He must conceive of a space of four mutually independent directions; a space, that is, having a direction at right angles to every direction that we know. We cannot point to this, we cannot picture it, but we can reason about it with a precision that is all but absolute. In such a space it would of course be possible to establish four axial lines, all intersecting at a point, and all mutually at right angles with one another. Every hyper-solid of four-dimensional space has these four axes.

The regular hyper-solids (analogous to the Platonic solids of three-dimensional space) are the "fantastic forms" which will prove useful to the artist. He should learn to lure them forth along them axis lines. That is, let him build up his figures, space by space, developing them from lower spaces to higher. But since he cannot enter the fourth dimension, and build them there, nor even the third—if he confines himself to a sheet of paper—he must seek out some form of representation of the higher in the lower. This is a process with which he is already acquainted, for he employs it every time he makes a perspective drawing, which is the representation of a solid on a plane. All that is required is an extension of the method: a hyper-solid can be represented in a figure of three dimensions, and this in turn can be projected on a plane. The achieved result will constitute a perspective of a perspective—the representation of a representation.

This may sound obscure to the uninitiated, and it is true that the plane projection of some of the regular hyper-solids are staggeringly intricate affairs, but the author is so sure that this matter lies so well within the compass of the average non-mathematical mind that he is willing to put his confidence to a practical test.

It is proposed to develop a representation of the tesseract or hyper-cube on the paper of this page, that is, on a space of two dimensions. Let us start as far back as we can: with a point. This point, a, [Figure 14] is conceived to move in a direction w, developing the line a b. This line next moves in a direction at right angles to w, namely, x, a distance equal to its length, forming the square a b c d. Now for the square to develop into a cube by a movement into the third dimension it would have to move in a direction at right angles to both w and x, that is, out of the plane of the paper—away from it altogether, either up or down. This is not possible, of course, but the third direction can be represented on the plane of the paper.

[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.

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