describing a circle--the plane projection of the hyper-sphere--about
the central point of intersection of the axes. (Figure 15, left.)
Where each of these intersects the circle an apex of the 16-hedroid
will be established. From each apex it is now necessary to draw
straight lines to every other, each line representing one edge of the
sixteen tetrahedral cells. But because the two ends of the fourth axis
are directly opposite one another, and opposite the point of sight,
all of these lines fail to appear in the left hand diagram. It
therefore becomes necessary to _tilt_ the figure slightly, bringing
into view the fourth axis, much foreshortened, and with it, all of the
lines which make up the figure. The result is that projection of the
16-hedroid shown at the right of Figure 15.[2] Here is no fortuitous
arrangement of lines and areas, but the "shadow" cast by an
archetypal, figure of higher space upon the plane of our materiality.
It is a wonder, a mystery, staggering to the imagination,
contradictory to experience, but as well entitled to a place at the
high court of reason as are any of the more familiar figures with
which geometry deals. Translated into ornament it produces such an
all-over pattern as is shown in Figure 16 and the design which adorns
the curtains at right and left of pl. XIII. There are also other
interesting projections of the 16-hedroid which need not be gone into
here.

[Illustration: Figure 15. DIRECT VIEW AXES SHOWN BY HEAVY LINES TILTED
VIEW APEXES SHOWN BY CIRCLES THE 16-HEDROID IN PLANE PROJECTION]

For if the author has been successful in his exposition up to
this point, it should be sufficiently plain that the geometry
of four-dimensions is capable of yielding fresh and interesting
ornamental motifs. In carrying his demonstration farther, and in