[Illustration:  Figure 14.  TWO PROJECTIONS OF THE HYPERCUBE OR TESSERACT, AND THEIR TRANSLATION INTO ORNAMENT.]

Let us represent it as diagonally downward 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 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.