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The Fourth Dimension and Matta

The Fourth Dimension in Early Twentieth-century Art and Matta

full text from ©Linda Dalrymple Henderson

The idea of a possible higher geometrical dimension “the fourth dimension of space,” has been a vital stimulus for artists since the late nineteenth and early twentieth century. That notion, which suggested that our familiar world might be only a shadow or section of a higher reality, had grown out of the field of n-dimensional geometry in the later 1870s. From the 1880s to the 1920s the widespread popular fascination with such an invisible dimension of space is readily apparent in the vast number of articles and books published on this topic, such as architect Claude Bragdon’s 1913 A Primer of Higher Space (The Fourth Dimension).’ Two plates from Bragdon’s book are useful in setting forth two of the basic wars of conceptualizing a higher spatial dimension. The first (fig. 1) focuses on the generation of higher dimensional forms by moving a lower dimensional figure perpendicular to itself- from line to plane to cube to hypercube; those resultant figures can then also be imagined as folded down into a lower dimension. The second process (fig. 2) is that of sectioning or slicing a higher dimensional form as it passes through a space of one less dimension. In both approaches, reasoning by analogy to the relationship of two to three dimensions is central to imagining the transition from three to four dimensions.

Although the first popularization of the idea had occurred in English theologian E.A. Abbott’ s Flatland: A Romance of Many Dimensions by a Square of 1884, the individual who definitively extended the “fourth dimension” beyond its mathematical roots was the Englishman Charles Howard Hinton. In his books A New Era of Thought (1888) and The Fourth Dimension (1904), Hinton developed the philosophical implications of four-dimensional space and secured its place in late nineteenth- and early twentieth-century culture. Hinton’s “hyperspace philosophy” was an idealist worldview based on his belief that by developing an intuitive apprehension of four-dimensional space, individuals would gain access to true reality and hence resolve the problems of materialist three-dimensional world. Hinton’s method for “educating the space sense” of his readers was a 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 power and transcend self-oriented perception, such as the senses of left/right and up/down or gravity.

1.”The Development of Corresponding Figures in One-. Two-, Three-, and Four-Space ” from Claude Bragdon A Primer of Higher Space ( The fourth Dimension) (Rochester. NY, 1913), pl. 30

Although Hinton achieved little personal success of 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 Dimension, for example, was reprinted in London in 1906, 1912, 1921,1934, and 1951. Those who subsequently built upon and/or promulgated his ideas included Bragdon in the United States, mystic philosopher Peter Demianovich Ouspensky in Russia, German Theosophist/Anthroposophist Rudolf Steiner in Germany, both mathematicians (Esprit Pascal Jouffret and Maurice Boucher) and Theosophists in France, Svmbolist writer Maurice Maeterlinck in Belgium, and Theosophist C.W. Leadbeater in England.’ In his book Tertium Organum, first published in St. Petersburg in 1911, 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.’

Writers such as Hinton and Bragdon particular, had a major impact on the way the public imagined and imaged the fourth dimension 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.

2. “The Projections Made by a Cube traversing a Plane.” from Claude Bragdon A Primer of Higher Space ( The fourth Dimension) (Rochester. NY, 1913), pl. 30

3. Frontispiece from Charles Howard Hinton, The fourth Dimension (London, 1906)

For artists, whose visual imaginations had been largely constrained by painting’s traditional allegiance to the visible world, the possibility that space was actually four dimensional revolutionary. The chiaroscuro modeling techniques and one-point linear perspective painters had relied upon since the Renaissance to create convincing three-dimensional form and space were irrelevant if the world had four dimensions.

There was another strong impetus tor artists to question the visible world in the early twentieth century: 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 spectrum then being identified. That discovery undoubtedly contributed to the continued popular interest in the fourth dimension, which might otherwise have remained the province of mathematicians, philosophers, and mystics. Once the X-ray established the inadequacy of the human eye, who could deny with certainty 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 in the 1890s of the electron and of radioactivity as well as the interest in the Hertzian waves of wireless telegraphy contributed further to a radical reconception of the nature of matter and space in this period. Beyond its possible four-dimensionalitv, matter wastransparent to the X-ray and, on the model of radioactivity, was often discussed as dematerializing into the space around it. Moreover, that space was never thought of as empty in this period. 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 itself was in the “electric theory of matter.” Widely popularized, the 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 sensitive artists-the successor to the visionary seer posited by the Symbolists in the 1890s-who would be required to envision 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 in the early twentieth century -Cubism, Suprematism, and the art of Marcel Duchamp- and then turns to the painting of Roberto Sebastián Antonio Matta Echaurren, who would make the most important subsequent innovations in this tradition, beginning in the later 1930s in the context of Surrealism. 

