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ephemeral lines on water (2003)
digital print,

mathematics and art

(max bill, 1949)

by a mathematical approach to art it is hardly necessary to say i do not mean any fanciful ideas for turning out art by some ingenious system of ready reckoning with the aid of mathematical formulas. so far as composition is concerned every former school of art can be said to have had a more or less mathematical basis. there are also many trends in modern art which rely on the same sort of empirical calculations. these, together with the artist individual scale of value, are just part of the ordinary elementary principles of design for establishing the proper relationship between component volumes; that is to say for imparting harmony to the whole. yet it cannot be denied that the same methods have suffered considerable deterioration since the time when mathematics was the foundation of all forms of artistic expression and the covert link between cult and cosmos. nor have they seen any progressive development from the days of the ancient egyptians until quite recently, if we except the discovery of perspective during the renaissance. this is a system which, by means of pure calculation and artificial reconstruction, enables objects to be reproduced in what is called "true-to-life" facsimile by setting them in an illusory field of space. perspective certainly represented an entirely new aspect of reality to human consciousness, but one of its consequences was that the artist's primal image was debased into mere replica of his subject. therewith the decadence of painting, both as a symbolic art and an art of free construction, may be said to have begun.

impressionism, and still more cubism, brought painting and sculpture much closer to what were the original elements of each: painting as surface design in colors; sculpture as the shaping of bodies to be informed by space. it was probably kandinsky who gave the immediate impulse towards an entirely fresh conception of art. as early as 1912, in his book on the spiritual harmony in art, kandinsky had indicated the possibility of a new direction which, if followed to its logical conclusion, would lead to the substitution of a mathematical approach for improvisations of the artist's imagination. but as he found other ways of liberating painting from romantic and literary associations he did not adopt this particular line in his own work.

if we examine a picture of klee or one of brancusi's sculptures we shall soon discover that, though the "subject" may be an indeterminable echo of something or other in the actual world about us, it is an echo which has been transmuted in a form that is original in the sense of being elemental. kandinsky confronted us with objects and phenomena which have no existence in ordinary life, but which might well have meaning or be portents on some unknown planet; a planet where we should be quite unable to gauge their purpose or relevance. yet it was undoubtedly mondrian who went furthest in breaking away from everything that had hitherto been regarded as art. if the technique of structural design may seem to have inspired his rhythms the resemblance is fortuitous and one which was not present in his own intention or consciousness, although the specific content of his work is constricted with the utmost discipline, the horizontal-vertical emphasis represents a purely emotional factor in his composition. it is not for any whimsical reasons that he called his latest pictures "broadway boogie-woogies" and "victory boogie-woogies", but simply to stress their affinity with jazz rhythms.

if we can agree that mondrian realized the ultimate possibilities of painting in one direction - that is by his success in eliminating most of the remaining elements which are alien to it - two others still lie open to us: either we can return to traditionalism (in its wider sense), or else we can continue the quest for subjects with a content of a new and altogether different nature.

let me take this opportunity to explain why it is impossible for many artists to go back to the old type of subjects. in the vast field of pictorial and plastic expression there are a large number of trends and tendencies which have more or less originated in our own age. different people look at modern painting and sculpture with different eyes because what they severally recognize as significant of our age is necessarily various. clergymen have a different idea of art from scientists. peasants and factory-hands live under radically different conditions. there are inevitable variations in standards of living and levels of culture. similar differences can be found among artists. they, too, come from different walks of life, and their work reflects different emotional and intellectual undercurrents. there is another attitude to modern art which must not be overlooked as its now numerous followers can always be relied upon to take their stand against every disinterestedly progressive movement. i mean the much-boosted school which demands that, if art itself cannot perhaps solve social and political problems, these shall at least be made dramatically "actual" and suitably glorified through its medium. we have good reasons to be skeptical about any "political art" - regardless of whether it emanates from right or left; especially when, under the cloak of antagonism to the prevailing social order, its aim is to bring about a new, but in all essentials, almost identical structure of society - because this is not art at all but simply propaganda.

after this digression into the potential alternatives which may be said to have existed prior to somewhere about 1910, let me try to make clear why some of us were unable to rest content with what had then been achieved. that would have meant going on marking time over the same ground during the last forty years and painting in one or the other of those manners which may be called "à la klee", "à la kandinsky", "à la mondrian", or, what was more usual "à la picasso", "à la braque", and "à la matisse". a great many educated and intelligent artists are still wearing out their talents in ringing the changes on these modern masters. in fact this sort of painting has now become something in the nature of a substitute for the masters themselves, and "à la" pictures begin to rank as interesting variants of their originals. to acquiesce in this state of stagnation was impossible because we have no right to allow a halt to be called in any genuinely creative field of human activity.

