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 s.p.a.ce, 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 a.n.a.lytic explanation of my phenomena," for the fact remains a fact, that in the mathematician's four-dimensional s.p.a.ce there is a s.p.a.ce 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 trans.m.u.tation 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 s.p.a.ce of four mutually independent directions; a s.p.a.ce, 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 s.p.a.ce 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 s.p.a.ce has these four axes.

The regular hyper-solids (a.n.a.logous to the Platonic solids of three-dimensional s.p.a.ce) 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, s.p.a.ce by s.p.a.ce, developing them from lower s.p.a.ces 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 const.i.tute 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 compa.s.s 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 s.p.a.ce 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.

[Ill.u.s.tration: 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 s.p.a.ce altogether--three-dimensional s.p.a.ce 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 s.p.a.ce 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.

[Ill.u.s.tration: PLATE XIII. IMAGINARY COMPOSITION: THE AUDIENCE CHAMBER]

These two projections of the tesseract upon plane s.p.a.ce 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 s.p.a.ce, would appear from several different directions. Each view would yield new s.p.a.ce-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 s.p.a.ce 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 s.p.a.ce.

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 s.p.a.ce. The fourth axis we will a.s.sume to pa.s.s 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 s.p.a.ce 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 s.p.a.ce upon the plane of our materiality.

It is a wonder, a mystery, staggering to the imagination, contradictory to experience, but as well ent.i.tled 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.

[Ill.u.s.tration: 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 ill.u.s.trations, 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 compa.s.s of the cleverest designer unacquainted with projective geometry.

[Ill.u.s.tration: 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 subst.i.tuted 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.

[Ill.u.s.tration: 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 compa.s.s 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!

[Ill.u.s.tration]

[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.]

HARNESSING THE RAINBOW

Reference was made in an antecedent essay to an art of light--of mobile color--an abstract language of thought and emotion which should speak to consciousness through the eye, as music speaks through the ear. This is an art unborn, though quickening in the womb of the future. The things that reflect light have been organized aesthetically into the arts of architecture, painting, and sculpture, but light itself has never been thus organized.

And yet the scientific development and control of light has reached a stage which makes this new art possible. It awaits only the advent of the creative artist. The manipulation of light is now in the hands of the illuminating engineers and its exploitation (in other than necessary ways) in the hands of the advertisers.

Some results of their collaboration are seen in the sky signs of upper Broadway, in New York, and of the lake front, in Chicago. A carnival of contending vulgarities, showing no artistry other than the most puerile, these displays nevertheless yield an effect of amazing beauty. This is on account of an occult property inherent in the nature of light--_it cannot be vulgarized_. If the manipulation of light were delivered into the hands of the artist, and dedicated to n.o.ble ends, it is impossible to overestimate the augmentation of beauty that would ensue.

For light is a far more potent medium than sound. The sphere of sound is the earth-sphere; the little limits of our atmosphere mark the uttermost boundaries to which sound, even the most strident can possibly prevail. But the medium of light is the ether, which links us with the most distant stars. May not this serve as a symbol of the potency of light to usher the human spirit into realms of being at the doors of which music itself shall beat in vain? Or if we compare the universe accessible to sight with that accessible to sound--the plight of the blind in contrast to that of the deaf--there is the same discrepancy; the field of the eye is immensely richer, more various and more interesting than that of the ear.

The difficulty appears to consist in the inferior impressionability of the eye to its particular order of beauty. To the average man color--as color--has nothing significant to say: to him gra.s.s is green, snow is white, the sky blue; and to have his attention drawn to the fact that sometimes gra.s.s is yellow, snow blue, and the sky green, is disconcerting rather than illuminating. It is only when his retina is a.s.saulted by some splendid sunset or sky-encircling rainbow that he is able to disa.s.sociate the idea of color from that of form and substance. Even the artist is at a disadvantage in this respect, when compared with the musician. Nothing in color knowledge and a.n.a.lysis a.n.a.logous to the established laws of musical harmony is part of the equipment of the average artist; he plays, as it were, by ear. The scientist, on the other hand, though he may know the spectrum from end to end, and its innumerable modifications, values this "rainbow promise of the Lord" not for its own beautiful sake but as a means to other ends than those of beauty. But just as the art of music has developed the ear into a fine and sensitive instrument of appreciation, so an a.n.a.logous art of light would educate the eye to nuances of color to which it is now blind.

[Ill.u.s.tration: PLATE XIV. SONG AND LIGHT: AN APPROACH TOWARD "COLOR MUSIC"]

It is interesting to speculate as to the particular form in which this new art will manifest itself. The question is perhaps already answered in the "color organ," the earliest of which was Bambridge Bishop's, exhibited at the old Barnum's Museum--before the days of electric light--and the latest A.W. Rimington's. Both of these instruments were built upon a supposed correspondence between a given scale of colors, and the musical chromatic scale; they were played from a musical score upon an organ keyboard. This is sufficiently easy and sufficiently obvious, and has been done, with varying success in one way or another, time and again, but its very ease and obviousness should give us pause.

It may well be questioned whether any arbitrary and literal translation, even though practicable, of a highly complex, intensely mobile art, unfolding in time, as does music, into a correspondent light and color expression, is the best approach to a new art of mobile color. There is a deep and abiding conviction, justified by the history of aesthetics, that each art-form must progress from its own beginnings and unfold in its own unique and characteristic way.

Correspondences between the arts--such a correspondence, for example, as inspired the famous saying that architecture is frozen music--reveal themselves usually only after the sister arts have attained an independent maturity. They owe their origin to that underlying unity upon which our various modes of sensuous perception act as a refracting medium, and must therefore be taken for granted.

