[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-solid all of its eight apexes will be equidistant from the centre of a containing hyper-sphere, whose “surface” these will intersect at symmetrically disposed points.  These apexes are established in our representation by 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.