MY MAGICAL ATTIC: THE TATE COLLECTION I: PASSION OF ARTS

THE TATE COLLECTION I: PASSION OF ARTS

THE IMAGE & IMAGINATION OF THE FOURTH DIMENSIONS IN TWENTIETH-CENTURY ART AND CULTURE

LINDA DALRYMPLE HENDERSON UNIVERSITY OF TEXAS AT AUSTIN

ABSTRACT

‘’ One of the most important stimuli for the imaginations of modern artists in the twentieth century was the concept of a higher, unseen fourth dimension of space. An outgrowth of the n-dimensional geom[1]etries developed in the nineteenth century, the concept predated the definition of time as the fourth dimension by Minkowski and Einstein in relativity theory. Only the popularization of relativity theory after 1919 brought an end to the widespread public fascination with the supra-sensible fourth dimension between the 1880s and 1920s. Ini[1]tially popularized by figures such as E. A. Abbott, Charles Howard Hin[1]ton, Claude Bragdon, and P. D. Ouspensky (as well as science-fiction writers), the fourth dimension was a multivalent term with associa[1]tions ranging from science, including X-rays and the ether of space, to idealist philosophy and mystical “cosmic consciousness.” This essay focuses on the differing approaches to higher spatial dimensions in the cubism of Pablo Picasso and Juan Gris, the suprematism of Ka[1]zimir Malevich, and The Large Glass project of Marcel Duchamp in the early twentieth century. It concludes by examining contemporary artist Tony Robbin’s thirty-year engagement with the mathematics of four-dimensional geometry and computer graphics, as well as his cur[1]rent work with knot theorist Scott Carter ‘’

In the wake of a short article on the four-dimensional hypercube titled “Visualizing Hyperspace,” published in the March 1939 issue f Scientific American, the journal’s editors found it necessary to re[1]spond in September 1939:

‘’ From time to time . . . the editors have received inquiries from puzzled readers who appear to be confused about a variety of questions suggested by this ar[1]ticle. Is not time the fourth dimension? How do mathematicians know that there are more than the three dimensions with which we are all daily familiar? . . . First, regarding time as the fourth dimension: True, time does figure in the so-called “space-time continuum,” but not as an extra dimension of space. Next, how do they know there are extra dimensions of space? They don’t! They play with them, however, just as if they did exist. . . . The mathematician is a whimsical fellow who deliberately enjoys creating a make-believe and then proceeding to show what would be the case if it were true. . . . What probably confuses the puzzled non-mathematician is the fact that the mathematician uses for his excursions into the imaginary the same word he uses in connection with something he and all the rest of us know to exist; that is, “dimensions.” If he would call them something else the confusion would promptly end for most of us.1’’

The critical role of the imagination in mathematics and geom[1]etry has long been acknowledged and given more serious discussion than in this description of mathematicians as “whimsical fellow[s].” In the twentieth century, popular books such as Edward Kasner and James Newman’s Mathematics and the Imagination of 1940 and David Hilbert and S. Cohn-Vossen’s Geometry and the Imagination of 1952 have made that connection apparent for the lay public. Hilbert em[1]phasized the importance of a specifically visual component of the imagination, declaring in his introduction: “With the aid of visual imagination, we can illuminate the manifold facts and problems of geometry, and beyond this, it is possible in many cases to depict the geometric outline of the methods of investigation and proof.”2 The beautiful drawings in Hilbert’s books certainly stimulated the creative imaginations of a number of artists who used the book, in[1]cluding sculptor Mark di Suvero and members of the Park Place Gal[1]lery, who responded to his sections on topology and “Polyhedra in Three and Four Dimensions.”3 In a similar way, H. S. M. Coxeter’s 1963 book, Regular Polytopes, served as a vital inspiration for painter Tony Robbin as he began to explore four-dimensional geometry dur[1]ing the early 1970s. Coxeter, noting that while “we can never fully comprehend” figures in four or more dimensions,” had declared: “In attempting to do so, however, we seem to peep through a chink in the wall of our physical limitations, into a new world of dazzling beauty.”4

Indeed, it was not simply geometry, but specifically the nine[1]teenth-century field of n-dimensional geometry and the concept of a possible fourth spatial dimension that emerged from it in the 1870s that proved crucial to the imaginations of twentieth-century artists. From the 1880s to the 1920s, popular fascination with an invisible, higher dimension of space—of which our familiar world might be only a section or shadow—is readily apparent in the vast number of articles and the books such as architect Claude Bragdon’s A Primer of Higher Space (The Fourth Dimension) (1913) published on the topic.5 Two plates from Bragdon’s book are useful in setting forth two of the basic ways of conceptualizing a higher spatial dimension: the gener[1]ation of the next higher-dimensional form by motion through space (fig. 1), and sectioning or slicing (fig. 2). In both approaches, reason[1]ing by analogy to the relationship of two to three dimensions is cen[1]tral to imagining the transition from three to four dimension

Just as Bragdon’s beautiful hand-lettered plates provide a time capsule of approaches to the fourth dimension in 1913, the 1910 book, The Fourth Dimension Simply Explained, collected the win[1]ning essays in a 1909 contest sponsored by Scientific American on the topic, “What Is the Fourth Dimension?” 6 Virtually all of the Scientific American essayists in 1909 treated the fourth dimension as a spatial phenomenon, because the widespread popularization of Einstein’s special and general theories of relativity (1905, 1916) would begin only in 1919 with the solar eclipse that established em pirically the curvature of light that Einstein’s theory had predicted.7 It was little wonder, then, that in 1939, Scientific American readers were confused, since during the course of the 1920s, Einstein and mathematician Hermann Minkowski’s earlier incorporation of time into the four-dimensional space-time continuum had gradually overshadowed cultural memories of the geometrical, spatial fourth dimension. During the 1930s through the 1950s, in fact, the fourth dimension of space essentially went underground, staying alive in nonmathematical culture primarily in science-fiction writing and in the mystical, philosophical literature that had developed around the idea.8

Mathematicians, of course, continued to study four-dimensional geometry, but even Kasner and Newman recognized the need to explain the idea to a 1940s audience in a chapter of Mathemat[1]ics and the Imagination on “Assorted Geometries—Plane and Fancy.” “Physicists may consider time to be the fourth dimension, but not the mathematician,” they assert at the start of their explication of the con[1]cept. While their discussion focuses on the geometrical properties of four-dimensional objects and the analogies by which we can reason about them, their conclusion takes the idea well beyond the realm of geometry to point out its larger significance in the history of hu[1]man thought: “No concept that has come out of our heads or pens marked a greater forward step in our thinking, no idea of religion, philosophy, or science broke more sharply with tradition and com[1]monly accepted knowledge, than the idea of a fourth dimension.”9[1]

From its first popularization in English theologian E. A. Abbott’s Flatland: A Romance of Many Dimensions by a Square of 1884, the fourth dimension had been linked to the enlarging or freeing of thought and imagination. Abbott dedicated his cautionary tale about a two-dimensional world oblivious of the larger three-dimensional space in which it existed “To the Inhabitants of Space IN GENERAL,” whom he hoped would “aspire yet higher and higher To the secrets of FOUR FIVE OR EVEN Six Dimensions Thereby contributing To the Enlargement of THE IMAGINATION.”10 This theme would be come a leitmotif of literature on the fourth dimension both in the context of mathematics and as it quickly acquired broader philo[1]sophical implications. As Casius Keyser wrote in a 1906 essay in The Monist titled “Mathematical Emancipations”: “The hyper-dimen[1]sional worlds that man’s reason has already created, his imagination may yet be able to depict and illuminate. . . . It is by creation of hyperspaces that the rational spirit secures release from limitation. In them it lives ever joyously, sustained by an unfailing sense of in[1]finite freedom.”11 Citing Keyser, H. P. Manning, in his 1914 textbook Geometry of Four Dimensions, argued that the “synthetic” study of the “forms and properties” of four-dimensional figures so “that it is almost as if we could see them” results in “greatly increas[ing] our power of intuition and our imagination.”12

he figure who definitively extended the fourth dimension be[1]yond its mathematical roots, while maintaining its geometrical core meaning, was the Englishman Charles Howard Hinton. In his books A New Era of Thought (1888) and The Fourth Dimension (1904), Hin[1]ton developed the philosophical implications of four-dimensional space and secured its place in late nineteenth- and early twentieth[1]century culture. Hinton’s “hyperspace philosophy” was an idealist worldview based on his belief that by developing an intuitive appre[1]hension of four-dimensional space, individuals would gain access to true reality and hence resolve the problems of the materialist three[1]dimensional world. According to Hinton,

‘’ w]hen the faculty is acquired—or rather when it is brought into conscious[1]ness, for it exists in everyone in imperfect form—a new horizon opens. The mind acquires a development of power, and in this use of ampler space as a mode of thought, a path is opened by using that very truth which, when first stated by Kant, seemed to close the mind within such fast limits. . . . But space is not limited as we first think.13 ‘’

Hinton’s method for “educating the space sense” of his readers was a set of exercises to be carried out with a block of multicolored cubes, such as those pictured in various colors on the frontispiece of The Fourth Dimension (fig. 3). By memorizing the relative positions and color gradations of cubes within large blocks, Hinton’s readers were to develop their mental powers and transcend self-oriented per[1]ception (e.g., the senses of left/right and up/down or gravity).14 With this knowledge, they would also be able to visualize the passage of the successive cubic sections of a four-dimensional hypercube through three-dimensional space. But this training was simply the practical prelude to what Hinton hoped would be “a new era of thought,” as he declared in that book of 1888: “I shall bring forward a complete system of four-dimensional thought—mechanics, science, art. The necessary condition is, that the mind acquire the power of using four-dimensional space as it now does three-dimensional.” Although Hinton never realized such a “system,” he extended his ideas into the realm of literature, writing a series of “scientific romances” pub[1]lished in 1884–1885 and 1896.15

Although Hinton achieved little personal success or recognition in his lifetime, his writings—with their message of a higher truth and the possibility of self-realization—were remarkably influential in the United States and Europe as well as in England. The Fourth Dimen[1]sion, for example, was reprinted in London five times, in 1906, 1912, 1921, 1934, and 1951. Those who subsequently built upon and/or promulgated his ideas included Bragdon in the United States, math[1]ematician and mystic Peter Demianovich Ouspensky in Russia, Ger[1]man theosophist/anthroposophist Rudolf Steiner in Germany, both mathematicians E. Jouffret and Maurice Boucher and theosophists in France, the symbolist writer Maurice Maeterlinck in Belgium, and theosophist C. W. Leadbeater in England.16 Ouspensky developed a mystical interpretation of the fourth dimension, associating it with infinity and the achievement of “cosmic consciousness” of a truer, four-dimensional reality.17 If Ouspensky was envisioning a liberating effect quite different from Hinton’s more pragmatic approach, the theme of the fourth dimension as a liberating agent of some kind ran through most all of its interpretations. As Bertrand Russell wrote in his review of The Fourth Dimension in Mind in October 1904:

‘’ The merit of speculations on the fourth dimension—a merit which the present work possesses in full measure—is chiefly that they stimulate the imagination, and free the intellect from the shackles of the actual. A complete intellectual liberty would only be attained by a mind which could think as easily of the non-existent as of the existent.18 ‘’

Writers such as Hinton and Bragdon, in particular, had a major impact on the way the public imagined and imaged the fourth di[1]mension during the twentieth century. Painters were particularly responsive to the idea, and many of the stylistic innovations in the first decades of the century were made in the context of attempts to represent or signify in some way the elusive fourth dimension. Russell’s reference to the “shackles of the actual” is especially tell[1]ing, because it points up the fundamental shift that the possibil[1]ity of a spatial fourth dimension produced in the visual arts. For artists, whose visual imaginations had been largely constrained by painting’s traditional allegiance to the visible world, the possibil[1]ity that space was actually four-dimensional was revolutionary. The chiaroscuro modeling techniques and one-point linear perspective painters had relied upon since the Renaissance to create convinc[1]ing three-dimensional form and space were irrelevant if the world had four dimensions. One of the pioneers of totally abstract art, the Russian suprematist painter Kazimir Malevich, was encouraged by his belief in four-dimensional space to leave behind completely all traces of the visible world, as discussed below.

