in the fourth dimension,
five cubes. Now
all of these extend to
no distance at all in
the fifth dimension.
To represent what
lies in the fifth dimension
we shall have to
draw, starting from
each of our cubes,
five similar cubes to
represent the four steps
on in the fifth dimension. By this assemblage we get a
catalogue of all the points shown in fig. 75, in which
L represents the fifth dimension.
Now, as we saw before, there is nothing to prevent us
from putting all the cubes representing the different
stages in the fourth dimension in one figure, if we take