we can safely assume that all the various forms of expression open to painting and sculpture at the present day are now sufficiently known, and that the techniques they postulate have been sufficiently demonstrated and clarified in the work of their respective pioneers (except perhaps for a very few which can be already anticipated, but which have not so far been realized). what, then, it may be asked, are the possibilities of future development? but there are two other important points which must be dealt with before that question can be answered: namely, whether the several idioms just referred to can claim general validity in the plastic arts; and whether there is reason to enlarge the existing limits of their content. careful study of those forms of expression has led me to the conclusion that all of them were the discoveries of individual artists, either born of their will to overcome particular problems or else expedients called forth by exceptional circumstances; and that therefore they cannot be considered universally applicable or appropriate. as regards content, most of the modern work which is often held to have been largely inspired by mathematical principles cannot, in point of fact, be identified with that entirely new orientation i have called the mathematical approach to art. and as this needs to be more nearly defined, i will now endeavor to elucidate it, and at the same time answer the question i have left in suspense.

i am convinced it is possible to evolve a new form of art in which the artist's work could be founded to quite a substantial degree on a mathematical line of approach to its content. this proposal has, of course, aroused the most vehement opposition. it is objected that art has nothing to do with mathematics; that mathematics, besides being by its very nature as dry as dust and as unemotional, is a branch of speculative thought and as such in direct antithesis to those emotive values inherent in aesthetics; and finally that anything approaching ratiocination is repugnant, indeed positively injurious to art, which is purely a matter of feeling. yet art plainly calls for both feeling and reasoning. in support of this assertion the example of johann sebastian bach may be credited; for bach employed mathematical formulas to fashion the raw material known to us as sound into the exquisite harmonies of his sublime fugues. and it is worth mentioning that, although mathematics had by then fallen into disuse for composition in both his own and the other arts, mathematical and theological books stood side by side on the shelves of his library.

it is mankind's ability to reason which makes it possible to coordinate emotional values in such a way that what we call art ensues. now in every picture the basis of its composition is geometry or in other words the means of determining the mutual relationship of its component parts either on plane or in space. thus, just as mathematics provides us with a primary method of cognition, and can therefore enable us to apprehend our physical surroundings, so, too, some of its basic elements will furnish us with laws to appraise the interactions of separate objects, or groups of objects, one to another. and again, since it is mathematics which lends significance to these relationships, it is only a natural step from having perceived them to desiring to portray them. this, in brief, is the genesis of a picture. pictorial representations of that kind have been known since antiquity, and, like those models at the musée poincaré in paris where conceptions of space have been embodied in plastic shapes or made manifest by colored diagrams, they undoubtedly, provoke an aesthetic reaction in the beholder. in the search for new formal idioms expressive of the technical sensibilities of our age these borderline examples had much the same order of importance as the "discovery" of native west african sculptures by the cubists, though they were equally inapt for direct assimilation into modern european art. the first result of their influence was the phase known as constructivism. this, together with the use of new materials such as engineering blueprints, aerial photographs, and the like, furnished the necessary incentive for further developments along mathematical lines. at about the same time mathematics itself had arrived at a stage of evolution on which the proof of many apparently logical deductions ceased to be demonstrable and theorems were presented that the imagination proved incapable of grasping. though mankind's power of reasoning had not reached the end of its tether, it was clearly beginning to require the assistance of some visualizing agency. aids of this kind can often be provided by the intervention of art.

as the artist has to forge his concept into unity his vision vouchsafes him a synthesis of what he sees which, though essential to his art, may not be necessarily mathematically accurate. this leads to the shifting or blurring of boundaries where clear lines of division would be supposed. hence abstract conceptions assume concrete and visible shape, and so become perceptible to our emotions. unknown fields of space, almost unimaginable hypotheses, are boldly bodied forth. we seem to be wandering through a firmament that has had no prior existence; and in the process of attuning ourself to its strangeness our sensibility is being actively prepared to anticipate still further and, as it were, as yet inconceivable expanses of the infinite.