There was another strong impetus for breaking the “shackles of the actual” in the late nineteenth- and early twentieth centuries: the discovery of the X-ray by Wilhelm Röntgen in 1895. X-rays proved definitively the limited nature of human vision, which perceives only the narrow band of visible light in the electromagnetic spec[1]trum then being identified.19 With an impact second only to that of the atomic bomb, the discovery of the X-ray undoubtedly con[1]tributed to the continued popular interest in the fourth dimension, which might otherwise have remained the province of mathemati[1]cians, philosophers, and mystics.20 Once the X-ray established the inadequacy of the human eye, however, who could deny with cer[1]tainty the possibility of a fourth spatial dimension simply because it was invisible?

In addition to the fourth dimension and the X-ray, the successive discoveries during the 1890s of the electron and of radioactivity, as well as the interest in the Hertzian waves of wireless telegraphy, con[1]tributed further to a radical reconception of the nature of matter and space in this period. 21 Beyond its possible four-dimensionality, matter was transparent to the X-ray and, on the model of radioac[1]tivity, was often discussed as dematerializing into the space around it. Moreover, during this period, that space was never thought of as empty; instead, it was understood to be filled with the impalpable ether of space traversed by various ranges of vibrating waves, and the ether itself was thought by some to be the source of matter, as in the “electric theory of matter.”22 Widely popularized, these new scientific discoveries, along with the possibility of a fourth spatial dimension, strongly suggested that an invisible reality existed just beyond the reach of human perception. And in the view of artists and critics, it was the sensitive artist possessed of intuition and imagination—the successor to the visionary seer posited by the symbolists during the 1890s—who would be required to evoke higher dimensions, as well as the newly fluid conceptions of matter and space.

This essay samples the techniques employed in three of the major artistic responses to the fourth dimension during the early twentieth century: cubism, suprematist abstraction, and the art of Marcel Du[1]champ, the early twentieth-century artist who engaged the fourth dimension most fully, albeit playfully. Only toward the end of the twentieth century would the advent of computer graphics make it possible for artists and geometers to navigate four-dimensional space with mathematical tools, but here also, artistic intuition and imagi[1]nation would play an important role. After briefly surveying the cul[1]tural understanding of the term “fourth dimension” at mid-century, when Einsteinian space-time dominated the layman’s awareness of the concept, the essay concludes with a look at the computer-era work of artist Tony Robbin, as well as his collaboration with math[1]ematician Scott Carter to explore the visual properties of braided surfaces and lattices in four and five dimensions.

CUBISM: WINDOWS ON INVISIBLE GEOMETRICAL COMPLEXITY

The cubist painter and theorist Jean Metzinger was the first artist to write about the importance of the new geometries for contempo[1]rary painters, and he and Juan Gris are said to have studied four-di[1]mensional geometry with the insurance actuary Maurice Princet.23 All three of these figures were close to Pablo Picasso, who in 1909, along with Georges Braque, developed the style that has come to be known as analytical cubism. While Picasso and Braque drew critical lessons from the art of Paul Cézanne and the conceptual nature of African sculpture, their mature cubism—with its faceted forms and fusion of figure and ground—was a response as well to the exhilarating new ideas about reality issuing from popularized science and mathemat[1]ics.24 If Picasso described his goal in cubism as “paint[ing] objects as I think them, not as I see them,” the more theoretically oriented Metzinger and the poet Guillaume Apollinaire, another of Picasso’s friends, touted the fourth dimension overtly to justify the cubist painter’s freedom both to deform objects and to reject perspective. “It is to the fourth dimension alone that we owe a new norm of the perfect,” Apollinaire declared in 1912, adding that the concept was part of the “language of the modern studios.”25 In his book Les Pein[1]tre Cubistes of 1913, the poet likewise dismissed perspective as “that miserable tricky perspective, that fourth dimension in reverse.”26

In the early 1970s, I suggested that plates from Esprit Pascal Jouf[1]fret’s 1903 book Traité élémentaire de géométrie à quatrième dimensions, such as that shown in figure 4, would have confirmed Picasso’s stylistic direction.27 Since that time, more discussion of Jouffret, Prin and Picasso has occurred, and Tony Robbin in his book Shadows of Reality: The Fourth Dimension in Relativity, Cubism, and Modern Thought argues convincingly that certain techniques in Picasso’s paintings of this period, especially his 1910 Portrait of Daniel-Henry Kahnweiler (Art Institute of Chicago), derive directly from Jouffret’s innovative drawing techniques.28 Robbin is particularly interested in the complex, rectangular areas in the head of Kahnweiler, which he compares to several other of Jouffret’s illustrations. “The odd way in which spaces are both inside and outside a four-dimensional fig[1]ure [with its three-dimensional bounding cells] is the subject of both Jouffret’s illustration and Picasso’s portrait of Kahnweiler,” Robbin concludes.29

For the purposes of this essay, figure 4 serves effectively to point up the general similarity between cubist paintings, including Gris’s Still Life Before an Open Window: Place Ravignan (fig. 5) and Jouffret’s techniques for presenting complex figures. Here, the geometer’s use of transparency, shifting overlays of differing views of an object, and the resulting spatial ambiguity are strikingly similar to Gris’s approach. Like other cubists, Gris combines multiple viewpoints, just as Henri Poincaré had suggested in his 1902 book La Science et l’hypothèse that a four-dimensional object could be rendered by means of “several perspectives from several points of view.” Given the “muscular sensations” accompanying the transition from view to view, Poincaré had concluded: “In this sense we may say the fourth dimension is imaginable.”30 While Gris’s view of trees and a building out the window may appear conventional enough (it is actually a distinct blue monochrome), the complex overlay of visual signs on the table—interacting with the wrought iron of the bal[1]cony—deny completely the possibility of reading the space or mat[1]ter as three-dimensional.

In addition, Still Life Before an Open Window effectively evokes the newest scientific ideas of matter and wave-filled space. Here, the interior and exterior of objects and of the room itself interpenetrate, producing the kind of clairvoyant, see-through vision of three-di[1]mensional forms that would be accessible to four-dimensional sight or an X-ray. Not only are spatial clues ambiguous, but Gris plays one kind of light off another, drawing on both visible and invisible light. The mauve and green palette in the central area of the still life contrasts markedly with the ultraviolet or “black light” that seems to illuminate the blue/black areas around the center. Although the bright, seemingly natural light in the central area does not cast shadows or give substance to the objects of the still life, it does re[1]fract and distort the Le Journal banner line dramatically. Only the curtained window in the upper left corner is painted convention[1]ally in light and shade. However, it is dwarfed by the other ranges of light in the painting, which thus makes a powerful commentary on the changed status of the window as source of visible light and, metaphorically, truth. Gris’s Still Life and other cubist paintings are testaments to the new paradigm of reality ushered in by the discov[1]ery of X-rays and interest in the fourth dimension. Such paintings are new kinds of “windows”—in this case, into a complex, invisible reality or higher dimensional world as imagined by the artist.

My 1983 book The Fourth Dimension and Non-Euclidean Geometry in Modern Art was written before I had studied the late-Victorian ether physics still prevalent during the early twentieth century. In the 1980s—and actually from the 1940s onward—the science with which cubism was associated in art historical literature was Ein[1]steinian relativity theory. That conflation was the result of a kind of “short circuit” in the 1940s when discussions of cubist references to the fourth dimension were erroneously linked to the only fourth dimension the public knew—namely, the space-time world of Ein[1]stein.31 But such debates over the supposed relationship of Picasso to Einstein also served to occlude study of the science to which Picasso, Gris, and others were responding in pre–World War I Paris. The re[1]covery of that science has been critical to a fuller history of the im[1]pact of the spatial fourth dimension, because the concept was rarely understood in isolation from contemporary ideas about space and matter; instead, it was regularly discussed against the backdrop of contemporary ether physics, beginning with Hinton, who focused attention on the fourth dimension’s possible relation to the ether itself.32

A case in point is the 1903 book by Maurice Boucher, Essai sur l’hyperespace: Le Temps, la matière et l’énergie, which Metzinger mentions in his memoirs. There, Boucher argues in support of the fourth dimension: “Our senses, on the whole, give us only deformed im[1]ages of real phenomena, some of which have long remained un[1]known, because none of our organs put us in direct contact with them.”33 As we shall see, the Russian avant-garde knew Boucher’s book, as did, quite certainly, Duchamp. Such a text makes clear the close connections of interpretations of the fourth dimension to a contemporary science that, while it dealt with invisible phenom[1]ena like the X-ray and the ether, was highly suggestive to the visual imaginations of artists.

Kazimir Malevich’s Suprematism: Sections Afloat in Infinite Space If the cubists created geometrically complex images that suggested the invisible reality beyond surface appearances, the abstract supre[1]matism of Kazimir Malevich utilized the method of sectioning to create geometrical planes moving in space.34 The two-dimensional analogy that lay behind Flatland and was illustrated in Bragdon’s Primer of Higher Space (fig. 2) had first been discussed extensively by Hinton, and both Ouspensky—Hinton’s Russian disciple—and Boucher in his Essai sur l’hyperespace followed Hinton’s model. That Malevich and his friend, musician and artist Mikhail Matyushin, knew Boucher’s Essai, with its unification of the fourth dimension and ether physics, is clear from a 1916 text by Matyushin in which he writes: “How to solve the question of ‘space,’ ‘where’ and ‘where to’? Lobachevsky, Riemann, Poincaré, Bouché, Hinton and Mink[1]ovsky provided the answer.”35 In addition to Boucher, Poincaré, Hinton, and Minkowski, Matyushin here cites the two major pio[1]neers of non-Euclidean geometries, Nikolai Ivanovich Lobachevsky and Georg Friedrich Bernhard Riemann.

