Art and Mathematics in the Thought of El Lissitzky

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Art and Mathematics in the Thought of El Lissitzky: ABSTRACT Lissitzkysanalogies between His to artand mathematics pertain to two Relationship Suprematism aspectsof modernity-namely, non- objectivityand the negation of per- and Constructivism spectivalspace. According to Lis- sitzky,non-objectivity inart was analogousto themodern concept of number,and the structure of a Esther Levinger workof artcorresponded to mathe- maticalconcepts such as functions andsystems. These theories dem- onstrateLissitzky's divergence fromboth Suprematism and Rus- sianConstructivism. Furthermore, theanalogies elucidate the artist's insistenceon order, his notion of L issitzky's writings on art contain many refer- but always active in the same theart-game and the relationship ences to mathematics, connoting a tight relationship ambience: the culture of their betweenintuition and intellect in the between the two This time." He further creativeact. An analysis of pictorial disciplines. relationship pertains explained spaceby analogy with real and im- especially to pictorial form and space and involves some of that he was taking the analogies aginarynumbers and a comparison the central issues of modern art: the ideas of game, order in their "essential sense" [6]. of artwith set theoryhelp clarify and objectivity; the relative importance of intuition and My aim in the present paper thesystem of the Proun paintings intellect in the creative and the of indeter- is to andthe theoretical base of Lis- process; concept clarify Lissitzky'sinterpre- exhibition minateness. tation of this 'essential' rela- sitzky's spaces.They alsoexpand our understanding of These and other issues, such as 'becoming' in opposition tionship between mathematics Lissitzky'srelationship to Male- to 'being', as well as terms like 'systems' or 'dematerializa- and art. I will show that the vich'sSuprematism. Inconclusion, tion', occurred often in the discussions and writings of analogies Lissitzky drew be- theanalogies provide an alternative theoretical Russian artists and theoreticians. However, Lis- tween the two in the baseto non-objective avant-garde disciplines art,different from the other 'isms' sitzky'suse of these terms was idiosyncratic in that it involved 1920s reveal an alternative of artin the 1920s. a mathematical sense. He elaborated a consistent theory of theoretical base for non-objec- art by analogy with mathematics [1 ]. tive art-a theory that differs Underlying his analogies was his belief that the modern from both Suprematism and era had abolished the barriers between the different spheres Russian Constructivism [7]. of knowledge and activity,for example, between technology, art and physics. In support of this thesis, Lissitzky evoked Minkowski's space-time continuum, the theoretical inter- changeability of the dimensions of space and time [2]. In using the Communist Revolution as a social example, Lis- sitzky observed that it had discredited old concepts that had set up barriers in society: the notions of classes, nations, patriotism and imperialism. In this vein, he argued that towns would be rebuilt in such a way as to abolish the sepa- ration between their different elements, since houses, streets, squares, bridges and the like were now linked by "underground metro, underground monorail, electricity trans- mitted under the ground and above the ground" [3]. Similarly, Lissitzky rejected as invalid the contradiction between spirit and matter, or 'soul and body', and hence ob- jected to the division between an artistic and a scientific understanding of space. 'The discoveries of new spatial per- ceptions," he held, "went hand in hand in painting and in mechanics" [4]. Aware, though, of the dangers of superfi- cial comparisons, Lissitzkystated that "the parallels between A.[rt] and mathematics must be drawn very carefully, for every time they overlap, it is fatal for A. [rt]" [5]. Thus, at the beginning of his lecture on the Prouns, given at the Moscow Institute of Artistic Culture (INKhUK), Lissitzky declared: "Wewill examine the movements of mathematics and art as 1. Proun (Untitled), and watercolor on two sets of not in Fig. gouache paper, graphs always advancing parallel planes 38.8 x 40.5 cm, ca. 1920. (Indiana University Art Museum, Bloomington, Jane and Roger Wolcott Memorial. Photo: Michael and Kevin The of a Proun Esther Levinger (university lecturer), Department of Art History, University of Haifa, Cavanagh Montague.) reading painting Mount Carmel-Haifa 31 999, Israel. varies according to shifting attentions and alternating analyses of Received 19 March 1987. the relationship between forms and colors.

? 1989 ISAST Pergamon Press pic. Printedin Great Britain. 0024-094X/89$3.00+0.00 LEONARDO, Vol. 22, No. 2, pp. 227-236, 1989 227 Malevich, therefore, was the first artist -;:~ :" 2. I Fig. Studyfor to start in a completely new direction. n e iims th oProunG 7,pencil Moreover, claimed that i ado and watercolor, Lissitzky plex forms were inventions ,i 78.4 x 62.7 cm, geometric pure of the mind to the e11 n e 1922. (Stedelijk [15], comparable van Abbemu- abstract terms of functional equations seum, Eindho- and the notations x, y, z, which imply The theme ven) a universally valid system of relation- of change and ships [16]. In the same way, the artist multiple read- had to find a system of ings pervades all relationships of Lissitzky's that would be valid for all artists. This, works. Proun G Lissitzky argued, happened in Supre- 7, for instance, matism: e c the l l ne E represents moa~terial idea of the 'artist The moment the square and the cir- pn cle are dissected and distributed over engineer', a the flat surface ... a relationship is human configura- formed between the individual parts. tion accompa- ... The result is not a affair nied personal by geometri- concerning one individual artist, but of j cal instruments, plexca a systemof universal validity [empha- which Lissitzky sis added] [17]. repeated in other paintings Thus art, like mathematics, consti- with either tuted a system of relationships. -ur . 1913 minor or major In his lecture on the Prouns, Lis- modifications. sitzky used the analogy of mathematical systems to define Suprematism. He between modern times, on the other hand, a distinguished proportions- THEORY OF that is, an of ratios between line is considered an unlimited com- equality two quantities-and the concept of NON-OBJECTIVE ART plex of points, and space is conceived functions in set theory. The latter has The term connotes as abstract: "no point in it can be seen 'non-objective' to do with the idea of dependence be- two of or measured, it is merely a relative interrelated characteristics tween any two sets of elements, inde- center" Hence, in one modernism: the work of art as an au- [11]. antiquity, of as the artist de- In pendent quantities; non-referential knew only what one saw and felt. tonomous, object [8] fined them: 'The ratio of quantities is modern times, by contrast, abstract and the specific 'non-objective' forms DEPENDENCE is the in PROPORTION; that make a or a notions prevail; analytic geometry, up painting sculpture. essence of function" (Lissitzky's em- believed that both character- for example, points in space can be Lissitzky phasis) [18]. In 1921, when Lissitzky istics well to modern identified by sets of numbers, and applied equally developed his theory of art, he felt that space can be n-dimensional. mathematical thought. He