it must not be supposed that an art based upon the principles of mathematics, such as i have just adumbrated, is in any sense the same thing as a plastic or pictorial interpretation of the latter. indeed, it employs virtually none of the resources implicit in the term "pure mathematics". the art in question can, perhaps, best be defined as the building up of significant patterns from the ever changing relations, rhythms and proportions of abstract forms, each one of which, having its own causality, is tantamount to a law in itself. as such, it presents some analogy to mathematics itself where every fresh advance had its immaculate conception in the brain of one or other of the great pioneers, this euclidian geometry no longer possesses more than a limited validity in modern science, and it has an equally restricted utility in modern art. the concept of a finite infinity offers yet another parallel. for this essential guide to the speculations of contemporary physicists has likewise become an essential factor in the thinking of contemporary artists. these, then, are the general lines on which art is daily creating new symbols: symbols that may have their source in antiquity but which meet the aesthetic-emotional needs of our time in a way hardly any other form of expression can hope to realize.

things having no apparent connection with mankind's daily needs - the mystery enveloping all mathematical problems; the inexplicability of space - space that can stagger us by beginning on one side and ending in a completely changed aspect on the other, which somehow manages to remain that selfsame side; the remoteness or nearness of infinity - infinity which may be found doubling back from the far horizon to present itself to us as immediately as at hand; limitations without boundaries; disjunctive and disparate multiplicities constituting coherent and unified entities; identical shapes rendered wholly diverse by the merest inflection; fields of attraction that fluctuate in strength; or again, the square in all its robust solidity; parallels that intersect; straight lines untroubled by any relativity and ellipses which form straight lines at every point of their curves - can yet be fraught with the greatest moments. for though these evocations might seem only the phantasmagorical figments of the artist's inward vision they are, notwithstanding, the projections of latent forces; forces that may be active or inert, in part revealed, inchoate or still unfathomed, which we are unconsciously at grips with every day of our lives; in fact that music of the spheres which underlies each man-made system and every law of nature it is our power to discern.

hence all such visionary elements help to furnish art with a fresh content. far from creating a new formalism, as is often erroneously asserted, what these can yield us is something far transcending face values since they not only embody form as beauty, but also form in which intuitions or ideas have taken visible substance. the primordial forces contained in those elements call for intimations of the occult controls which govern the cosmic structure; and these can be made to reflect a semblance of the universe as we have learned to picture it today: an image that is no mere transcript of this invisible world but a systemization of it ideographically conveyed to our senses.

it may, perhaps, be contented that the result of this would be to reduce art to a branch of metaphysical philosophy. but i see no likelihood of that for philosophy is speculative thought of a special kind which can only be made intelligible through the use of words. mental concepts are not as yet directly communicable to our apprehension without the medium of language; though they might ultimately become so by the medium of art. hence i assume that art could be made a unique vehicle for the direct transmission of ideas, because if these were expressed by pictures or plastically there would be no danger of their original meaning being perverted (as happens in literature, for instance, through printer's errors, or thank to the whim of some prominent executant in music) by whatever fallacious interpretations particular individuals chance to put on them. thus the more succinctly a train of thought was expounded and the more comprehensive the unity of its basic idea, the closer it would approximate to the prerequisites of the mathematical approach to art. so the nearer we can attain to the first cause or primal core of things by these means, the more universal will the scope of art become - more universal, that is, by being free to express itself directly and without ambivalence; and likewise forthright and immediate in its impact on our sensibility.

to which, no doubt, a further objection will be raised that this is no longer art; though it could equally well be maintained that this alone was art. such a structure would be like saying that euclid's was the only geometry, and that the new conception of geometry associated with the names of lobachevsky and riemann was not geometry at all. one claim would stand against the other and that would be that!

although this new ideology of art is focused on a spectral field of vision this is one where the mind can still find access. it is a field in which some degree of stability may be found, but in which, too, unknown quantities, undefinable factors will inevitably be encountered. in the ever-shifting frontier zones of this nebular realm new perspectives are continually opening up to invite the artist's creative analysis. the difference between the traditional conception of art and that just defined is much the same as exists between the laws of archimedes and those we owe einstein and other outstanding modern physicists. archimedes remains our authority in a good many contingencies though no longer in all of them. phidias, raphael, and seurat produced works of art that characterize their several epochs for us because each made full use of the means of expression as his own age afforded him. but since their days the orbit of human vision has widened and art has annexed fresh territories which were formerly denied to it. in one of these recently conquered domains the artist is now free to exploit the untapped resources of that vast new field of inspiration i have described with the means our age vouchsafes him and in a spirit proper to its genius. and despite the fact the basis of this mathematical approach to art is in reason, its dynamic content is able to launch us on astral flights which soar into unknown and still uncharted regions of the imagination.

(max bill, 1949)