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 lines connecting the four vertices of the one with those
of the other. The third dimension (the one beyond the plane of the
paper) is here conceived of as being not beyond the boundaries of the
first square, but _within_ them. We may with equal propriety conceive
of the fourth dimension as a "beyond which is within." In that case
we would have a rendering of the tesseract as shown in B, Figure 14:
a cube within a cube, the space between the two being occupied by six
truncated pyramids, each representing a cube. The large outside cube
represents the original generating cube at the beginning of its motion
into the fourth dimension, and the small inside cube represents it at
the end of that motion.

[Illustration: PLATE XIII. IMAGINARY COMPOSITION: THE AUDIENCE
CHAMBER]

These two projections of the tesseract upon plane space are not the
only ones possible, but they are typical. Some idea of the variety of