observed a funda- to the decisive Suprematism epitomized that mathematics was the "purest According Lissitzky, transformations in art were product of man's creativity: a creativ- analogous to the new of number-that ity which does not repeat (repro- concept is, art became divorced from duce), but creates (produces)" [9], totally and that it was unconcerned with con- material phenomena and free of the This fundamental crete magnitudes and quantities but physical object. occurred, in Lissitzky's view, dealt with the relationship between change with Malevich's Suprematist Black AKTIVISTfFOLYOIRIfT abstract concepts. As he described it: Square (ca. 1913). Prior to Suprema- The number in was antiquity always tism, artists depicted objects from concrete,only concrete; the number of nature, and all new movements-e.g. modern times is abstract,non-objective. For a Greek '3' always meant 3 Cubism, Futurism and Expression- columns, 3 sheep, 3 ribs; beyond the ism-were nothing more than at- object there was no number. In the tempts to endow the object with a new new mathematics of x, y, z there is no life [12]. The canvas, on definition of the are Suprematist quantity. They the other hand, existed signs of the connection between an indepen- infinite number of possible positions dently of anything outside it [13]. In within one and the same character; terms of the analogy with mathemat- taken as a a num- whole, they equal ics, "Suprematism transposed paint- ber.... x, y, z are numbers only in- ing from the condition of the ancient asmuch as the signs + or = are [10]. objective and concrete number to that The same occurred in the of the modern number changes abstracted Fig. 3. Cover for MA, No. 8 (August of line and In ancient concepts space. from the object. This number occu- 1922). This cover for the Hungarian Greece, Lissitzky explained, the line pies its own, independent place in avant-garde art journal MA presents one was considered the measurable variation on the theme of the 'artist edge nature alongside all objects" [14]. which started in of a body, and space was defined by the engineer', Lissitzky early 1922 with Proun G 7. object and its delimiting planes. In Studyfor

228 Levinger,Art and Mathematicsin the Thoughtof El Lissitzky mental change in the concept of rela- 4. Pro tionships, for this art form was built, Fig.- not on the idea of but on proportions, 5 from the notion of plate dependence: ErsteKeslner- From the canvas dis- mappe (First Suprematism . carded all depiction and represen- Kestner port- tationof and colors.It left be- folio), litho- .- : objects ';' hind pure color within pure form. graph, printed in Fromthese elements it beganto com- color, 60.4 x 44.2 pose whole classes,groups and equa- tionsof formalpossibilities according lection, The Mu- to their functionalinterdependence seum of Modem [19]. Art, NewYork) Lissitzky wished The definition of Suprematism in to incorporate-Sm' ': terms of functional dependencies re- time into his ' llSi ferred to the basic difference that the work. He initially : First Working Group of Constructiv- attempted to do .:ii!:^ ists had established between composi- so by trying to tion and construction [20]. Lissitzky capture different moments of agreed with the Group's definitions, movement " i'.ist and using mathematical terminology, through the repe- i ii he explained that tition of lines ratioscan be increasedor decreased. and forms in G F such a that- But functions can only be trans- way formed. Therein lies the essential these seem to their - change A.';,; , , . contrastbetween the old and the new posi- ',. : . * 'w"T . ;sW ;li:^^:^-; /- *."* ^*'^.: world, between old and new plastic tion in space. forms.Any proportion presupposes a constancy of each element (the CLASSICALORDERS), but any transformationpresupposes a varia- bility(SUPREMATISM). An old work of art can be increasedor decreased, whereasa modernwork of art has to intensity, nothing in the work would rhythm in art was an element of 'life'; be transformed[Lissitzky's empha- be changed" [23]. The Group's com- it was that which "Bergson, Rikkert sis] [21]. position verdict against Malevich and Spengler name 'becoming' in op- If one variable in a ratio changes, stood in blatant contradiction to Lis- position to the frozen.... Rhythm is the other variables have to be changed sitzky, who argued that the term ap- always movement, elan, surge" [28]. accordingly, but the ratio will remain plied equally, even primarily, to Supre- This nature-derived significance of be- the same. In artistic terms, following matism. coming is also present in Lissitzky's the Constructivists and Lissitzky, ratio This difference between Lissitzky writing; however, for him the similar- denotes composition, that is, if parts and the Constructivists was more than ity to mathematics was equally impor- of a composition are removed or col- a disagreement over terminology. For tant. In his opinion, the work of art was ors changed, the rest of the composi- Lissitzky, a Suprematist painting was, analogous to mathematical systems, tion would require some rearrange- by definition, constructive art [24]. and like them, it evolved during its ex- ment, but in essence it remains the Thus he repudiated the final decision ecution [29]. same. If one element of a function is taken by the First Working Group of Furthermore, even when the work changed, the whole function has to be Constructivists-namely, that a utili- had been completed, the process con- transformed. Hence functions define tarian element had to be immediately tinued, since the spatial relations be- construction; the term construction present in the idea of construction tween the forms and colors were came to denote the functional neces- [25]. Lissitzky distinguished between changeable, depending on the specta- sity of each part as in any mechanical the necessity to create new forms and tor's choice. Indeed, the reading of a construction. "In adding new ele- the question of direct utility. For him, Proun painting varies according to ments and transposing them", Lis- a new artistic form generated utilitar- shifting attentions and alternating sitzky said, "we transformed the can- ian forms, which, consequently, en- analyses of the relationship between vas" [22]. riched and modified art and caused forms and colors and their relation- The First Working Group of Con- further developments in both art and ship to the picture plane. For instance, structivists devoted one of its early ses- production [26]. in Proun (Untitled) (Fig. 1), a square is sions to a discussion of two paintings Another concept involved in the painted on a form that, because of the by Malevich, one Cubist and the other theory of systems was that of change. black contour shown in perspective on Suprematist. Their consensus was that Ernst Cassirer had noted that the its lower right-hand side, is either a Malevich's Suprematist canvas fol- mathematician was concerned with circle or a cylinder. On the upper edge lowed the laws of composition. Ac- the idea of 'becoming' rather than the of the 'circle', to the left, Lissitzky cording to Rodchenko, for example, idea of 'being' [27]. The opposition of placed a stereometric open cube-a "Malevichmakes no color resolutions. becoming to being was a recurring recurring theme in the Prouns. Its His form is strong, but if you were to theme in Russian Constructivist dis- black face on the left-hand side may be paint it a different color with the same cussions. Thus, for N. Tarabukin, read as either advancing or receding.