Matyushin, however, does not include in this list the figure who was even more central to Malevich’s invention of suprematism: Ous[1]pensky, the primary Russian advocate of the fourth dimension. By 1916, in fact, Malevich’s and Matyushin’s enthusiasm for Ouspensky had cooled somewhat, since, in the 1914 edition of his 1911 Tertium Organum, Ouspensky had criticized contemporary Russian artists for what he considered their wrong-headed approach to the fourth di[1]mension. Nonetheless, Ouspensky’s books, The Fourth Dimension of 1909 and Tertium Organum: A Key to the Enigmas of the World of 1911, which provided a full accounting of Hinton’s ideas, were criti[1]cal sources for Malevich and his colleagues Matyushin and the poet Alexei Kruchenykh.36 Most important for Malevich’s mature supre[1]matism, however, was Ouspensky’s discussion of the transition to four-dimensional “cosmic consciousness” and its relation to infinity. Indeed, Boucher’s chapter on infinity and the fourth dimension, as well as his dismissal of the visible world of the senses as illusion, may have been a stimulus for Ouspensky himself—as well as for Malevich.

When Malevich exhibited his first suprematist canvases at the 0.10 exhibition in St. Petersburg in December 1915, one canvas was titled Movement of Painterly Masses in the Fourth Dimension, and oth[1]ers bore the subtitles Color Masses in the Fourth Dimension and Color Masses in the Second Dimension (fig. 6). Malevich’s suprematist paint[1]ings with planes of one color only strongly suggest the two-dimen[1]sional sections or traces created when three-dimensional objects pass through a plane, as discussed in Hinton and Ouspensky and illustrated in Bragdon’s Primer of Higher Space. These “color masses in the second dimension” may have served Malevich as indirect signs of the fourth dimension by means of the well-known two-dimensional analogy.

Malevich’s Painterly Realism of a Football Player: Color Masses in the Fourth Dimension (fig. 7), however, is more typical of his suprema[1]tist works, which generally include multicolored overlapping planes that prevent a reading of the image as two-dimensional. Here, the artist evokes higher dimensions more directly by suggesting motion through an infinite, multidimensional white space. Eschewing three[1]dimensional form, Malevich sets two-dimensional planes of high[1]keyed color into motion, drawing on the theme of time and motion as provisional means of gaining higher spatial understanding. Both Hinton and Ouspensky understood time as a means toward a spatial end, as in its role in both the generation of high-dimensional forms (fig. 1) and their sectioning (fig. 2).37 Undoubtedly reflecting ideas he shared with Malevich, Matyushin wrote in his diary in May 1915: “Only in motion does vastness reside. . . . When at last we shall rush rapidly past objectness we shall probably see the totality of the whole world.”38

According to Ouspensky, a “sensation of infinity” and vastness would characterize the first moments of the transition to the new “cosmic consciousness” of four-dimensionality, and Malevich re[1]ferred specifically to the space of his suprematist paintings as the “white, free chasm, infinity.”39 Fascinated by flight, Malevich does not, however, paint his space blue; instead, it is a cosmic white ex panse in which variously colored elements float freely, without any specific left/right or up/down orientation, just as Hinton had argued that gaining independence from conventional orientation and the pull of gravity would be the initial step in educating one’s “space sense” to perceive the fourth dimension. Like a cubist painter, Malevich generally avoided any signs of the third dimension. However, in contrast to cubism’s geometrical complexity and suggestion of a window onto an invisible world, Malevich sought to convey the physiological experience of four-dimensional cosmic consciousness, relying on concepts long associated with the fourth dimension: spatial vastness and infinity, freedom from gravity and specific ori[1]entation, and implied motion.

MARCEL DUCHAMP: PLAYFUL GEOMETRY AND OTHER SIGNS OF THE FOURTH DIMENSION

Marcel Duchamp, who had begun his painting career in the con[1]text of cubism, was dedicated to realizing aspects of four-dimen[1]sional space in his art, but both his approach and his result were far removed from cubism and from Malevich’s suprematism. Du[1]champ’s nine-foot-tall work on glass, The Bride Stripped Bare by Her Bachelors, Even (1915–1923), known as The Large Glass (fig. 8), is a mathematical/scientific allegory of sexual quest, in which Duchamp worked to create an unbridgeable gap between the four-dimensional realm of the biomechanical Bride above and the three-dimensional Bachelor Machine below.40 His sources on the fourth dimension in[1]cluded Matyushin’s entire list of names, quoted earlier, with the sub[1]stitution of Jouffret for “Minkovsky.” But he also read many other sources, since he actually gave up painting for a time and took a job at the Bibliothèque Ste. Genèviève in 1912, determined as he was to “put painting at the service of the mind.”41 Disgusted by what he be[1]lieved was the mindless, “retinal” painting of his fellow artists, Du[1]champ found in the fourth dimension a topic tied closely to mental activity, including imagination, intuition, and reason (the latter a prominent theme in Boucher’s book), and thus a field in which he could define himself as a new kind of artist. Not only did he trade canvas and oil paint for glass and unconventional materials, such as lead wire, lead foil, and dust, but he developed the Large Glass as a text/image project, writing hundreds of preparatory notes that he considered to be as important as the object itself.4

Without Duchamp’s notes, we would be hard pressed to decipher the basic narrative of The Bride Stripped Bare by Her Bachelors, Even, as well as to appreciate the “playful physics” and geometry that un[1]derlie it. Basically, a series of operations begins at the left side of the Bachelors’ realm during which “illuminating gas” is gradually lique[1]fied into a semen-like “erotic liquid,” which is ultimately splashed onto the upper half to form the chance-determined “Nine Shots” at the right of the Bride’s realm. This is the closest the Bachelors come to making contact with the object of their desire. In order to establish insurmountable allegorical “collisions” between the desiring Bach[1]elors and the unreachable Bride, Duchamp drew on contemporary science as well as the four-to-three-dimensional contrast between their realms.43 Boucher’s Essai sur l’hyperespace would have been an es[1]pecially relevant source for him, since it treated the fourth dimension in relation to contemporary ideas on matter, energy, and the ether. In fact, wave-borne communication is a central theme of the Large Glass, in which the Bride, hanging gravity-free in her etherial, four[1]dimensional realm, issues commands to the Bachelors by means of her “splendid vibrations.” The Bride’s basic columnar form is rooted in X-ray images, and her vibratory communications are based on the latest wireless telegraphy and radio control via the ether. By con[1]trast, the laws of classical mechanics, playfully “stretched” by Duch[1]amp, rule the lower half of the work, where the Bachelors are further constrained by perspective and the relentless pull of gravity.44

Although Duchamp never published the comprehensive text he originally envisioned to accompany the Large Glass, his boxes of facsimiles of his notes, primarily The Green Box of 1934 and A l’infinitif (The White Box) of 1966, testify to the breadth of his study and his powers of verbal invention in creating his “hilarious pic[1]ture.”45 Given the fate of the spatial fourth dimension in the wake of Einstein’s emergence in the 1920s, Duchamp chose not to in[1]clude his notes on the fourth dimension in the Green Box. But by the 1960s, the subject was beginning to reemerge in culture, and his White Box notes on the subject display his rich imagination and wit as he played with the laws of four-dimensional geometry and ex[1]plored other means by which he might make the Bride’s realm four[1]dimensional. Duchamp’s notes and drawings offer highly inven[1]tive approaches to the topic, which, in the end, were unrealizable; nonetheless, his verbal invention in the notes stands as a significant counterpart to the Large Glass itself

Duchamp speculated extensively on four-dimensional geometry, working by means of analogy and developing his own playful laws n the subject.46 Although he considered Poincaré’s ideas on geo[1]metrical continua and cuts as well as the use of mirrors and virtual images as possible signs of the Bride’s four-dimensionality, he finally returned to the notion of shadows, as articulated by Jouffret: “The shadow cast by a 4-dim’l figure on our space is a 3-dim’l shadow.”47 Thus Duchamp painted the Bride to resemble a photograph of a three-dimensional figure, whom he thought of as the shadow of the true, four-dimensional Bride. However, he also took additional steps to augment the Bride’s four-dimensional otherness, creating for her a spatial realm he defined as beyond measure (in contrast to the Bachelor’s “mensurable” and “imperfect” forms).48 In her in[1]finite, immeasurable realm, the Bride, described as free of gravity, suggests qualities associated with expanded spatial perception in the tradition of Hinton. Yet Duchamp was far from Ouspensky’s and Malevich’s pursuit of mystical “cosmic consciousness”; instead, the self-proclaimed Cartesian was much closer to Boucher, the advocate of reason, in approaching the fourth dimension.4

Duchamp abandoned the execution of the Large Glass in 1923, leaving it unfinished and missing several components. Although he never added the Juggler or Handler of Gravity to the work itself, this key intermediary figure was to have stood symmetrically oppo[1]site the Bride, and to have facilitated communication between the Bride and the Bachelors.50 Drawn in the form of a spiral, the Juggler would have been able to function in both three and four dimen[1]sions, thus evoking the dimension-transcending associations of the spiral, which Hinton had utilized to demonstrate the illusion of a circling point created as a spiral passed through a plane.51 The spiral had a second link to the fourth dimension: advocates of a higher dimension pointed to right- and left-handed spiral growth in nature as “scientific” evidence for the existence of four-dimensional space. Such mirror symmetrical pairs, which also included right and left hands and right- and left-handed growing crystals as well as spirals, would need to be turned through a fourth dimension to be made to coincide with their opposites. Mirrors themselves were also preva[1]lent in popular literature on the fourth dimension from the start, in[1]cluding the mathematician Lewis Carroll’s Through the Looking Glass of 1872.52 Duchamp utilized mirror silver on the surface of the Large Glass to create the “Oculist Witnesses” (the circular eye chart–like forms at the right of the Bachelors’ realm), who were to “dazzle” up[1]ward a mirror reflection of the orgasmic splash to produce the Nine Shots. He had already played with the notion of mirror reversals and hinges in his hinged, semi-circular glass panel Glider Containing a Watermill of 1913 (Philadelphia Museum of Art), which offers the viewer mirror-reversed images of its front and back.

