ownward toward the right, namely,
y. In the y direction, then, and at a distance equal to the length
of one of the sides of the square, another square is drawn, a'b'c'd',
representing the original square at the end of its movement into the
third dimension; and because in that movement the bounding points of
the square have traced out lines (edges), it is necessary to connect
the corresponding corners of the two squares by means of lines. This
completes the figure and achieves the representation of a cube on a
plane by a perfectly simple and familiar process. Its six faces
are easily identified by the eye, though only two of them appear as
squares owing to the exigencies of representation.

Now for a leap into the abyss, which won't be so terrifying, since
it involves no change of method. The cube must move into the fourth
dimension, developing there a hyper-cube. This is impossible, for
the reason the cube would have to move out of our space
altogether--three-dimensional space will not contain a hyper-cube. But
neither is the cube itself contained within the plane of the paper;
it is only there _represented_. The y direction had to be imagined and
then arbitrarily established; we can arbitrarily establish the fourth
direction in the same way. As this is at right angles to y, its
indication may be diagonally downward and to the left--the direction
z. As y is known to be at right angles both to w and to x, z is at
right angles to all three, and we have thus established the four
mutually perpendicular axes necessary to complete the figure.

The cube must now move in the z direction (the fourth dimension)
a distance equal to the length of one of its sides. Just as we did
previously in the case of the square, we draw the cube in its new
position (ABB'D'C'C) and also as before we connect each apex of the
first cube with the corresponding apex of the other, because each of
these points generates a line (an edge), each line a plane, and
each plane a solid. This is the tesseract or hyper-cube in plane
projection. It has the 16 points, 32 lines, and 8 cubes known to
compose the figure. These cubes occur in pairs, and may be readily
identified.[1]

The tesseract as portrayed in A, Figure 14, is shown according to the
conventions of oblique, or two-point perspective; it can equally be
represented in a manner correspondent to parallel perspective. The
parallel perspective of a cube appears as a square inside another
square, with