Levinger, Art and Mathematics in the Thought of El Lissitzky 229 Thus, we see the outer envelope, the Moder mathematics was a game, lutionize the world, for when its aim two front faces of a receding cube or for, like games, it was an autonomous, and significance were fully grasped the two interior faces of an open and rule-governed activity [33]. Indeed, and integrated, "we shall give a new advancing cube. In fact, the theme of the modern formalist mathematicians face to this globe, we shall reshape it change and multiple readings per- of Lissitzky's time considered mathe- so thoroughly that the sun will no vades all of Lissitzky's work, since matics mere play with symbols. For longer recognize its satellite". Art, many Prouns are variations on a these mathematicians, mathematics then, was a supreme vehicle of world theme, and the same configurations had become an endless game of chess transformation, superseding even the may appear with either minor or that had no meaning beyond the rules social revolution; in the final analysis, major modifications. One example is of the game [34]. The aim of moder the future would be shaped, not by Study for Proun G 7 (Fig. 2), which was mathematics, including geometry, lay Communism, but by Suprematism: modified for the cover of the inside the actual mathematical Hungar- only In the furtherstages of development, ianjournal MA (Fig. 3) and then again operations. The real object of the it is communismwhich will have to re- slightly for Proun 43 [30]. mathematicians' investigations was main behind because suprema- the relational structure as such, the tism . . . will attract everyone away various of that from the domination of work... species dependency After the old testamentthere came GAME AND ORDER could subsist between numbers and the new-after the new the commu- forms The between modern art and [35]. nist-and afterthe communistthere analogy As saw had be- follows the testamentof modern mathematics induced Lis- Lissitzky it, art, too, finally supre- come a formal of matism[37]. to consider art a [31]: language signs-a sitzky game and game between forms and colors; The concept of art as game and its Archimedes would have regarded the relationships between the signs analogy with mathematics elucidate modern mathematicsas a clever,but were as rule-bound as those between curiousGAME (because its aim is not Lissitzky'sinsistence on order and ob- z in formal mathematics Un- an end result like three buns, forty- x, y, [36]. jectivity. His 'call to order', like that of like art- five kopecks,etc.... but the actual formal mathematics, however, other artists of the avant-garde, un- combination and con- operation, games had both a social and an ideo- doubtedly was motivated in part by the structionof dependences which we one could find with Riemann and Ein- logical significance: change example provided by moder technol- Gauss, the world art. The stein).... The resultof the Suprema- through Suprema- ogy and machines. But it was further tist was tist 'game' was a canvaswhich bore revolution, accordingly, equal sustained by his adherence to the idea withinit a kind of symbol[Lissitzky's to the Communist Revolution. It was, of necessary connections between art [32]. emphasis] in fact, Suprematism that would revo- and mathematics [38]. The very comparison of art with abstract mathematics and the definition of art as a system 5. Proun metal oil and on 98.6 x 98.6 1924. Fig. (8 Positions), foil, gouache canvas, cm, of relationships imply an idea of or- (The National of Canada,Ottawa) Another to incorporatetime into a workof Gallery way der, of which was surely con- artwas to makethe canvasrotatable, that is, equallylegible from anyof its four sides. Lissitzky scious [39]. The foregoing analysis of Lissitzky's theory of art, as paralleling changes in mathematics, demonstrates that he was not a Suprematist. Like Malevich, Lissitzky believed that forms and colors were autonomous creations of pure thought, totally unrelated to ex- terior objects. The two artists, though, differed completely on the conceptual foundation of the rejection of natural objects. Malevich refused representation on the basis of his idealist, transcendental philosophy, according to which "in reality form does not exist [and] nature ... does not reveal itself in things" [40]. Lissitzky,on the other hand, held that objects did exist, but in a sphere that was of no interest to the artist,just as it was of no interest to the moder mathematician. Both art and mathematics were concerned with abstract relationships governing the universe independently of concrete objects and magnitudes. Moreover, the very emphasis on relationships contradicted Malevich's philosophy of art, in which "each form is free and individual. Each form is a

230 Levinger,Art and Mathematics in the Thought of El Lissitzky world" [41]. In Lissitzky's Prouns, on geometry, although imagination and Non-Euclidean geometry occupied the contrary, relationships were im- the imaginary were necessarily an inte- an important place in Lissitzky's portant; the meaning of a Proun de- gral part of this construct. Moreover, thought in different contexts. It cor- pended on the relationships between 'A.' and 'F.' allude to the abstract, for- roborated the notion of non-objective the forms. mal nature of relationships between art, since it declared mathematics to the elements of the painting, compar- be a free creation of the mind, inde- able to x, y, z in mathematical func- pendent of our capacity to visualize INTUITION, tions. The square root of 1 and the [52]. It also negated the idea of 'abso- INTETLL.ECTAND square root of-1 referred to Lissitzky's lute truth', or 'undeniable truth', theory of space, developed further in through the coexistence of several INDETERMINATENESS his essay; therefore, as will be seen non-Euclidean geometries [53], thus no was intended old beliefs and that Another problem elucidated by the later, mystification nullifying proving and none should be inferred. humans could doubt [54]. comparison of art with mathematics is everything The that art, like math- re- Lissitzky'ssupposed Romanticism and recognition Discussing Lobachevsky, Lissitzky combined with marked that the mathematician "ex- vacillation between rationality and ematics, logic imagination reflects belief in the the absoluteness of Euclidean mysticism [42]. Alan C. Birnholz Lissitzky's ploded of intui- and added that claimed that, despite Lissitzky'sration- necessary interdependence space", Lobachevsky, tion and intellect in the creative Gauss and Riemann had that ality, the artist remained an idealist pro- proved cess. affirmed the im- Euclidean was one in- and a romantic. To prove his point, Indeed, Lissitzky geometry only role intuition in the stance other cor- Birnholz quoted from the essay "A. portant played by among many equally first of creation: the into rect [55]. to the and Pangeometry", in which Lissitzky phase "path possibilities Referring a from of he observed: "Mod- wrote: "A.