During the 1920s and ’30s, Duchamp combined his interest in the spiral with movement, setting spiraling disks into motion so that they seemed to pulsate outward and inward. These experiments in optics would subsequently link Duchamp, the early twentieth century’s most committed student of the spatial fourth dimension, to the kinetic art that developed during the early 1920s in response to the new focus on time in Einsteinian relativity theory. The Hun[1]garian artist László Moholy-Nagy was the primary advocate of the new space-time kinetic art, which he promulgated in books such as his Von Material zu Architektur (subsequently translated as The New Vision) of 1928 and Vision in Motion of 1947.53 By the later 1940s and ’50s, Duchamp was regularly grouped with Moholy-Nagy and Alexander Calder as a kinetic artist. Yet he had not forgotten the spatial fourth dimension that had been so central to the Large Glass, and in 1957, the artist and his wife Teeny were reading Kasner and Newman’s Mathematics and the Imagination, which was in its four[1]teenth edition.54 Duchamp must have been delighted by the au thors’ praise for the spatial fourth dimension (no “greater forward step”), as quoted earlier. And with stirrings of renewed interest in the idea during the later 1950s and ’60s, including in Martin Gard[1]ner’s Scientific American columns, Duchamp clearly decided that his playful musings on four-dimensional geometry might once again be intelligible and decided to publish them.55

Science fiction was one of the contexts in which the spatial fourth dimension had survived, and, recast as the “fifth dimension” (because time was now so widely linked to the fourth dimension), the idea achieved new exposure in fantasy literature (e.g., Madeleine L’Engle’s 1962 A Wrinkle in Time) and on television, beginning in 1959, in The Twilight Zone. There, Rod Serling’s memorable introduction touched upon many qualities earlier associated with the fourth dimension, in[1]cluding imagination. “There is a fifth dimension,” he intoned

‘’ beyond that which is known to man. It is a dimension as vast as space and as timeless as infinity. It is the middle ground between light and shadow, be[1]tween science and superstition, and it lies between the pit of man’s fears and the summit of his knowledge. It is the dimension of imagination. It is an area which we call . . . the Twilight Zone.56 ‘’

In his 1962 Profiles of the Future, Arthur Clarke recalled of the idea: “The fourth dimension has been out of fashion for quite a while: it was fashionable round the turn of the century, and perhaps it may come back into style some day.”57 That would certainly begin to happen subsequently during the 1960s, in the “space age” Clarke himself had foretold in his writings.

F or those artists who turned their attention to the spatial fourth dimension during the second half of the twentieth century, it was often an encounter with literature on the subject from the early years of the century that introduced them to the concept. This was true for Park Place Gallery artist Peter Forakis, who in 1957, while a student at the California School of Fine Arts, found copies of Bragdon’s Frozen Fountain of 1932 and Ouspensky’s Tertium Organum at an artist’s estate sale. In the age of Einstein, these books were akin to some kind of ancient wisdom that went against the grain of culture at large. During the 1960s, Forakis would go on to explore approaches to the fourth dimension in his geometrically oriented culpture, which also responded to Buckminster Fuller’s incorpora[1]tion of the idea into his “synergetic geometry.”58 Both Duchamp’s notes and Fuller’s ideas were important for Robert Smithson, for whom the spatial fourth dimension was a central concern during the latter half of the 1960s, and for whom mirrors and spirals were key signifiers of the idea.59

TONY ROBBIN: FOUR DIMENSIONAL ART GROUNDED IN MATHEMATICS AND PHYSICS

The later twentieth-century artist who has actually engaged four[1]dimensional geometry most fully—in the tradition of Duchamp though seriously, not playfully—is Tony Robbin. Robbin arrived in New York from graduate school at Yale University in 1969, two years after the Park Place Gallery had closed its doors. But in an art world dominated by minimalism and critic Clement Greenberg’s dogma of flatness in painting, space, in general, was not a topic of artistic dis[1]cussion, and he never heard anything of the Park Place artists’ inter[1]est in the fourth dimension.60 Robbin’s paintings of the early 1970s are considered part of the pattern and decoration movement, but he was particularly interested in the disjunctions between contrasting areas of subtly colored patterns in his works. In a text accompanying Robbin’s exhibit at the Whitney Museum of American Art in 1974, curator Marcia Tucker wrote that the “contradictory visual informa[1]tion” in Robbin’s paintings “suggests the complexity of four-dimen[1]sional geometry.”61 And in a scenario reminiscent of Forakis’s find, Robbin made contact with a mathematics professor at Trenton State College where he was teaching, and into his hands came a cache of sources on four-dimensional geometry and space, including early twentieth-century books by H. P. Manning and Duncan Sommerville, as well as Robert Marks’s Space, Time, and the New Mathematics. Additionally discovering Coxeter’s Regular Polytopes, Robbin was launched on his trajectory to become the most serious artist-scholar in four-dimensional geometry of the twentieth century.6

From that point, Robbin undertook the serious study of four[1]dimensional geometry, physics, and computer programming that would support his creation of works such as his twenty-seven-foot painting of 1980–1981, Fourfield (fig. 9), and the publication of his first book, Fourfield: Computers, Art, and the Fourth Dimension, in 1992. In his quest to convey the complexity of four-dimensional space as projected in three dimensions, it was Thomas Banchoff’s rendering of the four-dimensional, planar rotations of the hyper[1]cube in his 1978 film The Hypercube: Projections and Slicings that held the key. Combining his exquisite sense of color with sophisticated mathematical principles, Robbin has created a remarkable body of work over more than thirty years. In Fourfield, for example, he painted a richly textural, colored background of Necker-reversing, four- and six-sided figures. To this mutating ground he then added painted lines and three-dimensional rods extending from the can[1]vas surface, representing pairs of isometric projections of the eight bounding cubes of the hypercube in slightly altered positions. As a viewer walks from one end of Fourfield to the other, the painted lines and white metal rods, both shadows of the hypercube, shift and mu[1]tate, mimicking the distortions that occur in Banchoff’s projections of the hypercube’s planar rotation in four-dimensional space.63

As documented in his book Fourfield and his 2006 book Shadows of Reality, Robbin has worked over the years in close consultation with a number of mathematicians and physicists. In doing so, he has gained a level of expertise far beyond that of other artists and is recognized for his contributions in mathematics as well as for the engineering applications set forth in his 1996 book Engineering a New Architecture. 64 Robbin’s art continued to develop in new directions in tandem with his explorations in mathematics and physics, including wire sculpture reliefs illuminated by colored light and, subsequently, works grounded in the principles of quasi-crystal geometry. More recently, Robbin has returned to painting in a rich and sensuous pal[1]ette, combining mathematical structures with painterly execution (fig. 10).65 Having been in dialogue with topologist Scott Carter for the last several years, he now conceives of these paintings as four[1]dimensional knot diagrams—with three-dimensional lattices, com[1]posed of the polyhedra associated with quasi-crystals, interweaving with one another. Carter has likewise credited his seeing one of Rob[1]bin’s wire-rod paintings in the 1980s with helping him approach a problem in topology, and the more recent collaborations of the two are supporting Carter’s further topological investigations.66

As Robbin wrote in 2007, “[t]he artist using mathematical ideas should not merely illustrate them; mathematical models are to art as medical illustrations are to the work of Rembrandt. The goal is to see the higher-dimensional space, to get the feeling of being inside them, and to revel in their liberating possibilities.”67 Thirty years earlier, in 1977, he had declared in an article on “The New Art of 4-Dimensional Space”:

‘’ We are not in the least surprised . . . to find physicists and mathematicians work[1]ing simultaneously on a metaphor for space in which paradoxical three[1]dimensional experiences are resolved only by a four-dimensional space. Our read[1]ing of the history of culture has shown us that in the development of new metaphors for space artists, physicists, and mathematicians are usually in step.68 ‘’

Soon after Robbin wrote this, the field of computer graphics and the personal computer emerged as powerful new tools to stimulate the visual imaginations of mathematicians and artists alike. Yet whether by means of the computer or not, four-dimensional geometry and the multifaceted, popular fourth dimension have served as key sources for artists in the twentieth and now in the twenty-first century. Al[1]most a hundred years ago, Malevich’s friend Matyushin pointed to the centrality of space to the activity of the artist: “Artists have always been knights, poets, prophets of space in all eras.”69 The subsequent development of art proved Matyushin himself to be prophetic.

1 . A. G. Ingalls, “Hypergeometry and Hyperperplexity,” Scientific American 161 (1939): 131. For the essay in question, see Ralph Milne Farley, “Visualizing Hyperspace,” Scien[1]tific American 160 (1939): 148–149.

2. David Hilbert, “Introduction” to D. Hilbert and S. Cohn-Vossen, Geometry and the Imagination, trans. P. Nemenyi (New York: Chelsea Publishing, 1952), p. iii.

3. See ibid., secs. 23, 44–51 (chap. 6). Di Suvero noted his interest in the book in a telephone interview with the author on May 2, 2002. On the Park Place Gallery artists nd their interest in topology and the fourth dimension, see Linda Dalrymple Hender[1]son, “Park Place: Its Art and History,” in Reimagining Space: The Park Place Gallery in 1960s New York (Austin: Blanton Museum of Art, University of Texas, 2008), pp. 8–11, 14–15, 20–24.

4. H. S. M. Coxeter, Regular Polytopes, 2nd ed. (1963; reprint, New York: Dover Publica[1]tions, 1973), p. vi.

5. See Linda Dalrymple Henderson, The Fourth Dimension and Non-Euclidean Geometry in Modern Art (1983; new ed., Cambridge, MA: MIT Press, 2010), chap. 1, as well as ap[1]pendix B for a sampling of popular articles; see also Claude Bragdon, A Primer of Higher Space (The Fourth Dimension) (Rochester, NY: Manas Press, 1913).

6. See Henry P. Manning, ed., The Fourth Dimension Simply Explained (1910; reprint, New York: Dover Publications, 1960)

7 . On Einstein’s theories and their reception, see, for example, Helge Kragh, Quantum Generations: A History of Physics in the Twentieth Century (Princeton, NJ: Princeton Uni[1]versity Press, 1999).

8 For this history, see Henderson, “Reintroduction: The Fourth Dimension Through the Twentieth Century,” in Fourth Dimension (above, n. 5).

. Edward Kasner and James Newman, Mathematics and the Imagination (New York: Si[1]mon & Schuster, 1940), pp. 119, 131.

10. Edwin A. Abbott, Flatland: A Romance of Many Dimensions by a Square (Oxford: Basil Blackwell, 1950).

11. Casius J. Keyser, “Mathematical Emancipations: The Passing of the Point and the Number Three: Dimensionality and Hyperspace,” Monist 16 (1906): 83.

12. Henry Parker Manning, Geometry of Four Dimensions (1914; reprint, New York: Do[1]ver Publications, 1956), pp. 15–16.

13. Charles Howard Hinton, A New Era of Thought (London: Swan Sonnenschein, 1888), pp. 6–7. For a summary of Hinton’s ideas and the concept I termed “hyperspace phi[1]losophy,” see Henderson, Fourth Dimension (above, n. 5), pp. 26–31; see also “Reintro[1]duction” (above, n. 8).

14. See Charles Howard Hinton, The Fourth Dimension (London: Swan Sonnenschein, 1904).

15. For the “system” quote, see Hinton, New Era (above, n. 13), pp. 86–87. On Hinton’s Scientific Romances, which were issued by his publisher Swan Sonnenschein in London, see Bruce Clarke’s highly insightful discussions of Hinton, idealist philosophy, thermo[1]dynamics, and the ether in Energy Forms: Allegory and Science in the Era of Classical Thermodynamics (Ann Arbor: University of Michigan Press, 2001), pp. 28–30, 111–121, 175–178.

16. See Henderson, “Reintroduction” (above, n. 8), for this publishing history and a discussion of the impact of Hinton’s writings as greater than I had realized in 1983.

17. See P. D. Ouspensky, Tertium Organum: The Third Canon of Thought, a Key to the Engimas of the World, trans. Claude Bragdon and Nicholas Bessaraboff (New York: Alfred A. Knopf, 1922).