Cubism: Windows on Invisible, Fluid Geometrical Complexity

The Cubist painter and theorist Jean Metzinger was the first artist to write about the importance of the new geometries for contemporary painters and he and Juan Gris are said to have studied four dimensional geometry with the insurance actuary Maurice Princet ” All three of these figures were close to Pablo Picasso, who along with Georges Braque developed the style that has come to be known as Analytical Cubism during 1909 (fig. 4).

4 .Pablo Picasso. Portrait of Ambroise Vollard Spring 1910 Oil on canvas. 93 x 66 cm Pushkin State Museum of Fine Arts, Moscow

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 mathematics. I 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.’ In his book Les Peintres Cubistes of 1913. the poet likewise dismissed perspective as “that miserable tricky perspective. that fourth dimension in reverse.

In the early 1970s. I suggested that plates from Jouffret’s 1903 book Traité élémentaire de géométrie à quatre dimensions, such as figure 5, would have confirmed Picasso’s stylistic direction. Here the geometer’s use of transparency, shifting overlays of differing views of an object, and the resulting spatial ambiguity are strikingly similar to Picasso’s approach. Like other Cubists, Picasso 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 trom view to view.

5.”Perspective cavalière of the sixteen fundamental octahedrons of the isotetrahedroid ” from E. Jouffret traité élémentaire de la géométrie à quatres dimensions (Paris,1903) fig 41

Poincaré had concluded: “in this sense we may say the fourth dimension is imaginable.

Just as Picasso’s Portrait of Vollard denies the possibility of a reading of three dimensions, it also effectively evokes the newest scientific ideas of matter and ether-filled space. Here the sitter and the space around him interpenetrate in an ambiguous relationship. Picasso’s Divisionist brushstrokes suggest the emission of particulate matter grounded in the idea of universal radioactivity, at the same

time matter may be cohering from the ether. Likewise, the model of the X-ray’s transparency produces the kind of clairvoyant, see-through vision of three-dimensional forms that would be accessible to four-dimensional sight. 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 im Modern Art was written before I had studied the late Victorian ether physics still prevalent in 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 Einsteinian 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 in that period -i.e., the space time world of Einstein.” But such debates over the supposed relationship of Picasso to Einstein also served to occlude study of the science to which Picasso, Duchamp, and others were responding in pre-World War I Paris. The recovery of that science has been critical to a fuller history of the impact 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 the fourth dimension’s possible relation to the ether itself.”

A case in point is the 1903 book by Maurice Boucher, Essai l’hyperespace: Le Temps, la matière et l’énergie, which Metzinger mentions in his memoirs. There Boucher discusses the ether extensively and argues in support of the fourth dimension: “Our senses, on the whole, give us only deformed images of real phenomena, which have long remained unknown, because none of our organs put us in direct contact with them.” The Russian avant-garde also knew Boucher’s book, as did, quite certainly, Duchamp. 

Malevich’s Suprematism: Sections Afloat in Infinite Space

If the Cubists created geometrically complex and ambiguous images that suggested the invisible reality beyond surface appearances, the abstract Suprematism of Malevich (fig. 6) utilized the method of sectioning to create geometrical planes moving in space.”‘ The two-dimensional analogs that lay behind Flatland and was illustrated in Bragdon’s Primer (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 Minkovsky provided the answer.”(In addition to Poincaré, Boucher, Hinton, and Minkowski, Matyushin here cites two major pioneers of the curved non-Euclidean geometries that would so interest Matta, 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, Ouspensky, 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 1909 The Fourth Dimension Ouspensky had criticized contemporary Russian artists for what he considered their wrong-headed approach to the fourth dimension.” Nonetheless, Ouspensky’s books. The Fourth Dimension and Tertium Organum. A Key to the Enigmas of the World of 1911, which provided a full accounting of Hinton’s ideas, were critical sources for Malevich and his colleagues “Matvushin and the poet Alexei Kruchenykh.