[rt] is an invention of our the future ... is path leading theory relativity, creative intuition" [47]. He linked ern science ... has ab- spirit, a complex whole, combining destroyed many with the moment all con- solutes. The absolute of all measures the rational with the imaginary, the creativity scious control was abandoned: "There and standards has been physical with the mathematical, i- destroyed.... one out-the artist had The new mathematical has with /1l" [43]. Birnholz concluded was only way thought to himself into the in the assertions" [56]. that for Lissitzky art had an "inexpli- fling abyss destroyed many belief that when he reached the bot- Both ideas, the of old cable aura about it"; that by the use of suspension tom he would not be but would beliefs and the denial of absolutes- 'A.' for Art and 'F.' for Form, Lissitzky dead, wished to connote that art was "some- be newly born" [48]. A painting thus and consequently the inherent possi- out of the artist "as of doubt-were of im- thing mysterious, undefinable" [44]. grows organically bility primary as a flower from the soil" At the to Doubt as such In fact, the title of Lissitzky's essay [49]. portance Lissitzky. same art reflected our became the of the Prouns. It is referred to Lobachevsky's treatise on time, capacity subject as human to create, that is, to reflected in the re- non-Euclidean geometry, Pangeometry, beings ambiguous spatial order the of our sen- that Lissitzky established published in Kazan in 1855 [45]. Lo- voluntarily range lationships sations. the formal elements of between the forms in his bachevsky believed that the true math- Thus, paintings, a had to be con- such as fields ematics of space was Euclidean and painting organized overlapping transparent for without control over the of color, that his own non-Euclidean geometry sciously, equivocal tangencies, appar- of forms and colors, the ent and with fore-, was a totally imaginary construct of the arrangement symmetries play Lis- and All these mind; nevertheless, he considered result was a "rubbish heap" [50]. mid-, back-planes [57]. the Russian artists serve to arouse incertitude. Thus the 'pangeometry' a rational, even rigor- sitzkypraised young who order as the base of all between forms are inde- ous, theory of parallels [46]. Thus Lis- recognized relationships art, and he the clear terminate, and the cannot sitzky, through this reference to Lo- preferred geom- spectator of Laszl6 to other exhaust all the of a bachevsky, apparently was suggesting etry Moholy-Nagy possible readings which he Proun. that art was a rigorous, logical con- non-objective painting, termed [51]. The 'essential sense' of mathema- struction just like Lobachevsky's pan- 'jellyfish-like" tics to which Lissitzky referred in his lecture, and which initiated his study of correspondences between art and Table1. Lissitzky'sequations refer to the generaltheory of relativityand, mathematics, is closely allied with the throughthe theory,to the artist'sconcept of 'imaginary'space, thatis, a space denial of absolutes. Both mathematics created the rotationand vibration of a Inthe 1921 INKhUK through rapid body. and art were "symbolsof the new con- lecture,the equationsreferred more directly to the possibilityprovided by the of the world" [58] in that they axolnometricmode of representationof rotatinga Prounpainting [88]. cept abandoned old formulas and stated that humans forever had to consider new and new solutions. Proun possibilities Space The artist's and the mathematician's revolutionary act concerned not only their domains but also life FORMOUTSIDE SPACE = 0 separate and revolution. If it was possible to re- FORMOUTSIDE MATERIAL = 0 ject Euclid and Newton, if it was possible to the natural concrete ob- THERELATIONSHIP OFFORM TO MATERIAL ISTHE RELATIONSHIP OFMASS reject and Albertian it was, TOFORM ject perspective, perhaps, also possible to install a new social order. The way to the new social

Levinger,Art and Mathematics in the Thought of El Lissitzky 231 history of art: the planimetric, the per- BlackSquare he had arrived at "the zero spectival and the 'irrational'. Artistic of form" [63]. In Lissitzky's interpre- space was at first planimetric; the 'set tation, the Suprematist Black Square of art' was later enriched by perspec- functioned like the zero on the num- tive. Thus the set of art became com- ber line, since it marked a turning posed of two members: planimetric point between negative and positive. and perspectival space [60]. Lissitzky More precisely, it separated the old compared each of these artistic sys- "painterly culture" (negative) from tems to a set of real numbers. In plani- the new "material culture" (positive), metric space, the relation between that is, it presented the possibility of forms was organized according to the creating both new artistic and every- arithmetic progression of ordinal day objects [64]. numbers (1, 2, 3, .. .). In The second in Fig. 6. Reconstructionof Lissitzky's perspectival stage Suprematism SecondExhibition Room in Hanover space, the relation was arranged ac- was the disintegration of the Black Provinzialmuseum,1927-1928. (Nieder- cording to geometric progression (1, Squareinto colorful forms, the organi- sachsischeLandesgalerie, Hanover, 1968. 2, 4, 8, 16, 32, .. .), that is to say, an zation of which required a new con- Photo:Foto-Hoerner, Hanover.) Left- equality of ratios between two pairs of cept of space. Suprematism arrived at hand wall, seen from the entrance. Placed (1:2 = 2: 4), or the white which is The to the walls were steel quantities propor- plane, infinity. perpendicular infinite had "bro- 3 cm wide and at 2-cm inter- tions-hence, composition [61]. Suprematist space strips spaced What called 'irrational' ar- ken the blue of vals. The strips were painted black on one Lissitzky through lampshade side, white on the other, with the wall it- tistic space corresponded to the con- the firmament" [65]. This conclusion self painted gray. The black side of the cept of irrational numbers. The set of accorded with Lissitzky's comparison strips faced the door; seen thus from the positive integers, called natural num- of art to set theory. the wall looked all black. entrance, bers, was first extended into a class of Georg Cantor's set theory pre- integers by the adjuncts of zero, nega- sented a method of dealing with in- tive integers (-1, -2, -3, . . .) and finites, namely, the paradox of a finite rational numbers, that is, fractions of dimension composed of an infinite integers. The next step was the inclu- number of points devoid of any di- sion of irrationals, or numbers that mension. It was a problem that pre- cannot be expressed as the quotient of occupied mathematicians, as well as two integers (such as 2), to obtain the philosophers, from the time of the an- set of real numbers, which is made up cient Greeks (Zeno's paradox) [66]. precisely of rational and irrational Cantor solved the problem of the in- numbers [62]. finite set-the set of all integers, the According to Lissitzky,Suprematist set of points on a line-by pairing, i.e. space was 'irrational' in the sense of by a one-to-one correspondence be- Fig. 7. Reconstruction of Lissitzkys Second Exhibition Room in Hanover irrational numbers-that is, the dis- tween the objects of two sets. In set Provinzialmuseum, 1927-1928. (Nieder- tance between forms could not be rep- theory, then, the infinite can be de- sachsische I Pndesgalerie, Hanover, 1968. resented as a determinate ratio of two fined mathematically [67]. According Photo: Foto-Hoerner, Hanover.) Left- integers. This meant that Suprematist to Lissitzky, Suprematism, like Can- hand wall, seen from the front. The wall space was the first extension of the 'set tor's set theory, symbolically trans- itself was therefore, from painted gray; of art' after Renaissance perspective. gressed the the front, the wall seemed an almost theological injunction However, since the extension of the the of the uniform