18. Bertrand Russell, “New Books. The Fourth Dimension. By Charles Howard Hinton,” Mind 13 (1904): 573–574.

19. For this science, including X-rays, see Henderson, “Editor’s Introduction: II. Cub[1]ism, Futurism, and Ether Physics in the Early Twentieth Century,” Science in Context 17 (2004): 445–466

20. For the measure of the impact of the X-ray, see Lawrence Badash, Radioactivity in America: Growth and Decay of a Science (Baltimore: Johns Hopkins University Press, 1979), p. 9.

21. See again, for example, Henderson, “Editor’s Introduction: II” (above, n. 19); and Linda Dalrymple Henderson, “Modernism and Science,” in Modernism, ed. Astradur Eysteinsson and Vivian Liska (Amsterdam: John Benjamins Publishing, 2007), pp. 383– 403.

22. See, for example, Oliver Lodge, “Electric Theory of Matter,” Harper’s Monthly Maga[1]zine 109 (1904): 383–389.

23. See Herschel Chipp, ed., Theories of Modern Art: A Source Book by Artists and Critics (Berkeley: University of California Press, 1968), p. 223n1. Apart from Duchamp, Gris was the most mathematically oriented of the cubists; see William Camfield, “Juan Gris and the Golden Section,” Art Bulletin 47 (1965): 128–134.

24. For a useful introduction to cubism, see Mark Antliff and Patricia Leighten, Cubism and Culture (New York: Thames & Hudson, 2001).

25. Guillaume Apollinaire, “La Peinture nouvelle: Notes d’art,” Les Soirées de Paris 3 (1912): 90–91. Apollinaire slightly reworded his discussion of the fourth dimension in his 1913 Les Peintres Cubistes; see note following for the “language of the studios” reference in that context, as well as Henderson, Fourth Dimension (above, n. 5), pp. 74–81, where these texts are analyzed. For Picasso’s statement, see Ramón Gómez de la Serna, “Completa y verídica historia de Picasso y el cubismo,” Revista de Occidente 25 (1929): 100.

. Guillaume Apollinaire, The Cubist Painters: Aesthetic Meditations, ed. Robert Moth[1]erwell and trans. Lionel Abel, in The Documents of Modern Art series (New York: Wit[1]tenborn, 1944), p. 30; for his section on the fourth dimension, see p. 12.

27. Linda Dalrymple Henderson, “A New Facet of Cubism: ‘The Fourth Dimension’ and ‘Non-Euclidean Geometry’ Reinterpreted,” Art Quarterly 34 (1971): 410–433; see also [sprit Pascal] Jouffret, Traité élementaire de géométrie à quatre dimensions (Paris: Gauth[1]ier-Villars, 1903).

28. See Tony Robbin, Shadows of Reality: The Fourth Dimension in Relativity, Cubism, and Modern Thought (New Haven, CT: Yale University Press, 2006), pp. 30–33. Arthur Miller rightly connects Picasso to Poincaré versus Einstein in Einstein, Picasso: Space, Time and the Beauty That Causes Havoc (New York: Basic Books, 2001), but loses sight of Picasso’s artistic context, reducing him to Princet’s willing geometry student. For further discus[1]sion of Miller’s book and others, see Linda Dalrymple Henderson, “Four-Dimensional Space or Space-Time?: The Emergence of the Cubism-Relativity Myth in New York in the 1940s,” in The Visual Mind II, ed. Michele Emmer (Cambridge, MA: MIT Press, 2005), pp. 384–386n16. On the earliest usages of the term “fourth dimension” in Pari[1]sian art circles, which was not specifically geometric, see Henderson, “Reintroduction” (above, n. 8).

29. Robbin, Shadows (above, n. 28), p. 33.

30. Henri Poincaré, La Science et l’hypothèse (Paris: Ernest Flammarion, 1902), pp. 89–90. On the debt of Metzinger and his fellow artist-author Albert Gleizes to Poincaré’s ideas on tactile and motor sensations, including his assertion that “[m]otor space would have as many dimensions as we have muscles,” see Henderson, Fourth Dimension (above, n. 5), pp. 81–85.

31. See Henderson, “Four-Dimensional Space or Space-Time?” (above, n. 28), for the rise of the cubism–relativity myth. For a sampling of articles written on the supposed cub[1]ism–relativity connection, see Henderson, Fourth Dimension (above, n. 5), appendix A.

32. On this subject, see Henderson, “Editor’s Introduction: II” (above, n. 19), and “Modernism and Science” (above, n. 21); see also Henderson, “Reintroduction” (above, n. 8), which addresses Balfour Stewart and P. G. Tait’s The Unseen Universe (1875), the first source to link the ether to the fourth dimension.

33. Maurice Boucher, Essai sur l’hyperespace: Le Temps, la matière, et l’énergie (Paris: Félix Alcan, 1903), p. 64; see also Jean Metzinger, Le Cubisme était né (Chambéry: Editions Présence, 1972), p. 43.

34. For a fuller discussion of the Russian avant-garde and the fourth dimension, see Henderson, Fourth Dimension (above, n. 5), chap. 5; for an excellent study of Malevich’s art, see Charlotte Douglas, Kazimir Malevich (New York: Harry N. Abrams, 1994).

35. M. Matyushin, as quoted in Larissa A. Zhadova, Malevich: Suprematism and Revolu[1]tion in Russian Art 1910–1930 (London: Thames & Hudson, 1982), p. 32. In Ouspen[1]sky’s 1914 revised edition of Tertium Organum, the Russians would have heard briefly about Minkowski’s four-dimensional space-time continuum, since Ouspensky quoted from a 1911 lecture by physicist N. A. Umov on the subject. However, Ouspensky also critiqued Umov for failing to embrace his belief that time and motion were illusions that would fade away with the advent of higher-dimensional consciousness; see Ous[1]pensky, Tertium Organum (above, n. 17), chap. 11.

36. For an overview of Ouspensky’s philosophy, see Henderson, Fourth Dimension (above, n. 5), pp. 245–255. Initially, his advocacy of the practice of alogical logic in order to achieve higher, four-dimensional consciousness had supported Kruchenykh’s creation of his transrational zaum language in 1913 and Malevich’s alogist style of painting during 1913–1914; see ibid., pp. 269–279.

37. Hinton wrote: “All attempts to visualize a fourth dimension are futile. It must be connected with a time experience in three space” (ibid., p. 207). For Ouspensky’s dis[1]cussion of this issue, see Tertium Organum (above, n. 17), chap. 4.

38. Matyushin diary entry, May 29, 1915; see Henderson, Fourth Dimension (above, n. 5), p. 284n173.

39. Malevich, “Non-Objective Creation and Suprematism” (1919), in K. S. Malevich: Essays on Art 1915–1933, 2 vols., ed. Troels Andersen (Copenhagen: Borgen, 1971), p. 1:122. For Ouspensky’s discussion on infinity and cosmic consciousness, see Tertium Organum (above, n. 17), chap. 20.

0. The discussion of the Large Glass that follows is drawn from Linda Dalrymple Hen[1]derson, Duchamp in Context: Science and Technology in the Large Glass and Related Works (Princeton, NJ: Princeton University Press, 1998); for an overview, see Henderson, “The Large Glass Seen Anew: Reflections of Contemporary Science and Technology in Marcel Duchamp’s ‘Hilarious Picture,’” Leonardo 32:2 (1999): 113–126. Duchamp’s engage[1]ment with the fourth dimension (sans science) is the topic of a chapter in Henderson, Fourth Dimension (above, n. 5). The best general introduction to the artist is Dawn Ades, Neil Cox, and David Hopkins, Marcel Duchamp (London: Thames & Hudson, 1999).

41. Duchamp, as quoted in James Johnson Sweeney’s 1946 interview, Bulletin of the Museum of Modern Art; reprinted in The Writings of Marcel Duchamp, ed. Michel Sanouil[1]let and Elmer Peterson (1973; reprint, New York: Da Capo Press, 1988), p. 125.

42. For Duchamp’s rejection of “retinal” art in favor of “gray matter,” see James John[1]son Sweeney’s 1956 NBC television interview with Duchamp, reprinted in ibid., p. 136. For Duchamp’s notes published during his lifetime, see ibid.; for the preparatory notes discovered after his death, see Marcel Duchamp: Notes, ed. and trans. Paul Matisse (Bos[1]ton: G. K. Hall, 1983). These unpublished notes are particularly rich in scientific con[1]tent and are analyzed in detail in Henderson, Duchamp in Context (above, n. 40).

43. On the Large Glass and allegory, see Linda Dalrymple Henderson, “Etherial Bride and Mechanical Bachelors: Science and Allegory in Marcel Duchamp’s ‘Large Glass,’” Configurations 4 (1996): 91–120, and Duchamp in Context (above, n. 40), chap. 12.

44. For “splendid vibrations,” see Duchamp, The Green Box, in Writings (above, n. 41), p. 42; for these aspects of the Large Glass, see Henderson, “The Large Glass Seen Anew” (above, n. 40).

45. For “hilarious picture,” see Duchamp, The Green Box, in Writings (above, n. 41), p. 30.

46. See Duchamp, A l’infinitif, in ibid., pp. 84–101. For an overview of these notes, see Henderson, Fourth Dimension (above, n. 5), chap. 3. Craig Adcock has made the most extensive study of these particular notes, in Marcel Duchamp’s Notes from the “Large Glass”: An N-Dimensional Analysis (Ann Arbor, MI: UMI Research Press, 1983).

47. Duchamp, A l’infinitif, in Writings (above, n. 41), p. 89. The note continues: “(see Jouffret, Géom. à 4 dim., page 186, last 3 lines.).”

48. Duchamp, The Green Box, in ibid., pp. 44–45.

49. On Duchamp’s embrace of “logic and close mathematical thinking,” which he as[1]sociated with Cartesianism, see Henderson, Duchamp in Context (above, n. 40), pp. 77, 269n59; for Boucher’s advocacy of reason, see, for example, Boucher, Essai (above, n. 33), pp. 144, 170.

50. For the Juggler, see Duchamp, The Green Box, in Writings (above, n. 41), p. 65; for Jean Suquet’s drawing that superimposes the Juggler onto the Large Glass, see Henderson, “Etherial Bride” (above, n. 43), as well as Duchamp in Context (above, n. 40), fig. 111.

51. See Hinton, Fourth Dimension (above, n. 14), p. 27; and Henderson, Fourth Dimen[1]sion (above, n. 5), fig. 32.

52. On spirals or mirrors and the fourth dimension, see Henderson, Fourth Dimension (above, n. 5), index; on Lewis Carroll, pseudonym of mathematician Charles Dodgson, see ibid., pp. 21–22.

53. On this development as well as on Moholy-Nagy, see Linda Dalrymple Henderson, “Einstein and 20th-Century Art: A Romance of Many Dimensions,” in Einstein for the 21st Century: His Legacy in Science, Art, and Modern Culture, ed. Peter L. Galison, Gerald Holton, and Silvan S. Schweber (Princeton, NJ: Princeton University Press, 2007), pp. 101–129.

