Hinton - The Fourth Dimension.pdf

A FOUR-DIMENSIONAL FIGURE 131
of measurement. Take for instance the points 3120 to
3021, which in the first diagram (fig. 72) lie in the first
and second figures. Their actual relation is shown in
fig. 73 in the cube marked 2k, where the points in question
are marked with a * in fig. 73. We see that the
distance in question is the diagonal of a unit square. In
like manner we find that the distance between corresponding
points of any two hexagonal figures is the
diagonal of a unit square. The total figure is now easily
constructed. An idea
of it may be gained by
drawing all the four
cubes in the catalogue
figure in one (fig. 74).
These cubes are exact
repetitions of one
another, so one drawing
will serve as a
representation of the
whole series, if we
take care to remember
where we are, whether
in a 0h, a 1h, a 2h,
or a 3h figure, when
R
T V
P Q
R S
we pick out the points required. Fig 74 is a representation
of all the catalogue cubes put in one. For the sake
of clearness the front faces and the back faces of this
cube are represented separately.
The figure determined by the selected points is shown
below.
In putting the sections together some of the outlines
in them disappear. The line TW for instance is not
wanted.
We notice that PQTW and TWRS are each the half
of a hexagon. Now QV and VR lie in one straight line.
S
W
Fig. 74.