Tony Robbin wants us to see the invisible in all its actuality. Working variously as a painter, sculptor, writer and researcher, he has come to make his creative home in the fourth dimension, and beyond. In his 1992 book, Fourfield: Computers, Art, and the Fourth Dimension (one of several lucid and singular books to the author’s credit) Robbin, who was born in 1943, offers something of a personal credo in his opening chapter, which is titled “Einstein’s Cave”, a reference to Plato’s well-known parable in which higher-dimensional reality must be inferred from shadows. Robbin writes:
In the caves, we are forbidden by authority to turn and face the dancers directly, but in fact authority has no real power over us in this matter. We have the ability to see the dancers in their full dimensionality –– to accept the cultivated experience of seeing the fourth dimension as being “out there,” and it is our choice to do so. Failing to make this choice handicaps our ultimate understanding of reality. Our ability to apply four-dimensional geometry as a useful template for experience connects us to the multiplicity of spaces and points of view that implode upon us every day. If culture can teach us to see the third dimension as real, then just a little more culture can teach us to see the fourth dimension as real as well.
From Kazimir Malevich and Wassily Kandinsky, to Pablo Picasso and Salvador Dalí, avant-garde artists at the beginning of our epoch took the fourth dimension seriously, but without really sweating the details. Even Marcel Duchamp, who diligently worked through hypergeometry manuals, did so only up to the point of malicious drollery. Tony Robbin, on the other hand, holds a patent on the application of three-dimensional projections of six-dimensional quasicrystals to architecture. His best-known book, Shadows of Reality; the Fourth Dimension in Relativity, Cubism, and Modern Thought (just translated into Chinese), is a primer for climbing the dimensional ladder, from Flatland to esoterica such as entanglement and quantum geometry. The book also chronicles, from an insider’s perspective, the history of 4-d visualization: that is, as diagrams by mathematicians and pedagogues, and as works of art by literally all the important schools of early Modernism. (So far, historians have ignored Robbin’s scrupulously argued mot juste, that Cubism should properly be called “Hypercubism.”)
Nor was Robbin satisfied with a century of attempts to visualize the fourth dimension. In 1980, after mastering the theory but still hungering, as had so many generations of 4-d obsessives, to see the thing itself, he learned of a pioneering computer animation at Brown University: Thomas Banchoff, a mathematician, and Charles Strauss, an engineer had tamed the morphing 3-dimensional projection–– the solid “shadow”–– of a hypercube. They could rotate it at will in hyperspace. (Please note: time is not the fourth spatial dimension, more like an extra dimension. In the rotation of a hypercube, time is the fifth dimension.) Robbin got his hands on the interactive knobs of Banchoff’s million-dollar computer, as well as a copy of his hypercube film, which he took home and studied on a flatbed editing console frame by frame, back and forth, until it took.
The friendship with Banchoff also took, opening the door to the milieu of professional mathematics. Soon Robbin himself had become a pioneer of computer visualization, having learned to code four-dimensional geometry at off hours in computer research labs, and later on his own first-generation workstation. To visualize a tessellation of hypercubes (in which all four dimensions would be continuously packed, as three can be by cubes), Robbin consulted the most renowned geometer in the world, H.S.M. Coxeter, who was delighted to see what had never been seen. On Coxeter’s recommendation, Robbin was invited to present his research at a mathematics conference. Many conferences later, Robbin’s friends, correspondents and collaborators in the math and science realm have proliferated–– from cyberpunk mathematician Rudy Rucker, author of Infinity and the Mind, to maverick cosmologist Roger Penrose, the recent Nobel Prize winner, whom Robbin has consulted about quasicrystals and twistor theory.
Robbin’s collection of experts also includes art historians, such as eminent Leonardo scholar Martin Kemp, and most profoundly, Linda Henderson, whose book The Fourth Dimension and Non-Euclidean Geometry in Modern Art (1983) placed this mathematics at the very foundation of modern art, and of Robbin’s thinking. Of course, Robbin’s Rolodex is mainly filled with fellow artists–– “Held, Al” being an especially well-thumbed entry. When I asked Robbin if Held, a lifelong friend who had been Robbin’s teacher at Yale, was a mentor, he quipped, “I met him in 1965 and we stole from each other ever since.”