Most important for Malevich ‘s mature Suprematism, 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 Malevitch.

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 others bore the subtitles Color Masses in the Fourth Dimension or Color Masses in the Second Dimension (fig. 6). Malevich’s Suprematist paintings with planes of one color only strongly suggest the two-dimensional sections traces created when three-dimensional objects pass through a plane discussed in Hinton and Ouspensky and illustrated in Bragdon’s Primer of Higher Space (fig. 2). These “Color Masses in the Second Dimension” may have served Malevich as indirect signs of the fourth dimension of means of the wel-known two dimensional analogy.

6. Photograph of work by Malevich in The last Futurist Exhibition 0.10 Petrograd 1915

Malevich’s more typical Suprematist compositions (fig. 7), however, 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-dimensional form. Malevich sets two-dimensional planes 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: spatial end, as in its role in both the generation of higher dimensional forms (fig. 1) and their sectioning (fig. 2).” 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.”

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 referred specifically to the space of his Suprematist paintings as the “white, free chasm, infinity.” Fascinated by flight, Malevich does not however, paint his space blue. Instead, It is a cosmic white expanse in which variously colored elements float freely, without any specific left-right or up-down orientation, just as Hinton had argued that gaming 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 Cubisms 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 orientation, and implied motion.

7. Kazimir Malevich, Suprematism Oil on canvas. 80.5 x 81 cm State Russian Museum. St. Pétersburg

Marcel Duchamp: Playful Geometry and Other Signs of the Fourth Dimension 

Duchamp, who had begun his painting career in the context of Cubism. was dedicated to realizing aspects of four-dimensional space in his art, but both his approach and his result were far removed from Cubism and from Malevich’s Suprematism Duchamp’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. His sources on the fourth dimension included Matyushin’s entire list of names, quoted earlier, with the substitution 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.”- Disgusted by what he believed was the mindless. “retinal” painting of his fellow artists. Duchamp found in the fourth dimension a topic tied closely to mental activity 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 a: the object itself.”

8. Marcel Duchamp, The Bride Stripped Bare by Her Bachelors, Even (The large Glass).1915-1923 Oil, varnish, lead foil, lead wire and dust on two glass panels, 277.5 x 177.8 x 8.6 cm Philadelphia Museum of Art. Philadelphia, Bequest of Katherine S. Dreier

Without Duchamp’s notes, we would be hard-pressed to decipher the basic narrative of the The Bride Stripped Bare by Her Bachelors, Even as well as to appreciate the “playful physics” and geometry that underlie it. Basically, a series of operations begins at the left side of the Bachelors’ realm during which “illuminating gas” is gradually liquefied into a semen-like “erotic liquid,” which is ultimately splashed onto the upper half of the Glass to form the chance-determined “Nine Shots 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 Bachelors and the unreachable Bride, Duchamp drew on contemporary science as well as the four, to-three dimensional contrast between their realms.” Boucher’s Essai sur l’hyperespace would have been an especially relevant source for him, since it treated the fourth dimension in relation to contemporary ideas of 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-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 vin the ether. Bu contrast. the lass of classical mechanics, playfully “stretched” by Duchamp, rule the lower halt of the Glass, where the Bachelor are further constrained of perspective and the relentless pull of gravity.

Although Duchamp never published the comprehensive text he originally envisioned to accompany the Large Glass, his boxes of facsimiles of his notes, primarly 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 picture.” As discussed in the following section on Matta, the emergence of Einstein and Relativity Theory after 1919 would gradually bring an end to the widespread public interest in a spatial fourth dimension. Thus, it is not surprising that Duchamp chose not to include his extensive notes on the fourth dimension in the Green Box. But by the 1960s, the topic 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 explored other means by which he might make the Bride’s realm four dimensional. Duchamp’s notes and drawings offer highly inventive approaches to the topic, which, in the end, were unrealizable: nonetheless, his verbal invention in the notes stands as a significant counterpart to Glass itself.