Each art, like each individual, is unique and singular; in this singularity dwells its most thrilling appeal. We are likely to miss light's crowning glory, and the rainbow's most moving message to the soul if we preoccupy ourselves too exclusively with the ident.i.ties existing between music and color; it is rather their points of difference which should first be dwelt upon.

Let us accordingly consider the characteristic differences between the two sense-categories to which sound and light--music and color--respectively belong. This resolves itself into a comparison between time and s.p.a.ce. The characteristic thing about time is succession--hence the very idea of music, which is in time, involves perpetual change. The characteristic of s.p.a.ce, on the other hand, is simultaneousness--in s.p.a.ce alone perpetual immobility would reign.

That is why architecture, which is pre-eminently the art of s.p.a.ce, is of all the arts the most static. Light and color are essentially of s.p.a.ce, and therefore an art of mobile colour should never lack a certain serenity and repose. A "tune" played on a color organ is only distressing. If there is a workable correspondence between the musical art and an art of mobile color, it will be found in the domain of harmony which involves the idea of simultaneity, rather than in melody, which is pure succession. This fundamental difference between time and s.p.a.ce cannot be over-emphasized. A musical note prolonged, becomes at last scarcely tolerable; while a beautiful color, like the blue of the sky, we can enjoy all day and every day. The changing hues of a sunset, are _andante_ if referred to a musical standard, but to the eye they are _allegretto_--we would have them pa.s.s less swiftly than they do. The winking, chasing, changing lights of illuminated sky-signs are only annoying, and for the same reason. The eye longs for repose in some serene radiance or stately sequence, while the ear delights in contrast and continual change. It may be that as the eye becomes more educated it will demand more movement and complexity, but a certain stillness and serenity are of the very nature of light, as movement and pa.s.sion are of the very nature of sound. Music is a seeking--"love in search of a word"; light is a finding--a "divine covenant."

With attention still focussed on the differences rather than the similarities between the musical art and a new art of mobile color, we come next to the consideration of the matter of form. Now form is essentially of s.p.a.ce: we speak about the "form" of a musical composition, but it is in a more or less figurative and metaphysical sense, not as a thing concrete and palpable, like the forms of s.p.a.ce.

It would be foolish to forego the advantage of linking up form with colour, as there is opportunity to do. Here is another golden ball to juggle with, one which no art purely in time affords. Of course it is known that musical sounds weave invisible patterns in the air, and to render these patterns perceptible to the eye may be one of the more remote and recondite achievements of our uncreated art. Meantime, though we have the whole treasury of natural forms to draw from, of these we can only properly employ such as are _abstract_. The reason for this is clear to any one who conceives of an art of mobile color, not as a moving picture show--a thing of quick-pa.s.sing concrete images, to shock, to startle, or to charm--but as a rich and various language in which light, proverbially the symbol of the spirit, is made to speak, through the senses, some healing message to the soul.

For such a consummation, "devoutly to be wished," natural forms--forms abounding in every kind of a.s.sociation with that world of materiality from which we would escape--are out of place; recourse must be had rather to abstract forms, that is, geometrical figures. And because the more remote these are from the things of sense, from knowledge and experience, the projected figures of four-dimensional geometry would lend themselves to these uses with an especial grace. Color without form is as a soul without a body; yet the body of light must be without any taint of materiality. Four-dimensional forms are as immaterial as anything that could be imagined and they could be made to serve the useful purpose of separating colors one from another, as lead lines do in old cathedral windows, than which nothing more beautiful has ever been devised.

Coming now to the consideration, not of differences, but similarities, it is clear that a correspondence can be established between the colors of the spectrum and the notes of a musical scale. That is, the spectrum, considered as the a.n.a.logue of a musical octave can be subdivided into twelve colors which may be representative of the musical chromatic scale of twelve semi-tones: the very word, _chromatic_, being suggestive of such a correspondence between sound and light. The red end of the spectrum would naturally relate to the low notes of the musical scale, and the violet end to the high, by reason of the relative rapidity of vibration in each case; for the octave of a musical note sets the air vibrating twice as rapidly as does the note itself, and roughly speaking, the same is true of the end colors of the spectrum with relation to the ether.

But a.s.suming that a color scale can be established which would yield a color correlative to any musical note or chord, there still remains the matter of _values_ to be dealt with. In the musical scale there is a practical equality of values: one note is as potent as another. In a color scale, on the other hand, each note (taken at its greatest intensity) has a positive value of its own, and they are all different. These values have no musical correlatives, they belong to color _per se_. Every colorist knows that the whole secret of beauty and brilliance dwells in a proper understanding and adjustment of values, and music is powerless to help him here. Let us therefore defer the discussion of this musical parallel, which is full of pitfalls, until we have made some examination into such simple emotional reactions as color can be discovered to yield. The musical art began from the emotional response to certain simple tones and combinations, and the delight of the ear in their repet.i.tion and variation.

On account of our undeveloped sensitivity, the emotional reactions to color are found to be largely personal and whimsical: one person "loves" pink, another purple, or green. Color therapeutics is too new a thing to be relied upon for data, for even though colors are susceptible of cla.s.sification as sedative, recuperative and stimulating, no two cla.s.sifications arrived at independently would be likely to correspond. Most people appear to prefer bright, pure colors when presented to them in small areas, red and blue being the favourites. Certain data have been acc.u.mulated regarding the physiological effect and psychological value of different colors, but this order of research is in its infancy, and we shall have recourse, therefore, to theory, in the absence of any safer guide.