A snapshot of this relationship exists in an article for Arts, where Robbin gave his take on Held’s black and white paintings. With their elegant spatial contradictions, the paintings, wrote Robbin, are “exercises in omni-attentiveness, and the viewer’s capacity for experiencing and enjoying them grows with his tolerance for multiplicity.” Forgiving the skunked “his” (magazine standard of the day), few viewers of any gender could have brought as much tolerance for multiplicity to Held’s studio as Robbin. A few years later, Robbin was to be greatly influenced by the paintings he was writing about here; considering the overt spatial ambition of the work that resulted, mutual thievery might well be considered a factor in Held’s richly colored paintings of the mid-eighties with their whipsawing perspectives.
In 1971, at the time of the article, however, Robbin was not yet making pure geometric abstractions. He was, instead, at the center of a growing movement involving, among others, Robert Kushner, Joyce Kozloff, and Valerie Jaudon, who were meeting to discuss non-Western, feminist, and countercultural approaches that might invigorate contemporary abstraction. As Kushner put it in an essay on Robbin, “We were even willing to accept that taboo word–– decoration.” Robbin’s contributions to what was originally called Pattern Painting (which Robbin still prefers for his work) were sweeping abstract rebuses with motifs and textures derived from the artist’s immersion in Japanese and Persian aesthetics (he had lived in Japan and Iran until age 16). One of these, Japanese Footbridge (1972) is included in the exhibition “With Pleasure: Pattern and Decoration in American Art 1972–1985,” originating at LA MoCA, (and slated to travel, after a year’s delay, to Bard College in 2021). On a twelve-foot-long golden cloudscape reminiscent of a Zen folding screen, the painting asserts curving rhymes that suggest Islamic tilework, stenciled kimono fabric, and swooping, supersonic speed.
At any rate, Robbin mentions F-111, James Rosenquist’s epic military-consumerist montage, as an influence around this time, although not for its subject matter but for its abrupt transitions. Increasingly, Robbin, like Rosenquist, divided his canvasses into cinematic sequences that stand apart from the symmetrical, fabric-like flattenings common to the works of most of his P&D peers. In 1974-5 Robbin had a solo exhibition of these aggressively compositional paintings at the Whitney Museum, and for the remainder of the decade exhibited at the influential Tibor de Nagy Gallery. His career path was ascendent.
Ironically, it was an observation in Whitney curator Marcia Tucker’s catalogue essay that set Robbin on a new and, to judge by the artworld’s neglect of his later accomplishments, unfashionable trajectory. In “forcefully architectonic” works such as Tonikuni (1972), in which Shinto temple pillars are concatenated with patchwork and aerial views, Tucker detected that “contradictory visual information suggests the complexity of four-dimensional geometry.” Tucker’s inadvertent prophecy sparked some ready tinder in the artist’s mind. Soon Robbin was engaging a physics graduate student to tutor him, equation by equation, through the authoritative textbook on relativistic gravitation–– which is to say, in the four-dimensional reality of the world we live in.
But at the very beginning of his hyper-awakening, a fundamental change was also happening in Robbin’s paintings on their own terms. Even as the authenticity of the artist’s hand along with revisionist cultural politics had, by and large, come to define P&D, Robbin purged references to the non-Western and the handmade and began to compose, as Al Held had been doing, solely with precision lines, curves and planes.