Duchamp speculated extensively on four dimensional geometry, working by means of analogy and developing his own playful laws on the subject.” Although he considered for a time Poincare’s ideas on geometrical 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.’

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). » In her infinite, immeasurable realm of etherial vibrations 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.” ” A self proclamed Cartestan, his was an intellectual, albeit humorous, approach to the fourth dimension.

Matta: Envisioning Space-Time in the 1930s and 1940s and Beyond

Duchamp and the Large Glass project, along with the writings of Ouspensky, would be critical for the Chilean-born artist Roberto Matta as he developed his painting style, after he joined the Surrealist movement in 1937. Characteristic of Matta’s unique approach are works such as Morphologie Psychologique de L’angoisse (La Vielle de la Mort) of 1938 (cat. 8) and subsequent paintings such as Inscape of 1943 (fig. 11). Matta, however, came to his interest in complex space over twenty-five years after Duchamp had done so and in context in which Einstein had suddenly risen to prominence after a 1919 eclipse expedition that proved one of the postulates of his 1915 General Theory of Relativity.

9. Roberto Matta, Morphologie Psychologique 14, 1939 Colored pencils and graphite on paper, 33×49.8 cm Sandy Gross and Anne Conaway Collection

10. Roberto Matta, The Bachelors Twenty Years Later, 1943 Oil on canvas,  96.5 × 127 cm Philadelphia Museum of Art

11. Roberto Matta, Inscape, 1943 Oil on canvas, 55 x 66 cm Thomas Monahan Collection

“Space-time” was now the watchword. with the spatial fourth dimension having been redefined as time in the four-dimensional space-time continuum that Minkowski had proposed in 1908 for Einstein’s 1905 Special Theory of Relativity.

In the 1915 General Theory of Relativity, Einstein adopted the model of irregularly curved non-Euclidean geometry to explain effects of gravitation, enshrining such curved geometries, developed in the nineteenth century just as n-dimensional geometry had been, as an up-to-date feature of Relativity Theory. ” Matta would repeatedly affirm his interest in the non-Euclidean geometry of Einstein’s space-time world in later interviews, and he also asserted that “Einstein was as important as Freud for the modern artist.”

Paintings such as figures 9 and 10 make clear his new conception of space that leaves behind the rectilinear world of three-dimensional Renaissance perspective. Duchamp had likewise engaged the irregular curvature of non-Euclidean geometry in his 3 Standard Stoppages of 1913-1914. In this project, he had dropped 3 meter-length threads from the height of 1 meter onto narrow canvases to create three ironic, non-standard curved variations on the meter. The two artists interest would come together around non-euclidean geometry as well as the fourth dimension in the 1940s. (As noted earlier, Malevich’s colleague Matyushin had cited the pioneers of negatively and positively curved non-Euclidean geometries, Lobachevsky and Riemann, in a text of 1916.)

While non-Euclidean geometry rather than n-dimensional geometry was the nineteenth-century geometry that had gained new currency in the context of Relativity Theory in the 1920s and 1930s, there was, nonetheless, a specific reading of space-time that focused on the continuum as a four-dimensional geometrical (i.e, spatial) “block universe.” And this was the realm Surrealism’s founder, André Breton, evoked in a 1939 essay when he wrote of the voung new Surrealists, including Matta, and their “fundamental aspiration… to move beyond the universe of three dimensions.” He continued: “Although that was one of the leitmotifs of Cubism in its heroic period, it must be admitted that this question poses itself in a much more pointed manner since Einstein’s introduction of the notion of space-time into physics. The necessity of a suggestive representation of the four-dimensional universe asserts itself particularly in Matta (landscapes with several horizons) and in Onslow Ford.”