54. Jacqueline Monnier (Duchamp’s stepdaughter), letter to author, July 20, 2001.

55. For an “archeology” of the traces of the fourth dimension as they emerged during the 1960s, including Gardner, see Henderson, “Reintroduction” (above, n. 8).

56. See Marc Scott Zicree, The Twilight Zone Companion (New York: Bantam Books, 1992), p. 31.

57. Arthur C. Clarke, Profiles of the Future (New York: Harper & Row, 1962), p. 78.

58. See Henderson, “Einstein and 20th-Century Art” (above, n. 53), pp. 124–125; for a fuller discussion of Forakis, see Henderson, “Park Place” (above, n. 3).

59. On Smithson and the fourth dimension, see Henderson, “Reintroduction” (above, n. 8); for a concise version, see Linda Dalrymple Henderson, “Space, Time, and Space[1]Time: Changing Identities of the Fourth Dimension in 20th-Century Art,” in Measure of Time, ed. Lucinda Barnes (Berkeley: Berkeley Art Museum and Pacific Film Archive, 2007), pp. 95–99.

60. For Robbin’s early work and history, see Fourfield: Computers, Art, and the Fourth Dimension (Boston: Bullfinch Press, 1992); on the critical views that militated against artists’ interest in space during the later 1960s, see Henderson, “Park Place” (above, n. 3), pp. 35–41.

61. Marcia Tucker, Tony Robbin (New York: Whitney Museum of American Art, 1974). Tucker, who knew the Park Place artists, may well have encountered the idea of the spatial fourth dimension there.

62. Tony Robbin, e-mail message to author, October 12, 2003.

63. Banchoff’s film, which he showed around the world, accompanying it with lectures, was highly influential in spreading news of the spatial fourth dimension. Robbin discusses his work with Banchoff in Fourfield (above, n. 60), and addresses “The Computer Revolution in Four-Dimensional Geometry” in chapter 10 of his Shadows of Reality (above, n. 28).

64. Shadows of Reality (above, n. 28) contains a passionate argument for the model of projections versus slicing as a way to understand dimensional relationships in mathe[1]matics and physics; for example, Robbin makes a close rereading of Minkowski’s papers of 1907 and 1908 that interprets the space-time continuum as a geometry of projec[1]tion, rather than the common interpretation of a slicing of worldliness (chap. 4). His chapter 1 includes an unprecedented history of early techniques for rendering four[1]dimensional objects.

65. For the various phases of Robbin’s development, see Fourfield (above, n. 60) and Shadows of Reality (above, n. 28); he recounts his interactions with mathematicians and physicists in both Fourfield (numerous sidebars) and Shadows. Among the exchanges discussed in the latter work is that with quantum physicist P. K. Aravind, for whom four-dimensional projective geometrical figures have become important to his research on particle entanglement (pp. 85–92).

66. Scott Carter, e-mail message to author, August 3, 2004. According to Carter, “Tony’s painting spoke directly to me since I had seen glimpses of 4-space in my own research. He had escaped the plane of the canvas in order to explain escaping the plane of the 3-dimensional world.”

67. Tony Robbin, unpublished statement (2007).

68. Tony Robbin, “The New Art of 4-Dimensional Space: Spatial Complexity in Recent New York Work,” Artscribe 9 (1977): 20.

69. M. V. Matyushin, “Of the Book by Gleizes and Metzinger, Du Cubisme,” Union of Youth 3 (1913): 25, reprinted in Henderson, Fourth Dimension (above, n. 5), appendix C.

ROY LICHTENSTEIN 1923 – 1997

MODERN ART I 1996

Screenprint on Paper

Dimensions: Object: 1302 × 962 mm
Frame: 1412 × 1078 × 46 mm

Modern Art I 1996 was published by Gemini G.E.L. in Los Angeles and is recorded in the second volume of the catalogue raisonné of the artist’s prints (Corlett and Fine 2002). It is part of a series of Modern Art prints that Lichtenstein made during 1996, the year before he died. Another example, Modern Art II 1996 (Tate AL00382), is also in Tate’s collection. In Modern Art I Lichtenstein explored the refracted style of cubism found in the work of modern masters such as Pablo Picasso. However, in Lichtenstein’s hands, with his use of Benday dots, blocks of colour and stark black outlines, the dislocated features are transformed into highly stylised imagery more common to slick cartoons and comic books. In Modern Art II Lichtenstein explored the cubist style further, in particular the dislocated figurative imagery of Picasso. The nose, with its starkly painted striations, suggests a reference to the imagery Picasso used in paintings such as Les Demoiselles d’Avignon 1907 (Museum of Modern Art, New York). Tate’s copy of Modern Art I is number three of ten artist’s proofs aside from the edition of fifty.

From the early 1960s Lichtenstein made works that focused on the work of modern masters, such as the painting Woman with Flowered Hat 1963. Speaking about this painting, Lichtenstein explained:

Instead of using subject matter that was considered vernacular, or everyday, I used subject matter that was celebrated as art. What I wanted to express wasn’t that Picasso was known and therefore commonplace. Nobody thought of Picasso as common. What I am painting is a kind of Picasso done the way a cartoonist might do it, or the way it might be described to you, so it loses the subtleties of Picasso, but it takes on other characteristics: the Picasso is converted to my pseudo-cartoon style and takes on a character of its own. Artists have often converted the work of other artists to their own style.
(Roy Lichtenstein, ‘A Review of My Work Since 1961’, in Bader 2009, p.61.)

Elsewhere, Lichtenstein noted:

I’ve always been interested in Matisse but maybe a little more interested in Picasso. But they are both overwhelming influences on everyone, really. Whether one tries to be like them or tries not to be like them, they’re always there as presences to be dealt with. They’re just too formidable to have no interest. I think that somebody who pretends he’s not interested is not interested in art.
(Ibid., p.55.)

Lichtenstein was born in New York, and was a central player in American pop art. He came to prominence in the 1960s, making works based on imagery from comic strips, such as In the Car and Whaam! 1963 (Tate T00897). In these works he used the Benday dot, common to newspaper and magazine reproduction, to produce works that appeared mechanically reproduced, and which in fact are even more stylised than the cartoons Lichtenstein appropriated. Printmaking was an integral part of his practice throughout his career from the late 1950s through to the 1990s.

Further reading
Mary Lee Corlett and Ruth E. Fine, The Prints of Roy Lichtenstein: A Catalogue Raisonné: 1948–1997, New York and Washington D.C., revised and updated second edition 2002.
Graham Bader (ed.), OCTOBER Files 7: Roy Lichtenstein, Cambridge, Massachusetts and London 2009.
Gianni Mercuri (ed.), Roy Lichtenstein: Meditations on Art, Milan 2010.
James Rondeau and Sheena Wagstaff, Roy Lichtenstein: A Retrospective, exhibition catalogue, Tate Modern, London and Art Institute of Chicago 2012.

Lucy Askew
Senior Curator, Scottish National Gallery of Modern Art, Edinburgh
August 2014

Amended by Stephen Huyton
Assistant Collection Registrar, ARTIST ROOMS
September 2017

https://www.tate.org.uk/art/artworks/lichtenstein-modern-art-i-al00381

ROY LICHTENSTEIN 1923–1997

WHAAM! 1963

Acrylic Paint and Oil Paint on Canvas

Dimensions: Support: 1727 × 4064 mm
frame: 1747 × 4084 × 60 mm

ROY LICHTENSTEIN 1923–1997

BULL III - 1973

Lithograph, Screenprint and Line - Cut on Paper

Dimensions: Image: 685 × 890 mm

ROY LICHTENSTEIN 1923–1997

REFLECTIONS ON GIRL 1990

Lithograph, Screenprint on Paper and Metalised PVC on Paper

Dimensions: Object: 1146 × 1391 mm
Frame: 1302 × 1552 × 65 mm

ROY LICHTENSTEIN 1923–1997

REFLECTIONS ON THE SCREAM 1990

Lithograph, Screenprint, Woodcut on Paper and Metalised PVC on Paper

Dimensions: Object: 1238 × 1661 mm
Frame: 1396 × 1815 × 65 mm

FRANCIS BACON 1909 – 1992

PORTRAIT OF ISABEL RAWSTHORNE 1966

Oil Paint on Canvas

Dimensions: Support: 813 × 686 mm
Frame: 943 × 793 × 59 mm

FRANCIS BACON 1909–1992

THREE STUDIES FOR FIGURES AT THE BASE OF A CRUCIFIXION 1944

Oil Paint on 3 Boards

Dimensions: Support, each: 940 × 737 mm
Frame, Each: 1162 × 960 × 80 mm

FRANCIS BACON 1909 – 1992

SECOND VERSION OF TRIPTYCH 1944, 1988

Dimensions: Oil Paint and Acrylic Paint on 3 Canvases

Support, Each: 1980 × 1475 mm
Frame (Each): 2178 × 1668 × 100 mm

ROBERT MAPPLETHORPE 1946 - 1989

SELF PORTRAIT 1988

Photograph, Gelatin Silver Print on Paper

Dimensions: Support: 577 × 481 mm
Frame: 850 × 747 × 22 mm

ROBERT MAPPLETHORPE 1946 - 1989

SELF PORTRAIT 1985

Photograph, Gelatin Silver Print on Paper

Dimensions: Image: 384 × 386 mm
Frame: 684 × 662 × 31 mm

ROBERT MAPPLETHORPE 1946 -1989

LOUIS BOURGEOIS 1982, PRINTED 1991

Photograph, Gelatin Silver Print on Paper

Dimensions: Support: 375 × 374 mm
Frame: 645 × 620 × 38 mm

FERNAND LEGER 1881 – 1955

TWO WOMEN HOLDING FLOWERS 1954

Oil Paint on Canvas

Dimensions: Support: 972 × 1299 mm
Frame: 1100 × 1432 × 80 mm

Two Women Holding Flowers is a large oil on canvas painting by French artist Fernand Léger, made in 1954. Two stylised nude female forms, on occupy most of the picture plane. One sits while the other reclines, their forms intermingling and contorting, fitting neatly along the length and breadth of the canvas. It is unclear where one body ends and the other begins. Their limbs form a loose rectangle within the painting and both women gaze directly out at the viewer. The reclining woman holds a flower by its stem while the seated woman reaches out towards her companion. The figures have thick black outlines with touches of grey paint giving definition to an otherwise flat composition. Parts of their bodies, such as their breasts, faces and hair, have been delineated in a highly stylised manner. Flat, bold areas of red, blue, yellow and orange overlay parts of the figures. The background is plain white, apart from the inclusion of what may be a window at the right-hand edge of the image. Léger has signed and dated the painting in black paint in the bottom right corner.

It is likely that Léger painted this work in his studio in Gif-sur-Yvette, Chevreuse, to which he moved in 1952. It was produced by applying oil paint in decisive, bold brushstrokes. The contour lines of the figures have been painted over the flat areas of colour. Two earlier versions of this composition are known to exist, both from 1954: the first is a smaller oil painting showing some variation in colour, pattern and shading, and the second is a small gouache with a very similar composition to Two Women Holding Flowers, but a different arrangement of colours.

