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 "un- derground 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 un- derstanding 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 mathemati- cal 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-objec- tive. 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 .