A crucail event in Matta’s education about the spatial fourth dimension occurred in the summer of 1938 when he spent time with his fellow voung Surrealist Gordon Onslow Ford in Brittany, reading, among other texts, Ouspensky’s Tertium Organum. As Onslow Ford recorded later of Ouspensky: “His conception of space-time, and his creatures conscious of different dimensions were inspiring …” With his goal of developing human psychology to the level of four-dimensional “cosmic consciousness,” Ouspensky had advocated throwing off the tyranny of Aristotelian logic by practicing a form of alogical logic in order to free the mind from the limitations of reason.” One can readily understand his appeal to the Surrealists who likewise rejected reason and logic in their exploration of Freudian psychology and pursuit of “sur-reality” fusing waking and dreaming life. In a statement suggestive of Matta ‘s paintings, Onslow Ford wrote in Minotaure in January 1939 of the need to

“see… beyond the rational,” beyond “dogmatic theories (that) are opaque” to a point where “all becomes transparent, floating, and controlled by forces which act from a domain not perceptible to our five senses.” Echoing Ouspensky, he concluded: “We can say that matter is only the misshapen shadow of reality.”

As Onslow Ford’s reference to Ouspensky’s “conception of space-time” suggests, the second edition of Tertium Organum (1916) had included a chapter in which Ouspensky reported on a 1911 Russian scientific congress in which physicist N.A.

Umov discussed Minkowski’s space-time continuum ” Although Ouspensky himself considered time and motion to be illusions produced by inadequate spatial understanding, he seems to have embraced the spatialization of time that occurred in Minkowski’s model, never dreaming that in the coming decades Relativity Theory would largely occlude public awareness of his beloved spatial fourth dimension. The immediate impact of Relativity Theory on artists in the 1920s was to encourage a focus on time itself in kinetic art, such as the works of Laszlo Moholy-Nagy, with whom Matta actually worked briefly in 1937. Matta’s interest, however, was not in time alone. but was much closer to the subtle fusion of space and time in Ouspensky that suggested growth in time might be understood as a manifestation of higher space. Ouspensky (and Bragdon before him) had argued that the passage from birth to death could be considered as a “four-dimensional body produced by “the infinite number of moments of existence of the three-dimensional one, analogous to the Hindu idea of the Linga-Sharîra.

This conception of process and growth would have resonated with Matta, given his deep interest in botanical growth and morphology as studied by figures such as Edouard Monod-Herzen’s in his Principes de Morphologie générale (1927), a book of considerable interest to the Surrealists. was both external morphology and internal wwpsychology morphology that fascinated Matta and to which he sought to give form in his paintings of the later 1930s and early 1940s. As he wrote in a text of 1938-1939

Time will be for us a medium comparable to a gelatinous shield that receives optically transformations which operate with a varying speed. The eye is adjusted only to a certain speed… I’ call psychological morphology the graph of transformation due to the absorption and emission of energies in the object from the initial aspect until its final form in the geodesical psychological medium …

He continued, comparing this “psychological time medium” to a “Euclidean space in rotary and pulsatory transformation, in which the object at each risk of interpenetration can oscillate from point-volume to moment-eternity, from attraction-repulsion to past-future, from light-shadow to matter-movement.”

Here Matta fuses inner psychology and outer space-time as well as suggesting the electromagnetic spectrum and the varying speeds of wave vibrations, of which only visible light is perceived by the human eye. That had been a central theme in early literature on the fourth dimension as well as on X-rays. Matta’s “gelatinous shield” even suggests a medium tor those wave vibrations. That medium had earlier been understood as the ether of space, and it was not uncommon for such language to be transferred to the space-time continuum, even though Einstein had dismissed a mechanical ether as irrelevant to his new physics. For example, in the 1927 reprint of Boucher’s 1903 book L’Hyperespace, discussed earlier, he argued that he had foretold Minkowski’s continuum by proposing in 1903 an “Espace Temps à 4 dimensions”; Boucher’s ” espace-temps, however, was filled with ether and infinite in its extent. A number of other French science books had continued to argue for the relevance of the ether in the 1920s. It is thus not surprising to observe Matta in the early 1940s expressing his

interest in a reality “made up of oscillations, waves, beams, and the world as a nexus of vibrations” at the same time he was writing to Breton that he was making “a comparative study of time and space.” Indeed, the catalog of his 1942 exhibition at the Pierre Matisse gallery reproduced a diagram of Hertzian waves (radio waves).”

