But to make mathematics, and not the emotion which it expresses, the important thing, is not this to fall into the time-worn heresy of art for art's sake, that is, art for form's sake--art for the sake of mathematics? To this objection there is an answer, and as this answer contains the crux of the whole matter, embraces the proposition by which this thesis must stand or fall, it must be full and clear.

What is it, in the last a.n.a.lysis, that all art which is not purely personal and episodical strives to express? Is it not the _world-order_?--the very thing that religion, philosophy, science, strive according to their different natures and methods to express?

The perception of the world-order by the artist arouses an emotion to which he can give vent only in terms of number; but number is itself the most abstract expression of the world order. The form and content of art are therefore not different, but the same. A deep sense of this probably inspired Pater's famous saying that all art aspires toward the condition of music; for music, from its very nature, is the world-order uttered in terms of number, in a sense and to a degree not attained by any other art.

This is not mere verbal juggling. We have suffered so long from an art-phase which exalts the personal, as opposed to the cosmic, that we have lost sight of the fact that the great arts of antiquity, preceding the Renaissance, insisted on the cosmic, or impersonal aspect, and on this alone, just as does Oriental art, even today.

The secret essence, the archetypal idea of the subject is the preoccupation of the Oriental artist, as it was of the Egyptian, and of the Greek. We of the West today seek as eagerly to fix the accidental and ephemeral aspect--the shadow of a particular cloud upon a particular landscape; the smile on the face of a specific person, in a recognizable room, at a particular moment of time. Of symbolic art, of universal emotion expressing itself in terms which are universal, we have very little to show.

The reason for this is first, our love for, and understanding of, the concrete and personal: it is the _world-aspect_ and not the _world-order_ which interests us; and second, the inadequacies of current forms of art expression to render our sense of the eternal secret heart of things as it presents itself to our young eyes.

Confronted with this difficulty, we have shirked it, and our ambition has shrunk to the portrayal of those aspects which shuffle our poverty out of sight. It is not a poverty of technique--we are dexterous enough; nor is it a poverty of invention--we are clever enough; it is the poverty of the spiritual bankrupt trying to divert attention by a prodigal display of the smallest of small change.

Reference is made here only to the arts of s.p.a.ce; the arts of time--music, poetry, and the (written) drama--employing vehicles more flexible, have been more fortunate, though they too suffer in some degree from worshipping, instead of the G.o.d of order, the G.o.d of chance.

The corrective of this is a return to first principles: principles so fundamental that they suffer no change, however new and various their ill.u.s.trations. These principles are embodied in number, and one might almost say nowhere else in such perfection. Mathematics is not the dry and deadly thing that our teaching of it and the uses we put it to have made it seem. Mathematics is the handwriting on the human consciousness of the very Spirit of Life itself. Others before Pythagoras discovered this, and it is the discovery which awaits us too.

To indicate the way in which mathematics might be made to yield the elements of a new aesthetic is beyond the province of this essay, being beyond the compa.s.s of its author, but he makes bold to take a single phase: ornament, and to deal with it from this point of view.

The ornament now in common use has been gathered from the dust-bin of the ages. What ornamental _motif_ of any universality, worth, or importance is less than a hundred years old? We continue to use the honeysuckle, the acanthus, the fret, the egg and dart, not because they are appropriate to any use we put them to, but because they are beautiful _per se_. Why are they beautiful? It is not because they are highly conventionalized representations of natural forms which are themselves beautiful, but because they express cosmic truths. The honeysuckle and the acanthus leaf, for example, express the idea of successive impulses, mounting, attaining a maximum, and descending--expanding from some focus of force in the manner universal throughout nature. Science recognizes in the spiral an archetypal form, whether found in a whirlpool or in a nebula. A fret is a series of highly conventionalized spirals: translate it from angular to curved and we have the wave-band; isolate it and we have the volute.

Egg and dart are phallic emblems, female and male; or, if you prefer, as ellipse and straight line, they are symbols of finite existence contrasted with infinity. [Figure 1.]

[Ill.u.s.tration: Figure 1.]

Suppose that we determine to divest ourselves of these and other precious inheritances, not because they have lost their beauty and meaning, but rather on account of their manifold a.s.sociations with a past which the war makes suddenly more remote than slow centuries have done; suppose that we determine to supplant these symbols with others no less charged with beauty and meaning, but more directly drawn from the inexhaustible well of mathematical truth--how shall we set to work?

We need not _set_ to work, because we have done that already, we are always doing it, unknowingly, and without knowing the reason why. All ornamentalists are subjective mathematicians--an amazing statement, perhaps, but one susceptible of confirmation in countless amusing ways, of which two will be shown.

[Ill.u.s.tration: Figure 2.]