The traditional theme of the nude was a regular feature of Léger’s art and had played an important role in the cubist revival of neo-classicism spearheaded by Pablo Picasso after the First World War. Art historian Yve-Alain Bois has described how in the post-war period Léger, unlike many of his colleagues, ‘opted for the heroic–monumental genre’ (Foster, Krauss, Bois and others 2004, p.316). He often paired female nudes in an image in order to explore the rhythmic patterns of the body. An earlier example is The Two Sisters 1935 (Staatliche Museen Preussicher Kulturbesitz, Nationalgalerie, Berlin). Two Women Holding Flowers was produced at a time when Léger, a committed socialist and communist, was painting energetic images of builders, circus performers and lively scenes of modern life. Although Two Women Holding Flowers lacks a precise narrative, its boldly contorted figures can be regarded an equally modern exploration and celebration of shape and form.

The use of bright primary colours was, by 1954, an established feature of Léger’s work and his preoccupation with a ‘rigorously clear vision of forms and colours’ has been traced by art historian André Verdet to his La Femme en bleu 1912 (Kunstmuseum, Basel). Léger explained how his use of colour diverged from that of his close associate and founder of orphism, Robert Delaunay:

He continued the Impressionist tradition of juxtaposing complementary colours, red against green. I did not want to use two complementary colours together any more. I wanted to isolate the colours, to produce a very red red a very blue blue. If one places a yellow next to a blue, one immediately produces a complementary colour, green. Delaunay was moving in the direction of the modification of colour, while I strove to achieve clarity of colour and of mass and contrast.
(Quoted in Verdet 1970, p.14.)

Two Women Holding Flowers features clearly delineated areas of primary and secondary colours, which both enhance the black outlines of the figures but also exist outside these lines as forms in their own right, adding to the dynamic rhythm of the composition.

Two Women Holding Flowers was painted the year before Léger died – a time of international success and increasing commissions in the applied arts, most significantly his murals for the United Nations building in New York in 1952. A major retrospective of his work had been held at the Musée National d’Art Moderne, Paris, in 1949. Léger’s legacy has been widespread: his use of a bold graphic aesthetic, reduced forms and clear colours finds reference in the work of abstract artists such as Stuart Davis (1894–1964) and Frank Stella (b.1936) and pop artists such as Roy Lichtenstein (1923–1997).

Further reading
Andre Vérdet, Léger, London 1970.
Ronald Alley, Catalogue of the Tate Gallery’s Collection of Modern Art other than Works by British Artists, Tate Gallery and Sotheby Parke-Bernet, London 1981, p.420, reproduced p.420.
Hal Foster, Rosalind Krauss, Yve-Alain Bois and others, Art Since 1900: Modernism, Antimodernism, Postmodernism, London 2004.

Jo Kear
May 2016

Supported by Christie’s.

https://www.tate.org.uk/art/artworks/leger-two-women-holding-flowers-t00246

LOUIS BOURGEOIS1911–2010

MAMAN 1999

Steel and Marble

Dimensions: 9271 × 8915 × 10236 mm
Approx Weight: 3658kg

Collection: Tate

LOUIS BOURGEOIS1911–2010

MAMAN 1999

Steel and Marble

Dimensions: 9271 × 8915 × 10236 mm
Approx Weight: 3658kg

Collection: Tate

https://www.tate.org.uk/art/artworks/bourgeois-maman-t12625

AFTER STUART DAVIS 1894–1964

COMPOSITION 1964

Screenprint on Paper

Dimensions: Image: 282 × 360 mm

London/VAGA, New York 2021

CY TWOMBLY 1928–2011

QUATTRO STAGIONI: PRIMAVERA

Quattro Stagioni (A Painting in Four Parts)

Acrylic Paint, Oil Paint, Crayon and Graphite on Canvas

Dimensions: Support: 3132 × 1895 × 35 mm
Frame: 3230 × 1996 × 67 mm

CY TWOMBLY 1928–2011

QUATTRO STAGIONI: ESTATE 1993 - 1995

Quattro Stagioni (A Painting in Four Parts)

Acrylic Paint and Graphite on Canvas

Dimensions: Support: 3141 × 2152 × 35 mm
Frame: 3241 × 2250 × 67 mm

CY TWOMBLY 1928–2011

QUATTRO STAGIONI: AUTUNNO 1993 - 1995

Quattro Stagioni (A Painting in Four Parts)

Acrylic Paint, Oil Paint, Crayon and Graphite on Canvas

Dimensions: Support: 3136 × 2150 × 35 mm
Frame: 3230 × 2254 × 67 mm

CY TWOMBLY 1928–2011

QUATTRO STAGIONI: INVERNO 1993 - 1995

Quattro Stagioni (A Painting in Four Parts)

Acrylic Paint, Oil Paint and Graphite on Canvas

Dimensions: Support: 3135 × 2210 × 35 mm
Frame: 3229 × 2300 × 67 mm

Quattro Stagioni is a cycle of four paintings representing the four seasons. Tate’s version is the second of two cycles; the first is in the collection of the Museum of Modern Art, New York. Both cycles were begun in 1993 at Twombly’s studio in Bassano in Teverina (north of Rome) and completed in 1994 at another house owned by the artist in Gaeta on the Tyrrhenian Sea. Born and bred in Lexington, Virginia, Twombly was deeply influenced by Modern European art, particularly twentieth century European painting, and moved to Italy in 1957. Since that date he has worked in Rome and various locations in Italy and the United States as well as travelling widely around the Mediterranean. Throughout his career, Twombly’s paintings have been based on two components – line and paint. In such early works as Panorama 1955 (Daros Collection, Switzerland), a monotone grey canvas is covered in irregular chalk scribbles which hover on the verge of becoming recognisable as letters or ciphers. In the 1960s, daubs, smears and drips of colourful paint applied with a brush, the brush handle and the tips of the artist’s fingers begin to supersede the crayon and graphite marks of his earlier paintings. In some paintings, such as August Notes from Rome 1961 (Hirshorn Museum and Sculpture Garden, Smithsonian Institute, Washington DC), line is almost completely replaced by colourful patches of paint; in others, such as Leda and the Swan 1961 (collection the artist), it is a source of violent energy. Since the mid 1970s, the linear marks frequently take the form of text, introducing a third component: written language. Clumsy capitals or scrawled cursive letters are mixed with doodled shapes and indecipherable scribbles usually in compositional balance with painted elements. The tension between the graphic qualities of linear inscription and the sensual materiality of paint is central to the impact of the work. This runs parallel to a tension between intellectual cultural history and intuitive emotional expression enacted in Twombly’s paintings. Classical mythology, literature and historical works of art are appropriated and translated into a visual response which is tactile, visceral and aesthetic. His particular reference to Greek and Roman myths evokes an archaic symbolism, a subject he shares with the American Abstract Expressionists. A generation younger, he is further connected to this movement by his ‘gestural’ use of paint.

Twombly’s representations of the four seasons are typical to his production of the late 1980s and 1990s in which light has become a principal theme. His prominent use of white echoes that of French Impressionists such as Claude Monet (1840-1926) for whom it was an important ingredient in the depiction of light. A series of nine paintings, Untitled 1988 (Cy Twombly Gallery, Houston), portraying the green reflective surfaces of a watery pool, recalls Monet’s celebrated paintings of his water garden at Giverny, France created between 1899 and 1926. Plant life and the sea also recur in Twombly’s imagery of this period. A single work is frequently made up of several parts, as in Quattro Stagioni which is subtitled A Painting in Four Parts. The four seasons as symbols of the natural cycles of birth and death are a classical theme in poetry, music and painting. In Twombly’s Quattro Stagioni strong colours evoking the brilliance of the Mediterranean light are combined with scrawled poetic fragments from several sources. After pre-priming the canvases with cream-coloured gesso, the artist pinned them to the wall and applied individual colours, allowing the paint to dribble down in long, vertical lines. Inverno, or winter, represents the fourth season of the year. In this painting, the jagged forms made up of horizontal and vertical strokes which produced curved ‘boats’ in parts I and II of the cycle, Primavera (Tate T07887) and Estate (Tate T07888), are depicted in an altered state in black. Heavily painted over and blended with one another, they are virtually indistinguishable as discrete forms. On the right side of the painting, black boat shapes beginning at the centre expand upwards into a large black patch. This is balanced by a smaller black patch at the bottom left of the painting. Swathes of white and daubs of yellow have been mixed over the areas of black, breaking it up so that it evokes pine branches buffeted by rain. Marks made by the movement of the artist’s fingers and brush across the canvas in horizontal streaks has created a sense of sideways motion, echoing that made by horizontal strokes of red in Autunno (Tate T07889). Fragments of text and other marks on the cream canvas are covered by white paint. Several layers of this have been smeared over a large proportion of the canvas in a thin wash resulting in dribbles over much of the central area. Minimal blobs of light green in the centre and a patch of pale yellow on the right soften the harsh atmosphere of the image, which conveys a strong sense of winter’s harsh winds and bleak cold.

Further reading:
Heiner Bastian: Cy Twombly: Catalogue Raisonné of the Paintings, volume IV 1972-1995, Munich 1995, pp.34-5 and 178, reproduced p.179 in colour
Demosthenes Davvetas, Roberta Smith and Harald Szeemann, Cy Twombly: Paintings, Works on Paper, Sculpture, exhibition catalogue, Whitechapel Art Gallery, London and Städtische Kunsthalle, Düsseldorf 1987
Kirk Varnedoe, Cy Twombly: A Retrospective, exhibition catalogue, Museum of Modern Art, New York 1994, pp.162-5

Elizabeth Manchester
May 2003

https://www.tate.org.uk/art/artworks/twombly-quattro-stagioni-inverno-t07890

https://mymagicalattic.blogspot.com/2014/07/american-painter-cy-twombly.html

CY TWOMBLY 1928–2011

UNTITLED (BACCHUS) 2008

Acrylic Paint on Canvas

Dimensions: Support: 3175 × 4683 mm

CY TWOMBLY 1928–2011

UNTITLED (BACCHUS) 2008

Acrylic Paint on Canvas

Dimensions: Support: 3175 × 4737 mm

VICTOR VASARELY 1908–1997

NIVES II 1949 - 1958

Oil Paint on Canvas

Dimensions: Support: 1949 × 1143 mm

Presented by the Friends of the Tate Gallery 1961

VICTOR VASARELY 1908–1997

SUPERNOVAE 1959 - 1961

Oil Paint on Canvas

Dimensions: Support: 2419 × 1524 mm

Supernovae 1959–61 is a rectangular, vertically oriented black and white abstract painting by Hungarian-born artist Victor Vasarely. The work is composed of a network of 1,161 small black squares set inside a thin white vertical grid. At the top left of the composition, five rows down and five rows from the left, the black squares shift on their axis, becoming slightly larger and forming a clear black cruciform configuration. On the upper right side of the work the black squares become smaller, forming another internal white cross intersecting the grid on a diagonal axis. Occupying roughly the bottom two-thirds of the painting are two opposing vertical channels of small circles situated within white squares. The circles increase in size from left to right. A little above the mid-point of the work there is a singular horizontal bar that contains a row of floating black rhomboids. The work is inscribed ‘VASARELY | SUPERNOVAE | 152 x 242 | 1959–61’ on the back of the canvas.