One of the consistent themes in writing on Matta’s works of this period is the atmospheric quality of his paintings. In 1947, James Thrall Soby wrote, for example, that Matta seems to “reduce matter to blown vapor,” producing “vaporish landscapes” that he also describes as an “ethereal maze composed of “diaphanous screens “

William Rubin in 1957 used similar descriptors such as “vaporous,” “cosmic,” “diaphanous, and “vibration.” Matta’s unique approach to Surrealist “automatism” created such effects by techniques such as “rubbing of liquid color by a loosely held cloth,”

“as Rubin chronicles. Along with his interest in Yves Tanguy’s paintings, Soby notes Matta’s admiration for the paintings of Wassily Kandinsky, whose works such as his 1913 Composition VI have a similar fluidity that we now know was grounded specifically in his interest in

the ether and its vibrations.” Matta also admired the “Clavilux” of Thomas Wilfred, as Soby also mentions. Wilfred had been a colleague of Claude Bragdon, and his color organ produced amorphous patterns of moving color to which he gave titles such as “Multidimensional” or”Spacetime”, appropriate since the fourth dimension had been a goal in his and Bragdon’s early activities.” Matta would have had ample chances to observe Wilfred’s weekly “lumia” performances at his Art Institute of Light in the Grand Central Palace, where viewers were immersed in experiences of color free of any clear spatial orientation.”” Matta’s early works likewise deny three-dimensional space by their very ambiguity, so suggestive of the etherial fluidity of Cubist paintings and the dematerialized forms of Kandinsky.

To pursue the theme of the ether a bit further before turning to the next phase of Matta’s works the introduction of near elements defining non-Euclidean spaces- it is important to acknowledge Matta’s interest in the occult, where the ether had also played a prominent role.” Rubin in his 1957 catalogue notes Matta’s “deep affinity” for French occultist Eliphas Levi and quotes a passage from Levi’s discussion of the “astral light”: “There exists a mixed agent, natural and divine, corporeal and spiritual, a universal plastic mediator, a common receptacle of the vibrations of movements and the images of form …this universal agent of the works of nature is the astral light.” Theosophy’s founder Madame Helena Blavatsky in her foundational text Isis Unveiled of 1877 had compared the ether of space specifically to the “astral light” of Levi, and the ether as medium for vibrations was indeed a “universal plastic mediator.” akin to Matta’s “gelatinous shield.”* When Matta in 1943 painted his work The Bachelors Twenty Years After (fig. 10), it is as if in the upper left corner of that painting he created a landscape for the etherial, four-dimensional realm of the Bride in the Large Glass, discussed earlier.

Duchamp and Matta had met in 1938 in Paris, and it was the older artist who urged him to leave France for New York in October 1939. Once Duchamp arrived in New York in June 1942, the two became close friends and engaged in regular extended conversations. Matta was fascinated by Duchamp’s Large Glass: he owned copy of the Green Box notes and would co-author a 1944 text on the Glass with Katherine Dreier in 1944. The vounger artist was also intrigued by Duchamp’s painting The Passage from Virgin to Bride of 1912, in which he recognized a model of the morphological change to which he was so committed.” In Duchamp, Matta found an artist colleague deeply interested in both the spatial fourth dimension and non-Euclidean geometry, which now had a new currency in the context of Relativity Theory. As Duchamp wrote of Matta in his 1949 entry on the artist for the catalogue of the Société Anonyme Collection, he had “followed the modern physicist in the search for new space.

Non-Euclidean geometry was indeed a key sign of the physicist’s new space for Matta, and he regularly emphasized it in interviews and statements. As he tried to explain in 1965 to critic Max Kozloff of his painting The Vertigo of Eros (1944) : “The reference I was making once again, was to a non-Euclidian space, where all the ordinates and co-ordinates are moving in themselves, because the reference to the ‘wall,’ shall we say, of the space are constantly changing. They are not parallel to the Euclidean cube, to which most previous painting has referred.”• Kozloff’s response was a puzzled “I’m very confused,” and as distance from the 1940s and public discussion of curved space-time became less common, Matta would find fewer comprehending listeners.