Robbin’s paintings of the later 1970s superimpose four or five cleanly delineated layers which disagree about space. At first Robbin placed darkly contrasted or fully monochrome backgrounds behind vibrantly colored linework: electric blue squares receding like the bent coffers of a barrel vault, yellow double circles shooting across the screen like bullet holes, green L-shaped gnomons in fisheye view, and plenty more, all moving past each other like the multiple exposures of a Dziga Vertov film. In darkly arresting works such as 1976-6, and 1979-3, we may feel caught inside the celluloid itself, adrift in the unspooling frames. Gradually, however, Robbin brought the color of the orthographic (non-perspective) background patterns into dominance, so as to play games of hide and seek. Where the lines intersect, they interrupt and occlude; cut or join. And often the “wrong” background color fills these Boolean and/or/nor mutations, making for irrational, disorienting jumps back to front. Beautiful works such as 1978-3 and 1978-20 seem to compress deep space like Formica marquetry–– and yet they don’t let the viewer off so easily, in that disparate spatial cues warp past the point of integration in a way quite unlike Held’s black and white works, which crisply hold the picture plane however much sliced and reassembled. As critic Carter Ratcliff observed of Robbin’s work of the time in a 1978 essay in Arts, “The irreconcilability of the spatial systems in these paintings has to be recognized as deliberate; that is, Robbin has generated new intentions.”
Robbin was not yet making explicitly four-dimensional works, but he was upping the ante on the “P” of P&D. (Ratcliff: “Of course, there are patterns, and there are patterns.”) Robbin’s new intentions were not to confuse, per se–– although there is a skepticism of systems in all his work, a subject to which I’ll return. Rather, he was goading the viewer into seeing more, seeing multiply. Robbin compares these paintings to fugues whose dense chords contain a weave of melodic symmetries; listeners can learn to hear the independent voices. Taking this approach to impressive paintings such as 1979-8 or 1979-20, one can begin to understand what the artist was after. Imposing in scale (70”x120” and 72”x166” respectively), the faceted crystal logic of these works suggests the reverberations of a pipe organ in a cathedral. But Messiaen or Boulez, perhaps, rather than Bach; pattern is not so much fugal as fugitive. As Robbin had written of Held’s paintings, his own works were increasingly “exercises in omni-attentiveness” that captivate and disorient in equal measure.
It was in 1980 that Robbin broke through to the fourth dimension. More than a century of geometers, artists and spiritualists (Steiner and Ouspensky, for example, who seized on the fourth dimension as a portal to higher being) had strained to see it for themselves. For all of them, the animation of a hypercube in various rotations by Banchoff and Strauss, made on a supercomputer of the day, would have been a holy grail. And so it was to Robbin, who studied it until he could see the rubbery distortions and weird inversions of the spinning hypercube as 3-D projections, or shadows, of a rigid, unchanging figure passing through a dimension we only infer–– just as anyone would take for granted the solidity of a rotating cube from its distorting 2-D shadow (or for that matter, the full volume of the world from the flat projections on our eyeballs.) One of Oliver Sacks’s last books, The Mind’s Eye, includes a chapter about some extraordinary powers of visualization among the blind. Sightless topologist Bernard Morin solved, with his inner vision, the problem of turning a sphere inside out. Reports Dr. Sacks: he quite literally saw it. Inner or outer visualization, it’s the same neurons. Like Morin (if not, perhaps, to the same extent), Robbin had succeeded in rewiring his mental map, and he was determined to bring that map to bear on the propositions of Pattern Painting.
A square can spin on a point, a cube on a line, and a hypercube… on a plane, as would be obvious if you could see four-dimensionally. As Robbin explained in his 1992 book Fourfield, “to the person accustomed only to observation in three dimensions the properties of planar rotation are mysterious, even paradoxical (shapes appear and disappear, turn inside out, flex and reverse); but these paradoxes become the very means by which we see the fourth dimension.”
For the painting Fourfield, Robbin’s 27-foot long magnum opus of 1980-81, the artist welded steel rods projecting from the surface to simulate the paradoxes of planar rotation. Here is Robbin’s description of his ingenious hybrid technique, from an essay (written 30 years later) entitled 4-D and I:
I went to a lot of trouble to make sure that the painted lines and the painted metal rods look the same to a standing viewer. But as the viewer moves, strange things happen. As in any relief, planes can be hidden behind an edge of that plane (seen exactly edge first), and in my four-dimensional works, one has the sensation that whole three-dimensional structures are hidden behind open cubes. Space spins out of space as the viewer moves.