Consider first your calendar--your calendar whose commonplace face, having yielded you information as to pay day, due day, and holiday, you obliterate at the end of each month without a qualm, oblivious to the fact that were your interests less sordid and personal it would speak to you of that order which pervades the universe; would make you realize something of the music of the spheres. For on that familiar checkerboard of the days are numerical arrangements which are mysterious, "magical"; each separate number is as a spider at the center of an amazing mathematical web. That is to say, every number is discovered to be half of the sum of the pairs of numbers which surround it, vertically, horizontally, and diagonally: all of the pairs add to the same sum, and the central number divides this sum by two. A graphic indication of this fact on the calendar face by means of a system of intersecting lines yields that form of cla.s.sic grille dear to the heart of every tyro draughtsman. [Figure 2.] Here is an evident relation between mathematical fact and ornamental mode, whether the result of accident, or by reason of some subconscious connection between the creative and the reasoning part of the mind.

To show, by means of an example other than this acrostic of the days, how the pattern-making instinct follows unconsciously in the groove traced out for it by mathematics, the attention of the reader is directed to the design of the old Colonial bed-spread shown in Figure 3. Adjacent to this, in the upper right hand corner, is a magic square of four. That is, all of the columns of figures of which it is composed: vertical, horizontal and diagonal add to the same sum: 34.

An a.n.a.lysis of this square reveals the fact that it is made up of the figures of two different orders of counting: the ordinary order, beginning at the left hand upper corner and reading across and down in the usual way, and the reverse-ordinary, beginning at the lower right hand corner and reading across and up. The figures in the four central cells and in the four outside corner cells are discovered to belong in the first category, and the remaining figures in the second. Now if the ordinary order cells be represented by white, and the reverse ordinary by black, just such a pattern has been created as forms the decorative motif of the quilt.

It may be claimed that these two examples of a relation between ornament and mathematics are accidental and therefore prove nothing, but they at least furnish a clue which the artist would be foolish not to follow up. Let him attack his problem this time directly, and see if number may not be made to yield the thing he seeks: namely, s.p.a.ce-rhythms which are beautiful and new.

We know that there is a beauty inherent in _order_, that necessity of one sort or another is the parent of beauty. Beauty in architecture is largely the result of structural necessity; beauty in ornament may spring from a necessity which is numerical. It is clear that the arrangement of numbers in a magic square is necessitous--they must be placed in a certain way in order that the summation of every column shall be the same. The problem then becomes to make that necessity reveal itself to the eye. Now most magic squares contain a _magic path_, discovered by following the numbers from cell to cell in their natural order. Because this is a necessitous line it should not surprise us that it is frequently beautiful as well.

[Ill.u.s.tration: Figure 3.]

The left hand drawing in Figure 4 represents the smallest aggregation of numbers that is capable of magic square arrangement. Each vertical, horizontal, and corner diagonal column adds up to 15, and the sum of any two opposite numbers is 10, which is twice the center number. The magic path is the endless line developed by following, free hand, the numbers in their natural order, from 1 to 9 and back to 1 again. The drawing at the right of Figure 4 is this same line translated into ornament by making an interlace of it, and filling in the larger interstices with simple floral forms. This has been executed in white plaster and made to perform the function of a ventilating grille.

Now the number of magic squares is practically limitless, and while all of them do not yield magic lines of the beauty of this one, some contain even richer decorative possibilities. But there are also other ways of deriving ornament from magic squares, already hinted at in the discussion of the Colonial quilt.

[Ill.u.s.tration: Figure 4.]

[Ill.u.s.tration: Figure 5.]

Magic squares of an even number of cells are found sometimes to consist of numbers arranged not only in combinations of the ordinary and the reverse ordinary orders of counting, but involving two others as well: the reverse of the ordinary (beginning at the upper right hand, across, and down) and the reversed inverse, (beginning at the lower left hand, across, and up). If, in such a magic square, a simple graphic symbol be subst.i.tuted for the numbers belonging to each order, pattern spontaneously springs to life. Figures 5 and 6 exemplify the method, and Figures 7 and 8 the translation of some of these squares into richer patterns by elaborating the symbols while respecting their arrangement. By only a slight stretch of the imagination the beautiful pierced stone screen from Ravenna shown in Figure 9 might be conceived of as having been developed according to this method, although of course it was not so in fact. Some of the arrangements shown in Figure 6 are closely paralleled in the acoustic figures made by means of musical tones with sand, on a sheet of metal or gla.s.s.

[Ill.u.s.tration: Figure 6.]

[Ill.u.s.tration: Figure 7.]

The celebrated Franklin square of 16 cells can be made to yield a beautiful pattern by designating some of the lines which give the summation of 2056 by different symbols, as shown in Figure 10. A free translation of this design into pattern brickwork is indicated in Figure 11.

