Hinton - The Fourth Dimension.pdf

REMARKS ON THE FIGURES 195
face in an orange line with null points. This then is one
of the boundaries of the section figure.
Let now the cube be so turned that the pink face
comes in his plane. The points null r. and null wh.
are now visible. The line between them is pink
with null points, and since this line is common to
the surface of the cube and the cutting plane, it is
a boundary of the figure in which the plane cuts the
cube.
Again, suppose the cube turned so that the light
yellow face is in contact with the plane being’s plane.
He sees two points, the null wh. and the null y. The
line between these lies in the cutting plane. Hence,
since the three cutting lines meet and enclose a portion
of the cube between them, he has determined the
figure he sought. It is a triangle with orange, pink,
and light yellow sides, all equal, and enclosing an
ochre area.
Let us now determine in what figure the space,
determined by the four points, null r., null y., null
wh., null b., cuts the tesseract. We can see three
of these points on the primary position of the tesseract
resting against our solid sheet by the ochre cube.
These three points determine a plane which lies in
the space we are considering, and this plane cuts
the ochre cube in a triangle, the interior of which
is ochre (fig. 119 will serve for this view), with pink,
light yellow and orange sides, and null points. Going
in the fourth direction, in one sense, from this place
we pass into the tesseract, in the other sense we pass
away from it. The whole area inside the triangle is
common to the cutting plane we see, and a boundary
of the tesseract. Hence we conclude that the triangle
drawn is common to the tesseract and the cutting
space.