Hinton - The Fourth Dimension.pdf

194
THE FOURTH DIMENSION
point, or three points, determine a plane. And finally,
four points determine a space. We have seen that a
plane and a point determine a space, and that three
points determine a plane; so four points will determine
a space.
These four points may be any points, and we can take,
for instance, the four points at the extremities of the red,
white, yellow, blue axes, in the tesseract. These will
determine a space slanting with regard to the section
spaces we have been previously considering. This space
will cut the tesseract in a certain figure.
One of the simplest sections of a cube by a plan is
that in which the plane passes through the extremities
of the three edges which meet in a point. We see at
once that this plane would cut the cube in a triangle, but
we will go through the process by which a plane being
would most conveniently treat the problem of the determination
of this shape, in order that we may apply the
method to the determination of the figure in which a
space cuts a tesseract when it passes through the 4
points at unit distance from a corner.
We know that two points determine a line, three points
determine a plane, and given any two points in a plane
the line between them lies wholly in the plane.
Let now the plane being study the section made by
or.
Null r. l.y.
p.
Null y.
x
Null A Null wh.
Fig. 119.
B
a plane passing through the
null r., null wh., and null y.
points, fig. 119. Looking at
the orange square, which, as
usual, we suppose to be
initially in his plane, he sees
that the line from null r. to
null y., which is a line in the
section plane, the plane, namely, through the three
extremities of the edges meeting in null, cuts the orange