If these processes seem unduly involved and elaborate for the achievement of a simple result--like burning the house down in order to get roast pig--there are other more simple ways of deriving ornament from mathematics, for the truths of number find direct and perfect expression in the figures of geometry. The squaring of a number--the raising of it to its second power--finds graphic expression in the plane figure of the square; and the cubing of a number--the raising of it to its third power--in the solid figure of the cube. Now squares and cubes have been recognized from time immemorial as useful ornamental motifs. Other elementary geometrical figures, making concrete to the eye the truths of abstract number, may be dealt with by the designer in such a manner as to produce ornament the most varied and profuse. Moorish ceilings, Gothic window tracery, Grolier bindings, all indicate the richness of the field.

[Ill.u.s.tration: Figure 8.]

[Ill.u.s.tration: PLATE XII. IMAGINARY COMPOSITION. THE BALCONY]

[Ill.u.s.tration: Figure 9.]

Suppose, for example, that we attempt to deal decoratively which such simple figures as the three lowest Platonic solids--the tetrahedron, the hexahedron, and the octahedron. [Figure 12.] Their projection on a plane yields a rhythmical division of s.p.a.ce, because of their inherent symmetry. These projections would correspond to the network of lines seen in looking through a gla.s.s paperweight of the given shape, the lines being formed by the joining of the several faces. Figure 13 represents ornamental bands developed in this manner. The dodecahedron and icosahedron, having more faces, yield more intricate patterns, and there is no limit to the variety of interesting designs obtainable by these direct and simple means.

[Ill.u.s.tration: Figure 10.]

If the author has been successful thus far in his exposition, it should be sufficiently plain that from the inexhaustible well of mathematics fresh beauty may be drawn. But what of its significance?

Ornament must _mean something_; it must have some relation to the dominant ideation of the day; it must express the psychological mood.

What is the psychological mood? Ours is an age of transition; we live in a changing world. On the one hand we witness the breaking up of many an old thought crystal, on the other we feel the pressure of those forces which shall create the new. What is nature's first visible creative act? The formation of a geometrical crystal. The artist should take this hint, and organize geometry into a new ornamental mode; by so doing he will prove himself to be in relation to the _anima mundi_. It is only by the establishment of such a relation that new beauty comes to birth in the world.

[Ill.u.s.tration: Figure 11.]

Ornament in its primitive manifestations is geometrical rather than naturalistic. This is in a manner strange, that the abstract and metaphysical thing should precede the concrete and sensuous. It would be natural to suppose that man would first imitate the things which surround him, but the most cursory acquaintance with primitive art shows that he is much more apt to crudely geometrize. Now it is not necessary to a.s.sume that we are to revert to the conditions of savagery in order to believe that in this matter of a sound aesthetic we must begin where art has always begun--with number and geometry.

Nevertheless there is a subtly ironic view which one is justified in holding in regard to quite obvious aspects of American life, in the light of which that life appears to have rather more in common with savagery than with culture.

[Ill.u.s.tration: Figure 12.]

[Ill.u.s.tration: Figure 13.]

The submersion of scholarship by athletics in our colleges is a case in point, the contest of muscles exciting much more interest and enthusiasm than any contest of wits. We persist in the savage habit of devouring the corpses of slain animals long after the necessity for it is past, and some even murder innocent wild creatures, giving to their ferocity the name of sport. Our women bedeck themselves with furs and feathers, the fruit of mercenary and systematic slaughter; we perform orgiastic dances to the music of horns and drums and cymbals--in short, we have the savage psychology without its vital religious instinct and its sure decorative sense for color and form.

But this is of course true only of the surface and sunlit shadows of the great democratic tide. Its depths conceal every kind of subtlety and sophistication, high endeavour, and a response to beauty and wisdom of a sort far removed from the amoeba stage of development above sketched. Of this latter stage the simple figures of Euclidian plane and solid geometry--figures which any child can understand--are the appropriate symbols, but for that other more developed state of consciousness--less apparent but more important--these will not do.

Something more sophisticated and recondite must be sought for if we are to have an ornamental mode capable of expressing not only the simplicity but the complexity of present-day psychology. This need not be sought for outside the field of geometry, but within it, and by an extension of the methods already described. There is an altogether modern development of the science of mathematics: the geometry of four dimensions. This represents the emanc.i.p.ation of the mind from the tyranny of mere appearances; the turning of consciousness in a new direction. It has therefore a high symbolical significance as typifying that movement away from materialism which is so marked a phenomenon of the times.

Of course to those whose notion of the fourth dimension is akin to that of a friend of the author who described it as "a wagon-load of bung-holes," the idea of getting from it any practical advantage cannot seem anything but absurd. There is something about this form of words "the fourth dimension" which seems to produce a sort of mental-phobia in certain minds, rendering them incapable of perception or reason. Such people, because they cannot stick their cane into it contend that the fourth dimension has no mathematical or philosophical validity. As ignorance on this subject is very general, the following essay will be devoted to a consideration of the fourth dimension and its relation to a new ornamental mode.

[Ill.u.s.tration]

II

THE FOURTH DIMENSION

The subject of the fourth dimension is not an easy one to understand.