gray. against representation set of numbers to include irrational infinite. numbers did not eliminate naturals, The foregoing analysis clarifies Lis- order-which, for Lissitzky, was the integers and rationals, Lissitzky em- sitzky's special interpretation of Su- ultimate aim-passed through art. ployed all three spatial systems-plan- prematism. He stripped Malevich's "Life", he wrote, "is now building a imetric, perspectival and irrational- philosophy of art from its transcen- new, reinforced concrete, Communist sometimes in the same painting. dental idealism and presented it as a foundation for the peoples of the In addition, Lissitzkycompared Su- logical system in accord with modern earth. Through the Proun we will prematist space to the extended line mathematics-hence, as acceptable to come to build upon this universal of real numbers, which comprises in- the international community of avant- foundation for a single world city-for tegers, rational and irrational num- garde artists in Germany and in cen- all the people of the earthly globe" bers and also infinites. In this rep- tral and eastern Europe [68]. [59]. resentation, the Suprematist picture plane functioned like the zero on the number line: like numbers, Suprema- IMAGINARY SPACE THEORY OF SPACE tist space could be projected not only Movement played an important role forward from the plane but also back- Lissitzky developed his definition of in Lissitzky'sart, as it did in the art of ward into depth. In other words, it pictorial space through analogy with other avant-garde artists. Speed, dy- possessed an infinite extensibility into set theory, postulating that art was a namics and energy were associated the foreground and the background. 'universal set' and that each spatial sys- with technological progress and thus On the other hand, the Black Square tem constituted a subset. He distin- characterized the modern era; by also to zero, in accord- three successive in the corresponded movement, artists believed guished systems ance with Malevich's idea that with the depicting

232 Leinger, Art and Mathematics in the Thought of El Lissitzky that they were participating in the new no existence without movement. plained the theory with his own group world of technology. For Lissitzky, Space unfolded in time and did not of equations (see Table 1). however, as for Tarabukin, the inclu- exist without it, "like the notes of mu- In Lissitzky'simaginary space, as in sion of time-movement in art demon- sic" [77]. However, he considered the the general theory of relativity, space strated its connection with modern film insufficient, since "itwas only a de- was not an absolute; it was generated mathematics [69]. 'The mathematics materialized projection of the plane" by matter, without which it would not of the ancients is stereometry. It ap- and exploited only the fact "that dis- have existed. The image created prehends objects as quantities,outside connected movements, separated by space, and space in its turn created the time. It's a kind of mathematical periods shorter than 1/30, seem to illusory image. Lissitzky himself had statics" (Lissitzky's emphasis). Mod- create the illusion of continuous never constructed a kinetic sculpture, ern mathematics, on the other hand, movement" [78]. Instead, Lissitzky but he was familiar with the works of was dynamic; it took time into prime suggested rapid rotation and body both Naum Gabo and Moholy-Nagy. It consideration. In Einstein's physics, vibration: is possible that he was also familiar space and time were inseparable. with Marcel When it [this body] is motionlessit Duchamp's experiments "Modern science", Lissitzky said, "has formsa unitin our threedimensional with the perception of the passage seen that the world lives in time and it space,and when set in motion it gen- from one dimension to the other, as has introduced time as the fourth co- eratesan entirelynew object, ... a new in RotaryGlass Plates (PrecisionOptics) ordinate. It has become and expressionof spacewhich is there as dynamic as the movement lasts and is [80]. has absolutes" [70]. long realized the from destroyed many thereforeimaginary [79]. Lissitzky passage Accordingly, Lissitzky wished to in- the real number line to imaginary The thus created was corporate time into his work. At first, space imagi- space in his Exhibition Rooms. In the since it was and he did so by capturing different mo- nary, ephemeral Berlin Proun Room (1922), the space immaterial, and abstract, since it coin- ments of movement through the repe- was still 'linear', although time-move- cided with the mathematical defi- tition of lines and forms (Fig. 4) and ment was an integral part of both the nition of a set of through a rotatable canvas, that is, space: points together conception and the realization of the with a set of relations. At the making the canvas equally legible same room. Visitors were induced to follow time, referred to time from any side, as in Proun (8 Positions) imaginary space a predetermined linear itinerary, as as the fourth dimension-the three- (Fig. 5) [71]. However, wanting to was stipulated by Lissitzky himself: make real time and real movement a dimensional concrete object together "The first form, which 'leads in' some- with the fourth time. component part of art, Lissitzky dimension, one coming from the large hall, is related therefore added a fourth spatial sys- Furthermore, imaginary space placed diagonally and 'leads' him to tem to the three described above and to the concept of matter and space in the broad horizontals of the front wall named it 'imaginary'. The mathemati- the general theory of relativity. In his and from there to the third wall with lecture on the cal analogy was imaginary numbers- Prouns, Lissitzky ex- the verticals" [81]. No such itineraries like the square root of-1, designated by i for 'imaginary' (-1 = i) [72]. Since the square of any number, posi- Fig. 8. Recon- structionof tive or negative, must, by definition, be Lissitzky's positive, there is no possible square Second Exhibi- ~ root for a negative number on the real tion Room in line [73]. Thus, by terming the new Hanover Provin- space 'imaginary', Lissitzky reinstated w7ilmlseum, his former analogy of planimetric, per- 1927-1928. spectival and irrational with (Niedersach- spaces sische Irndes- real numbers. It was as though the galerie, Hanover, three had completed the real line of 1968. Photo: the 'set of art', and hence any new Foto-Hoerner, space had to be termed 'imaginary'. Hanover.) Left- Furthermore, the exceptionality of hand wall, seen from the win- the number i is that it is not greater i dow. The white than, less than or equal to zero and side of the strips cannot be represented on the real line faced the source intended to [74]. Similarly, Lissitzky of light; there- abandon 'irrational' space, which was fore, from the still on the real line, and to move on window, the wall looked to abstract space [75]. The imaginary all white. number was also involved in time as the fourth dimension; 4-1 was used by Minkowski in his formulation of the space-time continuum to make the time dimension imaginary [76]. The first realization of abstract 'imaginary' space was, for Lissitzky,Vick- ing Eggeling's filmic space, in which the forms projected on the screen had