But Duchamp understood what Matta was saying. Not only had he produced the 3 Standard Stoppages in 1913-1914, but he had also created a 1914 painting titled Network of Stoppages, which incorporated the variable curvilinear measures the Stoppages on its surface. This was the painting Duchamp displayed in the First Papers of Surrealism exhibition of 1942, in which Matta’s work was also included. For this exhibition Duchamp also created the installation known as Sixteen Miles of String. in which he strung about a mile of string (not sixteen) among the freestanding panels on which paintings hung, creating a curvilinear web that filled much of the space. Soby first mentioned this

installation in 1947 as a stimulus for Matta’s introduction of linear elements into his paintings as of 1943-1944, such as his 1944 The Vertigo of Eros; the same effect is apparent in Children’s Fear of Idols of 1943 (cat. 12).”* Indeed, in the 1942 exhibition Duchamp may well have been commenting on the notion of curved space-time as well as highlighting his own prescient interest in non-euclidian geometry by including his Network of Stoppages.

Against this backdrop, it is also possible that in the lower areas of his 1943 The Bachelors Twenty Years After Matta liberated the Bachelors from their three-dimensional, Euclidean realm in the Large Glass. Instead, they now inhabit an energy-filled, non-Euclidean field suggestive of curved space-time, where black planar elements are also torqued by forces.” Matta pointed up this contrast by juxtaposing his painting with a reproduction of the lower half of the Glass at the conclusion of his and Dreier’s 1944 text on Duchamp’s Glass.

Drawing had been a key activity for the young Matta as he left behind architectural practice and joined the Surrealist movement. Matta and his friend Onslow Ford, along with other Surrealists, were fascinated by the mathematical models at the Institut Poincaré and the Palais de la Découverte in Paris, many of which included linear elements, Similar curvilinear, often biomorphic forms fill Matta’s early drawings, such as Sans titre and Untitled, both 1939 (cat. 9). Such outlined forms also occasionally appear at small scale in paintings of 1940-1941, such as the 1941 Invasion of the Night, but that was to change as lines came to play a more prominent role for him. While Matta’s paintings from the later 1930s onward had denied three-dimensional readings primarily by their amorphous, etherial quality, he must have realized that to suggest the new kinds of complex spaces/space-time that interested him, it would be necessary to provide more overt spatial clues.

Beginning in 1941, more prominent black linear elements started to appear and then, in the wake of the First Papers of Surrealism installation, white lines would come to define multidimensional curvilinear complexity in many paintings of 1942-1944, including Children’s Fear of Idols of 1943.

Gradually, Matta’s lines also began to define transparent planes in works like Children’s Fear of Idols. In addition to such open planes, however, he also begun to utilize solid black planes in works such as Inscape of 1943 (fig. 11) and others. Here, instead of the white, web-like lines of previous works, black lines (and a fer white lines) connect these black planes, which are oriented in multiple directions. Rather than Bragdon’s characteristic individual black plane sections (fig. 2) or Malevich’s black or multicolored rectilinear planes floating freely in space (figs. 6 and 7), Matta’s irregularly shaped planes are held in tension in a relational, curved space defined by the interlocked forms. As Matta told Kozloff in 1965, he wanted “[to make visible, to give a vision of, the structure of events,” and these works do suggest a new form for the “structure of events” embodied in the space-time continuum.

This was the period, too, when Matta was interacting with the visionary designer/architect Buckminster Fuller, whom he had met in 1939. Fuller may have been yet another stimulus for Matta’s continued interest in the spatial fourth dimension and, initially, its relation to

line in the tradition of Ouspensky. Fuller had published his book 4D Time Locke in 1928, and there he had set forth his rejection of ” cubical” architecture in favor of circular suspended structures whose radial distance from a central mast he associated with time. Although Fuller identified the fourth dimension as time in his book, linking his new architecture to Einstein and Relativity Theory, he had sent a copy of 4D Time Lock to Bragdon, thanking him for his writings as well as his Manas Press’s publication of Ouspensky’s Tertium Organum in 1920. 4D Time Locke makes clear that Fuller shared Matta’s interest in the interconnections of space with time on Ouspensky’s model, including the theme of temporal processes like growth and decay.

