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
aspects may be gained by imagining how a nest of inter-related cubes
(made of wire, so as to interpenetrate), combined into a single
symmetrical figure of three-dimensional space, would appear
from several different directions. Each view would yield new
space-subdivisions, and all would be rhythmical--susceptible,
therefore, of translation into ornament. C and D represent such
translations of A and B.

In order to fix these unfamiliar ideas more firmly in the reader's
mind, let him submit himself to one more exercise of the creative
imagination, and construct, by a slightly different method, a
representation of a hexadecahedroid, or 16-hedroid, on a plane. This
regular solid of four-dimensional space consists of sixteen cells,
each a regular tetrahedron, thirty-two triangular faces, twenty-four
edges and eight vertices. It is the correlative of the octahedron of
three-dimensional space.

First it is necessary to establish our four axes, all mutually
at right angles. If we draw three lines intersecting at a point,
subtending angles of 60 degrees each, it is not difficult to
conceive of these lines as being at right angles with one another
in three-dimensional space. The fourth axis we will assume to pass
vertically through the point of intersection of the three lines,
so that we see it only in cross-section, that is, as a point. It is
important to remember that all of the angles made by the four axes
are right angles--a thing possible only in a space of four dimensions.
Because the 16-hedroid is a symmetrical hyper