In order to exhibit the other regions of the tesseract we must remember
that now the white line runs in the unknown dimension. Where shall we
put the sections 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
_wh__{0}, _wh__{1}, _wh__{2}, _wh__{3}, _wh__{4}, 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 _wh__{1}, _wh__{2},
_wh__{3}, _wh__{4}, fig. 108. It is the same cube that is represented
in the sections _b__{1}, _b__{2}, _b__{3}: in fig. 107 the red and
white axes are in our space, the blue out of it; in the other case, the
red and blue are in our space, the white out of it. It is evident that
the face pink _y_, opposite the pink face in fig. 107, makes a cube
shown in squares in _b__{1}, _b__{2}, _b__{3}, _b__{4}, on the opposite
side to the _l_ purple squares. Also the light yellow face at the base
of the cube _b__{0}, makes a light green cube, shown as a series of
base squares.
The same light green cube can be found in fig. 107. The base square in
_wh__{0} is a green square, for it is enclosed by blue and yellow axes.
From it goes a cube in the white direction, this is then a light green
cube and the same as the one just mentioned as existing in the sections
_b__{0}, _b__{1}, _b__{2}, _b__{3}, _b__{4}.
The case is, however, a little different with the brown cube. This cube
we have altogether in space in the section _wh__{0}, fig. 108, while
it exists as a series of squares, the left-hand ones, in the sections
_b__{0}, _b__{1}, _b__{2}, _b__{3}, _b__{4}. The brown cube exists as a
solid in our space, as shown in fig. 108. In the mode of representation
of the tesseract exhibited in fig. 107, the same brown cube appears as
a succession of squares. That is, as the tesseract moves across space,
the brown cube would actually be to us a square—it would be merely
the lasting boundary of another solid. It would have no thickness at
all, only extension in two dimensions, and its duration would show its
solidity in three dimensions.
Public-domain text, read in full here on John Shaqi.
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