Levinger,Art and Mathematics in the Thought of El Lissitzky 233 space in art distanced Lissitzky from Russian Constructivism, and also from Malevich. On the one hand, Lissitzky proposed a dematerialized object and an abstract, dematerialized space in place of the Constructivist culture of materials; on the other hand, imaginary space differed from Malevich's idealist transcendental space, since Lissitzky never wished to reveal to the viewer a transcendent reality, unless one considers Einsteinian four-dimensional space unreal [83]. For Lissitzky, dematerialized imaginary space was comparable to modern mathematics and physics. Just as modern mathematics transcended matter, monumental art, hewn in granite and cast in bronze, was replaced by a transitory art, an art deprived of its corporeality. He had arrived, in his own words, at "an a-material materiality" [84]. That 9. Reconstruction of Second Exhibition Room in Hanover Provinzial- Fig. Lissitzky's is, tran- 1927-1928. 1968. Photo: Foto- although imaginary space museum, (Niedersachsische Landesgalerie, Hanover, scended material Lis- Hoerner, Hanover.) Right-hand wall, seen from the window. Next to the door, Lissitzky experience, ultimate interest was real placed a vertical frame containing a plate that could be moved to cover one of three paint- sitzky's space ings inside the frame, so that only two paintings were visible at any time. Another such and real material life. He felt that the vertical frame was installed on the left-hand wall (see Figs 7 and 8). development of abstraction could be detected in modern technology, for instance, in the new communication systems [85]; and he foresaw the 'dematerialization' of books as well: "The printed sheet, the infinity of the book, must be transcended. THE ELECTRO LIBRARY"(Lissitzky's emphasis) [86]. The analogy between imaginary numbers and abstract space, on the one hand, and Lissitzky's project for imaginary space, on the other hand, elucidates the theoretical base of his exhibition spaces and provides a theoretical explanation of the kinetic art of his time. Furthermore, it adds yet another facet to his relationship with Malevich. Imaginary space, far from repudiating Suprematism, was indeed its logical continuation. Art, for Lissitzky,was not limited to the representation of concrete, ob- servable phenomena; rather, it was- like the modern notion of numbers, Fig. 10. Reconstruction of Lissitzky's Second Exhibition Room in Hanover Provinzial- and construc- 1927-1928. Hanover, 1968. Photo: Foto- concepts space-a pure museum, (Niedersachsische Landesgalerie, tion of the mind. Aware that his Hoerner, Hanover.) Right-hand wall, seen from the entrance. On the right-hand wall, theory of art with its to mathematics Lissitzky placed a horizontally sliding plate that covered either of the two paintings analogy placed inside the frame. might be misinterpreted, Lissitzkyex- plained that he was neither a meta- were given in Dresden (1926) or in to gray to white (Figs 6, 7 and 8). physician nor a pseudo-mathemati- Hanover (1927); on the contrary, in Moreover, the sliding plates inside the cian. His concept of moder art, he these spaces, visitors were expected to horizontal and vertical frames encour- argued, was indissociable from the move about freely, retracing their aged the spectator to move the plates moder understanding of the world steps or standing in front of a wall and to and fro and thus to interrupt the and consistentwith new developments moving their eyes, head and body to straight continuity of the wall line in science and technology. It was, the left and to the right. It was only by (Figs 9 and 10). In so doing, the spec- moreover, a battle against death, for moving that the spectator could ex- tator was describing an imaginary stopping at concrete numbers and de- perience and appreciate the changes space [82]. picting objects meant death: the arrest in the colors of the wall-from black The very notion of an imaginary

234 Levinger,Art and Mathematics in the Thought of El Lissitzky sian Constructivism of the march of human thought and isvast; however, the following Suprematism, with reference to other aspects of two works are particularly helpful for assessing the his see M. Perniola, L'alienation human creativity [87]. thought, artistique essential differences between these movements (Paris: Union generale d'bditions, collection Although the formal vocabulary of and Lissitzky: E. Martineau, Malevitch et la philoso- 10/18, 1977) pp. 294-295; and A. Turowski, and Existe-t-il un art de de 7Est? Lissitzky's art was related to Supre- phie (Lausanne: L'Age d'Homme, 1977); 'lEurope I Utopie et ideologie Lodder [6]. (Paris: Les editions de la Villette, 1986) pp. 117- matism and Constructivism, his theory 119. 8. "The meaning of the new art is not to repre- of art, informed by his knowledge of sent, but to form something independent of any 25. On the utilitarian idea that emerged from the the exact sciences, distinguished him conditioning factor" (El Lissitzky, "From a Let- composition and construction debates at IN- in re- see Lodder from both these schools. Lissitzky's inter", Lissitzky-Kuppers [3] p. 358). Lissitzky KhUK, [6] pp. 94-98. frained from using the term abstract, for "it is difference to 26. "New Russian Art" [17] p. 340. metaphysical problems, impossible to make a direct equation between his concept of the necessary interde- mathematical, extra-material abstractions and 27. E. Cassirer, Substance andFunction &Einstein's that which in has the unfor- York: Dover pendence of the spiritual and the painting acquired Theory of Relativity (New Publications, tunate name of'abstract'." Lissitzky [2] p. 61. 1953) p. 90. scientific, of intuition and intellect, 9. [2] 61. 28. Taraboukine 128. and his refusal to sacrifice art on the Lissitzky p. [11] p. 10. It should be altar of industrial Lissitzky [2] pp. 61-62. noted, 29. "We acknowledge works which contain a sys- production ap- however, that even though classical Greek math- tem within themselves, a system which has not pealed to many eastern and central ematics was concerned with concrete magni- been evolved before the work started but has artists. His tudes, the importance of a book like Euclid's Ele- evolved in the course of it" (El Lissitzky, "Nacsi", European subsequent stay ments lies in the fact that the in in in from concepts employed Lissitzky-Kfppers [3] p. 351). Germany (and Switzerland) were totally abstract. Greek mathematicians were 30. For a reproduction of Proun 43 (1924), see 1922 to 1925, the crucial unconcerned with practical problems. See C. B. during years Lissitzky-Kuppers [3] Fig. 35. when International Constructivism Boyer, The History of Mathematics (New York: John Wiley & Sons, 1968) p. 111. 