By the 1940s Fuller was also developing his system of Energetic-Synergetic Geometry, based on the tetrahedron, with which he believed he could model four-dimensional space. Fuller would subsequently become a prominent advocate of the spatial fourth dimension in a world focused on Relativity Theory at mid-century.” Fuller and Matta shared a rejection of the Euclidean cube in favor. in Fuller’s case, of the tetrahedron and, in Matta’s case, of curved, multidimensional spaces. In his interview with Hans-Ulrich Obrist, Matta explained: “Fuller was interested in spaces like these (draws). But me, on the contrary, I was Interested in entering into these spaces (draws) spaces that had to do with forms drawn from non-Euclidean geometry. ” Those drawings would have reflected the contrast of Fuller’ tetrahedron-based language and Matta’s devotion to curved non-Euclidean geometry.

As Matta learned of the atrocities of the World War II, having also seen Picasso’s mural Guernica in 1937, he began to introduce totemic figures into his paintings to express his response to these horrors. He would explain this as a shift from his personal “psychological morphology” to a “social morphology” engaged with the world around him.”

The scale of Matta s works enlarged as well, but their spatial effects remained as complex and indefinite with their continued denial of three-dimensional, Euclidean readings. In the formalist oriented art world of New York in the 1940s, however these paintings were often criticized for their figuration and for their continued emphasis on complex space by critics such as Clement Greenberg, who equated advanced modern painting, with abstraction and the fatness of the canvas surface.” Matta left New York in 1948 for France and then settled in Rome during 1949-1954, returning to Paris that year. In Europe, he continued to combine potent social commentary win complex space in even larger mural-sized pantings launching yet another phase of his career.”

Matta was devoted to revealing a “world of as-yet-unseen realities.” according to Jimmy Ernst, who was part of the community of Surrealists in 1940 New York.” The artist would later state that he wanted to “conceive reality in terms of all directions.'” Rather than seeing through the square associated with the window in traditional painting, Matta came to believe a viewer should experience seeing as it at the center of a cube, which has an up, down, right, left, front, and back.” “On top is the infinite,” he continued.” In works of 1944, such as Composition (cat.18.19. 20).

Matta had begun to use individual, transparent toreshortened planes to suggest spatial recession. That he was rethinking the cube’s relation to its bounding two-dimensional planes is clear in his 1947 drawing Femme pliée. Here a yellow line defines the form of an unfolded cube on the model of Bragdon’s folded-down cube (fig. 1), a source he may have known through Fuller. Ten years later, in 1957, Matta made the painting Cube ouvert in this same configuration, and the idea of an art wore in the form of a folded cube would continue to interest him.

In paintings of the 1950s and 1960s Matta’s foreshortened planes would increasingly introduce spatial cues and even suggest the multiple sides of exploded cubes. That goal of defining a new kind of seeing is clear in such works as The Unthinkable of 1957 (at. 58). Matta wanted his viewers to see more completely and “grasp the picture” from multiple directions, including even “reality at different speeds.”” Here he was in the tradition of earlier explorers of the fourth dimension who sought to expand perception, but he was unique in trying to make seeing itself a multidimensional process, signified by the contrasting models of seeing via a three-dimensional cube versus the two-dimensional canvas surface. His one precursor may have been Malevich’s colleague Matyushin and his pursuit of 360-degree seeing in his “See Know” system in 1920s Russia.

From the viewpoint of the twenty-first century, we can now observe the larger history of the fourth dimension in the twentieth century and recognize the concept as one of the leitmotifs of the century as a whole. For example, artists such as Robert Smithson and the members of the Park Place Gallery group in the 1960s were deeply engaged with the idea, although the Park Place painters met the same kind of critical resistance as Matta to their spatially complex paintings.”

However, by the 1970s and 1980s the emergence of computer graphics and of string theory in physics began to inject discussion of higher dimensional spaces back into popular culture, with many of the books of the early twentieth century experiencing new currency in digitalized versions- from Flatland and to the texts of Bragdon and Ouspensky.” The moment is rise for a new consideration of Marta’s painting, which can now be approached from a far broader viewpoint than simply the mid-century American formalism that so limited the practice of and critical writing about spatially complex art.

There is no doubt that Matta’s painting stands as a highly important contribution to the history of the creative responses of artists to the spatial fourth dimension as well as to non-euclidean geometry through the twentieth century.