Hinton - The Fourth Dimension.pdf

184
THE FOURTH DIMENSION
presented to us in another aspect, it would not be a mere
surface. But it is most convenient to regard the cubes
we use as having no extension at all in the fourth
dimension. This consideration serves to bring out a point
alluded to before, that, if there is a fourth dimension, our
conception of a solid is the conception of a mere abstraction,
and our talking about real three-dimensional objects would
seem to a four-dimensional being as incorrect as a twodimensional
being’s telling about real squares, real
triangles, etc., would seem to us.
The consideration of the two views of the brown cube
shows that any section of a cube can be looked at by a
presentation of the cube in a different position in fourdimensional
space. The brown faces in b1, b2, b3, are the
very same brown sections that would be obtained by
cutting the brown cube, wh0, across at the right distances
along the blue line, as shown in fig. 108. But as these
sections are placed in the brown cube, wh0, they come
behind one another in the blue direction. Now, in the
sections wh1, wh2, wh3, we are looking at these sections
from the white direction—the blue direction does not
exist in these figures. So we see them in a direction at
right angles to that in which they occur behind one
another in wh0. They are intermediate views, which
would come in the rotation of a tesseract. These brown
squares can be looked at from directions intermediate
between the white and blue axes. It must be remembered
that the fourth dimension is perpendicular equally to all
three space axes. Hence we must take the combinations
of the blue axis, with each two of our three axes, white,
red, yellow, in turn.
In fig. 109 we take red, white, and blue axes in space,
sending yellow into the fourth dimension. If it goes into
the positive sense of the fourth dimension the blue line
will come in the opposite direction to that in which the