31. Numerous other artists of the avant-garde was formed, made Lissitzky a central were developing the notion of the of art; 11. [2] 62. Nisbet notes that Lissitzky's game Lissitzky p. since no other artist besides figure of the avant-garde. mathematical are verbatim however, Lissitzky examples repeated with formalist from Oswald The Decline the West. compared art-games mathematics, Spengler's of their theories are not relevant to the I References and Notes Indeed, the latter's theories of a fundamental re- argument present here. lationship between art and mathematics were 1. Lissitzky's analogies between art and mathe- widely known in Moscovite Constructivist circles, 32. Lissitzky [2] pp. 64-65. matics have been undervalued recently; Peter Nis- and Nikolai Tarabukin also referred to them 33. For the definition of as an autonomous bet claims that the artist's references to mathe- more than once. It is nevertheless certain that Lis- game rule-governed activity, see G. Boss, "Jeu et matics and to Einstein's physics were modern sitzky used the knowledge gained through Spen- Philosophie", Revue de Metaphysique et deMorale 84, metaphors, mere substitutes for images from na- gler to elaborate a theory of non-objective art with No. 4 (1979) p. 495. Boss's definition is based on ture. I disagree with his opinion that "underlying analogies to mathematics totally independent of Roger Caillois, Man, Play, and Games (New York: most of these (references) is a predilection for in- Spengler. See Nisbet [1] p. 29 and p. 49 n. 56. For Free Press ofGlencoe, 1961) 9-10. vocations of natural growth and biological Tarabukin, see N. Taraboukine, "Pour une pp. "An Introduction to El processes" (Peter Nisbet, theorie de la peinture", in Le dernier tableau, A. B. 34. E. T. Bell, TheDevelopment of Mathematics, 2nd Lissitzky", in El Lissitzky 1890-1941 [Cambridge, Nakov, ed., A. B. Nakov and M. Petris, trans. Ed. (New York: McGraw-Hill, 1945) p. 182. MA: Harvard University Art Museums, Busch- (Paris: Editions Champ Libre, 1972). 35. Cassirer [27] pp. 83, 93. Reisinger Museum, 1987] p. 28). 12. Lissitzky's interpretation of Malevich's use of 36. On the element of game in Lissitzky's work, 2. El Lissitzky, "Proun", trans.John E. Bowlt, in El abstract geometric forms is only one possibility. see my paper, "El Lissitzky's Art-Games", Neoheli- Lissitzky (Cologne: Galerie Gmurzynska, 1976) p. For a study of the influence of P. D. Ouspensky con 14, No.1 (December 1987) pp. 177-191. 60. On Lissitzky's understanding of Minkowski, and other related theories of the fourth dimen- see Linda Dalrymple Henderson, The Fourth Di- sion, see Henderson [2] pp. 274-294. 37. "Suprematism in World Reconstruction" [3] mension and Non-Euclidean Geometry in Modern Art pp.331-333. 13. Lissitzky [2] pp. 63-64. (Princeton, NJ: Princeton University Press, 1983) 38. At the Diisseldorf Congress of International pp.294-299. 14. Lissitzky [2] p. 64. Progressive Artists (1922), Lissitzky stated: 'The 3. El Lissitzky, "Suprematism in World Recon- 15. Geometric forms are the creations of human new art is founded not on a subjective, but on an struction", in Sophie Lissitzky-Kfppers, El Lis- beings; in a letter toJ.J. P. Oud, Lissitzky wrote objective basis. This, like science, can be de- sitzky, Life, Letters, Texts, 2nd Ed. (London: that he did not understand the Dutch point of scribed with precision .. ." (S. Bann, The Tradition Thames and Hudson, 1980) p. 331. view and that he disagreed with the equation of Constructivism [London: Thames and Hudson, "Universal = Straight Line + Vertical". In the uni- 1974] p. 63). At another time, he stated that 4. Lissitzky [2] p. 67. verse, he wrote, "there are only curvatures and no "emptiness, chaos, the unnatural, become space, 5. El Lissitzky, "A. and Pangeometry", in Lissitzky- straight lines" (El Lissitzky [2] p. 73). that is: order, certainty, plastic form" (El Lissitzky, Kuppers [3] p. 355. "Proun", in Lissitzky-Kuppers [3] p. 347). 16. Lissitzky [2] p. 62. 6. Lissitzky [2] p. 61. I agree with Alan C. Birn- 39. In his Principles of Mathematics, Bertrand Rus- 17. El Lissitzky, "New Russian Art", in Lissitzky- holz that this text must have been the lecture Lis- sell explained that "quantity... does not occur in Kuppers [3] p. 338. sitzky delivered at INKhUK on 23 September pure Mathematics.... The notion which does oc- 1921, not, as suggested byJohn E. Bowlt, on 23 18. Lissitzky [2] p. 62. cupy the place traditionally assigned to quantity October 1924. To Birnholz's arguments should is orde'r (as quoted in Cassirer [27] p. 95). 19. Lissitzky [2] p. 64. Since Lissitzky's intention be added the simple fact that the latest possible in this sentence is clearly connected with the 40. K. Malevich, Essays on Art, 1915-1928, Vol. 1, meeting of INKhUK was at the end of March of sets, the mathematical term 'classes' T. Xenia Glowacki-Prus and 1924. See Alan C. Birnholz, "Review of El Lis- theory Andersen, ed., seems more here than the term 'cate- Arnold McMillin, trans. sitzky", TheStructurist, No. 15/16 (1975) p. 176. For appropriate (Copenhagen: Borgen, used E. Bowlt in his translation. I 190, 192. For Malevich's idealist tran- the list of INKhUK lectures and their dates, see C. gories' byJohn 1971) pp. wish to thank K. of the Galerie Gmur- scendental see M. Lamach and Lodder, Russian Constructivism (New Haven and Rubinger philosophy, J. zynska for providing me with the original Russian Padrta, "The Idea of Malevich[23] London: Yale University Press, 1983) pp. 93-94 Suprematism", text. and p. 281 n. 85. There exist three different ver- pp.134-180. sions of "Proun". One is the text of the lecture 20. For a detailed account of these debates and 41. Malevich [40] p. 38. translated into English and German and pub- the differences between composition and con- 42. According toJean Leering, Lissitzky,despite lished in the Galerie Gmurzynska cataglogue (see struction, see Lodder [6] pp. 83-89; and M. his rationality, saw in art a creation beyond the Lissitzky [2]). Another, published by Lissitzky in Rowell and A. Zander Rudenstine, Art oftheAvant- rational. See Jean Leering, "Lissitzky's Impor- De Stijl 5, No. 6 (June 1922), was translated into Garde: Selections from the George Costakis Collection tance Today", in El Lissitzky[2] p. 35. See also F. English and published in Lissitzky-Kuppers [3] (New York: The Guggenheim Museum, 1981) pp. Pluchart, "Lissitzky,romantique et ingenieur de pp. 347-348. A third version, presumably the ear- 25-27,226-227. futur", ArtInternational No. 2, 16-18 liest, has not yet been translated from the Russian; 10, (1966). 21. Lissitzky [2] p. 62-63. it was published as "Tezisy k prounu" (Theses on 43. "A.and Pangeometry" [5] p. 352. the Proun [From Painting to Architecture]), in 22. Lissitzky [2] p. 64. 44. Alan C. Birnholz, "Time and in the Art M. Barkhin et al., eds., Mastera sovetskoi arkhitektury Space 23. On the debate on Malevich, see V. Rakitin, and of El The Structurist, No. ob arkhitekture (Soviet master architects on archi- Thought Lissitzky", "Malevich and INKhUK", in Malevich (Cologne: 15/16 (1975) p. 90. tecture), Vol. 2 (Moscow: Iskusstva, 1975) pp. Galerie Gmurzynska, 1978) pp. 288-294. 133-135. 45. N. I. Lobachevsky, Pangeometry (Kazan, 1855). 24. On Constructivist of 7. The literature on both Suprematism and Rus- Lissitzky's interpretation

