Hinton - The Fourth Dimension.pdf

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THE FOURTH DIMENSION
points LMN (fig. 128), and so in the region of our space
Red axis
before we go off into
N
the fourth dimension,
we have the plane
represented by LMN
E F
extended. This is what
is common to the
slanting space and our
L C M space.
White axis Yellow axis
Fig. 128.
This plane cuts the
ochre cube in the triangle EFG.
Comparing this with (fig. 72) oh, we see that the
hexagon there drawn is part of the triangle EFG.
Let us now imagine the tesseract and the slanting
space both together to pass transverse to our space, a
distance of one unit, we have in 1h a section of the
tesseract, whose axes are parallels to the previous axes.
The slanting space cuts them at a distance of five units
along each. Drawing the plane through these points in
1h it will be found to cut the cubical section of the
tesseract in the hexagonal figure drawn. In 2h (fig. 72) the
slanting space cuts the parallels to the axes at a distance
of four along each, and the hexagonal figure is the section
of this section of the tesseract by it. Finally when 3h
comes in the slanting space cuts the axes at a distance
of three along each, and the section is a triangle, of which
the hexagon drawn is a truncated portion. After this
the tesseract, which extends only three units in each of
the four dimensions, has completely passed transverse
of our space, and there is no more of it to be cut. Hence,
putting the plane sections together in the right relations,
we have the section determined by this particular slanting
space: namely an octahedron.