The work forms part of a series made by the artist entitled Black and White in which he investigated the principles of geometry, perception and movement. Supernovae are stars which suddenly increase greatly in luminosity, then undergo various changes, including casting off a considerable proportion of their mass. Supernovae is designed to appear to visually alter as the viewer moves in front of it. The work plays optical tricks such as seeming to surge or retreat in areas, flip orientation and change in chromatic density depending on the spectator’s angle of vision. Motion, the artist explained in 1971, is not implied by depicting the object as moving. Rather, it is ‘the aggressiveness with which the structures strike the retina’ (Vasarely in Robert Sandelson Gallery 2005, p.9).

The Black and White series also draws heavily on Vasarely’s parallel interest in cinema and photography in the 1960s. At their simplest, these technical applications allowed Vasarely to reproduce quickly both the positive and negative versions of an image. In more advanced darkroom experiments they also permitted the artist to manipulate the tonal intensity and configuration of his images. Each work in the Black and White series is executed in both its positive and negative form. A ‘partner’ work to Supernovae appeared on the cover of the Swiss edition of Art International in May 1965.

Supernovae was originally conceived as a mural designed to be integrated into the external architecture of a building. In a letter to Tate in 1965 Vasarely described the strong relationship Supernovae has to architecture: ‘Like all of my cinematic compositions, the theme SUPERNOVAE is a “starting prototype” eminently integrable into architecture’ (translated from the French by the author, see Tate Archive TG/4/2/1058/1). Vasarely worked on numerous architectural schemes, notably at the University of Caracas with architect C.R. Villanueva and in Paris with J. Ginsberg.

Further Reading
Ronald Alley,Catalogue of the Tate Gallery’s Collection of Modern Art other than Works by British Artists, London 1981, p.745, reproduced p.745.
Vasarely in Black and White, exhibition catalogue, Robert Sandelson Gallery, London 2005, pp.6–13.

Judith Wilkinson
May 2016

Supported by Christie’s.

https://www.tate.org.uk/art/artworks/vasarely-supernovae-t00676

SIR EDUARDO PAOLOZZI 1924–2005

COVER FOR A JOURNAL

Screenprint on Paper

Dimensions: Image: 380 × 255 mm

Presented by Rose and Chris Prater through the Institute

of Contemporary Prints 1975

NAUM GABO 1890 – 1977

KINETIC CONSTRUCTION ( STANDING WAVE ) 1919-1920,REPLICA 1985

Metal, Wood and Electric Motor

Dimensions: Object: 616 × 241 × 190 mm

The Work of Naum Gabo © Nina & Graham Williams / Tate, London 2021

Kinetic Construction (Standing Wave) is a mechanical sculpture consisting of a plain steel rod emerging from a small black wooden base, now encased for protection in a clear acrylic box. When activated by the press of a button, the machine springs to life: through the rapid oscillations caused by a hidden electric motor in the base, it forms the illusion of a sinuously twisting, three-dimensional shape. The image generated through these movements, with its quivering transparency, is that of a ‘standing wave’: a term taken from the field of physics, familiar to Gabo through his studies in natural science and engineering.

Gabo and his brother, fellow artist Antoine Pevsner, had been inspired by the 1917 Russian revolution to move back from Europe to their native Russia. In Moscow Gabo was exposed to the fervid political and aesthetic debates of the post-revolutionary period and became closely acquainted with avant-garde artists such as Vladimir Tatlin and Kasimir Malevich. According to the artist, Kinetic Construction (Standing Wave) was initially produced for students in 1919–20 as a demonstration of the constructivist ideas expressed in his Realistic Manifesto (Gabo 1969, p.89). Written by Gabo and published jointly with Pevsner in August 1920, the manifesto proclaimed the need for art to connect with the political and industrial transformations of the era by establishing a more active relationship to space and time (reprinted in Gabo: The Constructive Idea: Sculpture, Drawings, Paintings, Monoprints, exhibition catalogue, South Bank Centre, London 1987, pp.52−4). The title alludes to the ‘kinetic rhythms’ advocated in the manifesto and the subtitle, Standing Wave, was introduced by the artist in around 1966.

It took Gabo almost three-quarters of a year to realise his concept. In the chaos of civil war, finding it difficult to source the basic machine parts that he needed, he conducted extensive experiments using salvaged materials in a mechanical workshop in the Polytechnicum Museum. The artist wrote in 1969: ‘It was done in a primitive way, but the only way I could have done it at that time, when conditions were such that looking for elaborate mechanisms was to search for a golden plate from the moon!’ (Gabo 1969, p.89).

Kinetic Construction (Standing Wave) was among Gabo’s earliest abstract works, a dramatic departure from the intersecting planes of the figurative works that he had been creating since 1915. As perhaps the first motorised sculpture (Natalia Sidlina, Naum Gabo, London 2012, p.52), it was a distinctive response to the non-objective forms and utopian ambitions of his avant-garde Russian contemporaries. Although Gabo often emphasised the work’s genesis as a demonstration model, it was exhibited as a work of art, appearing in Moscow in 1920 as well as in the landmark First Russian Art Exhibition (Erste russische Kunstausstellung) in Berlin in 1922. Frustrated by the technical difficulties of electrical constructions, Gabo did not pursue kinetic sculpture in the early 1920s, although he did explore the potential of newly developed plastics and continued to work with abstraction for the rest of his life.

Kinetic art experienced a remarkable international revival in the 1960s, and Gabo was among those early twentieth-century artists hailed as a pioneer by a generation exploring the aesthetic possibilities of machinery. Gabo donated Kinetic Construction (Standing Wave) to Tate following his successful Tate Gallery retrospective in 1966, where one critic described it as ‘the exquisite little kinetic rod of 1920, quivering with hummingbird delicacy (a Brancusi-like essence of kineticism which makes all subsequent efforts look mutton-fisted)’ (Nigel Gosling, ‘Structures in Space’, Observer, 20 March 1966, p.25). Because of its fragility many of the work’s components had, by then, been gradually replaced and its overall effect subtly altered. Archival records from the late 1960s and early 1970s show that in subsequent repairs and replicas, the artist was concerned not only to achieve a precise formal effect but also to maintain the ‘primitive’ mechanism of the 1920 motor, even though it is hidden from view. He also suggested adding a switch button to protect the delicate structure from wear, adding a new element of interactivity.

Further reading
Naum Gabo, ‘The Kinetic Construction of 1920’, Studio International, September 1969, vol.178, no.914, p.89.
Steven A. Nash and Jörn Merkert (eds.), Gabo: Sixty Years of Constructivism, exhibition catalogue, Dallas Museum of Art, Dallas 1985, pp.20−1, 205–6.
Martin Hammer and Christina Lodder, Constructing Modernity: The Art and Career of Naum Gabo, New Haven and London 2000, pp.69−72.

Hilary Floe
March 2016

Supported by Christie’s.

https://www.tate.org.uk/art/artworks/gabo-kinetic-construction-standing-wave-t00827

REBECCA HORN 1944

CONCERT FOR ANARCHY 1990

Piano, hydraulic rams and compressor

Dimensions: Unconfirmed: 1500 × 1060 × 1555 mm

A grand piano is suspended upside down from the ceiling by heavy wires attached to its legs. It hangs solidly yet precariously in mid-air, out of reach of a performer, high above the gallery floor.
A mechanism within the piano is timed to go off every two to three minutes, thrusting the keys out of the keyboard in a cacophonous shudder. The keys, ordinarily the point of tactile contact with the instrument, fan disarmingly out into space. At the same time, the piano’s lid falls open to reveal the instrument’s harp-like interior, the strings reverberating at random. This unexpected, violent act is followed between one and two minutes later by a retraction as the lid closes and the keys slide back into place, tunelessly creaking as they go. Over time, the piano repeats the cycle. A mounting tension to the moment of release is followed by a slow retreat to stasis as the piano closes itself up like a snail withdrawing into its shell.

Concert for Anarchy is one of a series of mechanised sculptures Horn began making in the late 1970s. Dancing tables, a suitcase trying in vain to climb a pole, startled hammers pecking against their reflections: machines that mimic the mechanisms of desire, they betray the longings beneath the surface of everyday things. Often erotically charged, these works express anthropomorphic anxiety and sensuality.

Horn’s work is truly interdisciplinary. In addition to sculptures and installations her practice has included performance, painting, writing and filmmaking. Before it was adapted to make Concert for Anarchy, this piano was used as a prop in her film Buster’s Bedroom, 1990 (Tate T11851) set in a California mental asylum. Horn has described how, ‘having freed itself from the psychiatric clinic, [the piano] is now composing its own music, developing a new tonality’ (Rebecca Horn: The Glance of Infinity, p.250).

Horn takes the piano away from its normal setting, right side up, in a concert hall or indeed sanatorium, and gives it the means to create its own discordant recital. Its performance is visual and aural and introduces an element of delay to the full experience of the artwork. Horn has stated her intention that this work should ‘trigger a new form of interaction with the visitors of an exhibition’ (Rebecca Horn: The Glance of Infinity, p.15).

Horn is one of a generation of German artists who came to international prominence in the 1980s. While her work is indebted to Surrealism, particularly Meret Oppenheim’s fetishistic objects, Horn’s desiring automatons provide a feminine response to both Marcel Duchamp’s bachelor machines (see The Bride Stripped Bare by her Bachelors, Even (The Large Glass), 1915-23, Tate T02011) and the kinetic art of Jean Tinguely (see Metamechanical Sculpture with Tripod, 1954, Tate T03823). There are also strong links between Horn’s work at that of Louise Bourgeois (see Cell (Eyes and Mirrors),1989-93, Tate T06899) who hints at a similar mix of eroticism and violence in her emotionally potent installations.

Further reading:
Germano Celant, Bruce W. Ferguson, Heiner Müller, Elizabeth A.T. Smith and Rebecca Horn, Rebecca Horn: Diving through Buster’s Bedroom, exhibition catalogue, Museum of Contemporary Art, Los Angeles 1990.
Guiliana Bruno, Germano Celant, Stuart Morgan, Katharina Schmidt, Nancy Spector and Rebecca Horn, Rebecca Horn, exhibition catalogue, Guggenheim Museum, New York 1993, reproduced no.74 in colour.
Carsten Ahrens, Lynne Cooke, Doris van Drathen, Bruce W. Ferguson, Carl Haenlein, Katharina Schmidt and Rebecca Horn, Rebecca Horn: The Glance of Infinity, Hanover 1997, reproduced no.135 (detail), 136, 140 and 142 in colour.

Rachel Taylor
July 2003

https://www.tate.org.uk/art/artworks/horn-concert-for-anarchy-t07517

JEAN DUBUFFET 1901 – 1985

VICISSITUDES 1977

Acrylic Paint on Paper and Canvas

Dimensions: Support: 2100 × 3390 mm
Frame: 2134 × 3424 × 65 mm