Levinger, Art and Mathematics in the Thought of El Lissitzky 235 On Lobachevsky's fame and importance in Rus- centuries have brought painting right up to the Jones, ElementaryConcepts of Mathematics,3rd Ed. sia at the beginning of the twentieth century, see square, so that here they can find their way down. (London: Macmillan, 1970) pp. 136-138. On the Henderson [2] pp. 241-245. See also R. Crone, We are saying that if on the one side the stone of introduction of imaginary numbers for the solu- "Malevichand Khlebnikov: Suprematism Reinter- the square has blocked the narrowing canal of tion of equations, see Bourbaki [46] pp. 96-97. preted", Artforum 17, No. 4, 38-48 (1978). painting, then on the other side it becomes the 74. On complex numbers in advanced calculus, foundation-stone for the new spatial construction 46. N. Bourbaki, Elementsd'histoire des mathema- see Boyer [10] pp. 562-564. of reality" ("New Russian Art" [17] p. 338). The tiques(Paris: Hermann, 1969) pp. 26-27. same idea is found in Lissitzky [2] p. 64. 75. Abstract space has been defined as "a set of 47. "Suprematism in World Reconstruction" [3] objects, usually called with a set 65. "A. and Pangeometry" [5] p. 354. Here Lis- points, together p. 333. of relations in which these are involved" sitzky echoes Malevich: "The system points Suprematist (H. Eves, An Introductionto the Mathemat- 48. "New Russian Art" [17] p. 337. has conquered the blue of the firmament, heaven Historyof ics, 3rd Ed. [New York:Holt, Rinehart and Wins- is pierced and has entered white as the true rep- 49. "New Russian Art" [17] p. 338. ton, 1969] p. 370). resentation of infinite.... I have pierced the blue 50. "New Russian Art" [17] p. 339. lampshade of coloured limitation, I have come 76. On this subject, see Henderson [2] p. 297. out into the white ... "(quoted in Le miroir 51. El Lissitzky, "Exhibitions in Berlin", in Lis- supre- 77. El matiste,J. C. Marcade, ed., V. and J. C. Marcade, Lissitzky, "Vicking Eggeling", Izvestiya [3] 345. It is to note No. 1 in S. and sitzky-Kiippers p. interesting trans. [Lausanne: d'Homme, 1977] 82- Asnova, (1926), Lissitzky-Kfippers that views are shared researchers L'Age pp. El Proun und Lissitzky's by 84). Jen Lissitzky, Lissitzky, Wolkenbiigel, currently studying the psychology of artistic crea- Schriften,Briefe, Dokumente (Dresden: VEB Verlag tivity. According to Anton Ehrenzweig, creativity 66. Zeno's paradox is as follows: One cannot get der Kunst, 1977) p. 206. involves a conflict between two kinds of sensi- to the end of a racecourse, for one must traverse 78. "A.and [5] 356. bility: unconscious intuition and conscious intel- half of any given distance before one traverses the Pangeometry" p. lect. Creation starts in the unconscious as an whole, and half of that again, and so on, ad in- 79. "A.and Pangeometry" [5] p. 357. undifferentiated primary process, but a creative finitum; hence, there is an infinite number of 80. never mentioned but he work is the result of the successful coordination on line, and one cannot touch Lissitzky Duchamp, points any given knew Man work of unconscious undifferentiation and conscious an infinite number one one within a finite Ray's well. Since Man Ray photo- by Glass differentiation. See A. The Hidden time. See Bell [34] 64. graphed Duchamp's Rotary Plates, Lissitzky Ehrenzweig, p. could have heard of it. On Orderof Art and Los Lissitzky's acquain- (Berkeley Angeles: University 67. An infinite set is one that can be into a tance with Man see of California Press, 1971) p. 44. put Ray, Lissitzky-Kuppers[3] pp. one-to-one correspondence with a proper subset 66-67. 52. Bell offers the following comments on the ab- of itself; a finite set is one that cannot be put into 81. El Lissitzky, "Proun Room, Great Berlin Art stract nature of geometry: "The logical analysis of one-to-one correspondence with any of its proper Exhibition", in [3] 365. geometry... made it plain to all... that geometry subsets. See Armstrong [54] pp. 98-100. See also Lissitzky-Kfippers p. is an abstract hypothetico-deductive system G. Cantor, Contributionsto theFoundingofthe Theory 82. For a detailed study of Lissitzky's exhibition without intrinsic content other than that implied of TransfiniteNumbers, Notes and Introduction by spaces in Dresden and Hanover and of his later by arbitrarily prescribed sets of postulates" (Bell P. E. B. Jourdain (New York:Dover Publications, Soviet pavilions at international exhibitions, see [34] p. 330). See also "A. and Pangeometry" [5] 1915). On the historical problem of infinites and Birnholz [69] pp. 378-393, 430-432. p. 355. Cantor's theory of sets, see Bourbaki [46] pp. 39- 50. 83. For a different view of Lissitzky'sabstract-de- 53. James W. Armstrong, Elementsof Mathematics materialized space, this time in connection with (London: Macmillan, 1970) p. 9. 68. On the difference between Malevich's con- Gabo's and Moholy-Nagy's ideated spaces, see cept of "an absolute of infinite, white Rosalind E. in 54. According to Bell, "the deflation of older 'receptacle' Krauss, Passages ModernSculpture space" and of infinite in the and London: MIT beliefs comprises the main contribution of non- Lissitzky's concept (Cambridge, MA, Press, 1981) sense of Einstein's curved space-time continuum 55-67. Euclidean geometry to mathematical thought as pp. and the artist's use of the in the Prouns, a whole, and also perhaps, the principal contribu- concept 84. "What is for how- see Henderson [2] pp. 294-295. See also Alan C. particularly important us, tion of mathematics to the progress of civiliza- ever, is that this A.F. [Art should effect the Birnholz, "El Lissitzky"(Ph.D. dissertation, Yale Form] tion" (Bell [34] p. 330). destruction of the old idea of art, that of 'monu- University, 1973) pp. 78-79. On axonometry and for us monumental does not 55. "A. and Pangeometry" [5] p. 355. On the other aspects of Lissitzky's of Su- mentality'... imply interpretation a work which will stand there for a or a relationship between non-Euclidean geometries prematist space, see Y. A. Bois, "Lissitzky, year hundred years or a thousand but the and the relativityof knowledge, and hence doubt, Malevich and the Question of Space", in Supre- years, per- petual of human achievement" see Henderson [2] p. 17. matisme(Paris: GalerieJean Chauvelin, 1977) pp. expansion ("A. 29-48. and Pangeometry" [5] p. 357). 56. Lissitzky [2] p. 63. 85. El "Our Book", in 69. According to Tarabukin, the problem of time Lissitzky, Lissitzky-Kuippers 57. On these traits in Lissitzky'swork, see Alan C. [3] 360-361. in painting was evoked for the first time (in rela- pp. Birnholz, "'For the New Art': El Lissitzky's tionship with the introduction of a fourth 86. El of in Prouns", Artforum8,No. 2, 65-67 (1969) and No. Lissitzky,'Topography Typography", dimension in Minkowski, Rie- 3, 68-73 (1969). See also Birnholz, "Notes on the mathematics) by Lissitzky-Kiippers [3] p. 359. The notion of the mann and Lobachevsky. Tarabukin adds that dematerialization of culture was Chronology of El Lissitzky's Proun Composi- contemporary had never mentioned this, since he much discussed in tions", ArtBulletin 55, No. 3, 437-439 (1973). Spengler the Constructivist circles in "treated all manifestations of contemporary art Moscow, especially in three lectures given by Boris 58. "Suprematism in World Reconstruction" [3] with a contemptuous silence" (Taraboukine [11] Kushner at INKhUK in March 1922. Kushner and p.331. p. 132). Tarabukin saw dematerialization as part of the new industrial as in, for 59. Lissitzky [2] p. 70. 70. Lissitzky [2] p. 63. For other production, example, interpretations electric where the end is not a of the importance of time in work, see energy, product 60. "A.and [5] 353. Lissitzky's material but etc. For Kushner's Pangeometry" p. Birnholz [44]. object light, heat, lectures, see Lodder 100-101. For 61. "A.and Pangeometry" [5]. [6] pp. 93, 71. The reversibility of the Prouns is explored by Tarabukin's views, see N. Taraboukine, "Du che- 62. Bell 220. Birnholz in [34] p. [68] pp. 97-98, and in [44] p. 92. For valet a la machine", in Lederniertableau[11] p. 61. the theoretical importance of reversibility, see 63. On zero in Malevich's thought, see Y. A. Bois, 87. [2] 71. also Bois [68] p. 38. Lissitzky p. "Malevitch,le carre, le degre zero", Macula, No. 1 88. The are from [2] 67. (1978) pp. 28-48. 72. "A.and Pangeometry" [5] p. 355. equations Lissitzky p. 64. Lissitzkywrote that "it has been said that the 73. On imaginary (complex) numbers, see B. W.

236 Levinger,Art and Mathematics in the Thought of El Lissitzky