II
THE FOURTH DIMENSION
The subject of the fourth dimension is not an easy one to understand. Fortunately the artist in design does not need to penetrate far into these fascinating halls of thought in order to reap the advantage which he seeks. Nevertheless an intention of mind upon this "fairy-tale of mathematics" cannot fail to enlarge his intellectual and spiritual horizons, and develop his imagination—that finest instrument in all his chest of tools.
By way of introduction to the subject Prof. James Byrnie Shaw, in an article in the Scientific Monthly, has this to say:
Up to the period of the Reformation algebraic equations of more than the third degree were frowned upon as having no real meaning, since there is no fourth power or dimension. But about one hundred years ago this chimera became an actual existence, and today it is furnishing a new world to physics, in which mechanics may become geometry, time be co-ordinated with space, and every geometric theorem in the world is a physical theorem in the experimental world in study in the laboratory. Startling indeed it is to the scientist to be told that an artificial dream-world of the mathematician is more real than that he sees with his galvanometers, ultra-microscopes, and spectroscopes. It matters little that he replies, "Your four-dimensional world is only an analytic explanation of my phenomena," for the fact remains a fact, that in the mathematician's four-dimensional space there is a space not derived in any sense of the term as a residue of experience, however powerful a distillation of sensations or perceptions be resorted to, for it is not contained at all in the fluid that experience furnishes. It is a product of the creative power of the mathematical mind, and its objects are real in exactly the same way that the cube, the square, the circle, the sphere or the straight line. We are enabled to see with the penetrating vision of the mathematical insight that no less real and no more real are these fantastic forms of the world of relativity than those supposed to be uncreatable or indestructible in the play of the forces of nature.
These "fantastic forms" alone need concern the artist. If by some potent magic he can precipitate them into the world of sensuous images so that they make music to the eye, he need not even enter into the question of their reality, but in order to achieve this transmutation he should know something, at least, of the strange laws of their being, should lend ear to a fairy-tale in which each theorem is a paradox, and each paradox a mathematical fact.
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]
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.
[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 multiplying illustrations, he would only be going over ground already covered in his book Projective Ornament and in his second Scammon lecture.
Of course this elaborate mechanism for producing quite obvious and even ordinary decorative motifs may appear to some readers like Goldberg's nightmare mechanics, wherein the most absurd and intricate devices are made to accomplish the most simple ends. The author is undisturbed by such criticisms. If the designs dealt with in this chapter are "obvious and even ordinary" they are so for the reason that they were chosen less with an eye to their interest and beauty than as lending themselves to development and demonstration by an orderly process which should not put too great a tax upon the patience and intelligence of the reader. Four-dimensional geometry yields numberless other patterns whose beauty and interest could not possibly be impeached—patterns beyond the compass of the cleverest designer unacquainted with projective geometry.
[Illustration: Figure 16.]
The great need of the ornamentalist is this or some other solid foundation. Lacking it, he has been forced to build either on the shifting sands of his own fancy, or on the wrecks and sediment of the past. Geometry provides this sure foundation. We may have to work hard and dig deep, but the results will be worth the effort, for only on such a foundation can arise a temple which is beautiful and strong.
In confirmation of his general contention that the basis of all effective decoration is geometry and number, the author, in closing, desires to direct the reader's attention to Figure 17 a slightly modified rendering of the famous zodiacal ceiling of the Temple of Denderah, in Egypt. A sun and its corona have been substituted for the zodiacal signs and symbols which fill the centre of the original, for except to an Egyptologist these are meaningless. In all essentials the drawing faithfully follows the original—was traced, indeed, from a measured drawing.
[Illustration: Figure 17. CEILING DECORATION FROM THE TEMPLE OF
DENDERAH]
Here is one of the most magnificent decorative schemes in the whole world, arranged with a feeling for balance and rhythm exceeding the power of the modern artist, and executed with a mastery beyond the compass of a modern craftsman. The fact that first forces itself upon the beholder is that the thing is so obviously mathematical in its rhythms, that to reduce it to terms of geometry and number is a matter of small difficulty. Compare the frozen music of these rhymed and linked figures with the herded, confused, and cluttered compositions of even our best decorative artists, and argument becomes unnecessary—the fact stands forth that we have lost something precious and vital out of art of which the ancients possessed the secret.
It is for the restoration of these ancient verities and the discovery of new spatial rhythms—made possible by the advance of mathematical science—that the author pleads. Artists, architects, designers, instead of chewing the cud of current fashion, come into these pastures new!
[Illustration]
[Footnote 1: The eight cubes in A, Figure 14, are as follows: abb'd'c'c; ABB'D'C'C; abdDCA; a'b'd'D'C'A'; abb'B'A'A; cdd'D'C'C; bb'd'D'DB; aa'c'C'CA.]
[Footnote 2: The sixteen cells of the hexadehahedroid are as follows:
ABCD: A'B'C'D': AB'C'D': A'BCD: AB'CD: A'BC'D: ABC'D: A'B'CD': ABCD':
A'B'C'D: ABC'D': A'B'CD: A'BC'D: AB'CD': A'BCD': AB'C'D.]