Hinton - The Fourth Dimension.pdf

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THE FOURTH DIMENSION
square in the transference along the blue axis by which
this cube is generated from the orange face. This
purple square made by the motion of the red line is
the same purple face that we saw before as a series of
lines in the section b1, b2, b3. Here, since both red and
blue axes are in our space, we have no need of duration
to represent the area they determine. In the motion
of the tesseract across space this purple face would
instantly disappear.
From the orange face, which is common to the initial
cubes in fig. 107 and fig. 108, there goes in the blue
direction a cube coloured brown. This brown cube is
now all in our space, because each of its three axes run
in space directions, up, away, to the left. It is the same
brown cube which appeared as the successive faces on the
sections b1, b2, b3. Having all its three axes in our
space, it is given in extension; no part of it needs to
be represented as a succession. The tesseract is now
in a new position with regard to our space, and when
it moves across our space the brown cube instantly
disappears.
In order to exhibit the other region of the tesseract
we must remember that now the white line runs in the
unknown dimension. Where shall we put the section
at distances along the line? Any arbitrary position in
our space will do; there is no way by which we can
represent their real position.
However, as the brown cube comes off from the orange
face to the left, let us put these successive sections to
the left. We can call them wh0, wh1, wh2, wh3, wh4,
because they are sections along the white axis, which
now runs in the unknown dimension.
Running from the purple square in the white direction
we find the light purple cube. This is represented in the
sections wh1, wh2, wh3, fig. 108. It is the same cube