How much math does one need to know to appreciate Robbin’s paintings? By watching Banchoff and Strauss’s film and others now widely available on the internet, I have become somewhat conversant with the hypercube’s gemlike symmetries, which appear when axes align, and with its inversions and flexions as it rotates across a hidden plane. Sometimes I can recognize these familiar landmarks in Robbin’s works, like red rhomboidal capes waved by matadors. I haven’t, so far, experienced the full higher-dimensional consciousness that Robbin wants to impart, but the fascinating manner in which space spins out of space in Fourfield is something new in the history of painting and sculpture.
In color, rhythm and hybrid technique, Lobofour (1982, 96” x144” x 24”) seems similar to Fourfield at first glance; but it is less regular, non-orthographic, subtly wilder. According to one of Robbin’s collaborators, mathematician George Francis, in Lobofour, “the four-dimensional lattice is no longer constrained to flat Euclidean geometry.” (The painting’s title acknowledges Nikolai Lobachevsky, the Russian pioneer of curved space.) The complications of this geometry are beyond my intuition, but clearly some higher order lies behind the painting’s darting spatiality, constantly in motion like sparkles of reflection on a lake. If you look for space in Robbin’s work, you will find it endlessly.
For Primary Structures artists and the conceptual artists of the 1960s, geometry meant the eternal (if slippery) truths of simple grids, boxes, and counting numbers. For Robbin, the uses of geometry are open-ended and dynamic–– in a word, baroque. Not long after Fourfield and Lobofour, Robbin stopped painting for almost a decade, focusing his efforts on sculpture and research, but when he began painting again in the mid-1990s, this open-endedness became more and more pronounced.
First, however, Robbin committed himself to the implications of the steel rods, making wall sculptures that added curving, wiggly forms and rhomboids of tinted plexiglass to the projecting geometry. Like Man Ray’s rope dancer, these reliefs accompany themselves with their shadow. When lit by red and blue bulbs, the colors combine into near white except where the metal rods cast pairs of diverging shadows on the wall, one blue, one red, encoding the spatial relations of lights, sculpture and wall. For a viewer wearing 3-D red-blue glasses, the parallax of these shadow lines integrates into a stereoscopic image. Geometry now seems to project into the wall, while the actual projections–– the translucent panels, along with their skewed, tinted shadows, and the metal rods–– hover ambiguously in space. As with Fourfield and Lobofour, movement by the viewer allows for an experience of four-dimensional unfolding, while the interplay of dimensions–– one, two, three, and four; real, simulated and virtual–– glues together and flies apart.
As Robbin’s ambitions for sculpture grew, so did his grasp of cutting-edge research by Roger Penrose and others about irregular space-packing patterns, or quasicrystals. Robbin saw in quasicrystals a way to produce an infinitude of deep, fractal-like patterns–– patterns that exhibit simultaneous 2-fold, 3-fold, and 5-fold symmetry and yet, paradoxically, never repeat. Even better, quasicrystals turn out to be shadows of more regular figures from higher dimensions. How Platonic can you get?
Robbin’s involvement with quasicrystals climaxed with a permanent installation at a technical university in Copenhagen, where Robbin made an assemblage of rods and colored plates to hang from the roof of a three-story atrium. It was precisely engineered to unfold its layers of symmetry with viewers’ movements and to project animating quasicrystalline colors as the sun arcs low through the northern sky. COAST, installed in 1994 with great success, was summarily destroyed in 2003 by a new administration. With mathematician Francis’s help, however, Robbin has made a 3-D digital version of a quasicrystal, full-scale and interactive–– an aptly innovative memorial that compensates, somewhat, for bureaucratic vandalism.
In 1995, after the complex logistics of COAST, Robbin returned to painting, but this time with an eye to the native virtuality of the medium, its built-in dimensional depths. Where his paintings and hybrids, impressive as they were, had tended toward dryness and a certain claustrophobia–– with improvisation concealed afterward or restricted to the cranium–– now Robbin applied himself fully to the flat, unshadowed picture plane, allowing for improvisation to flow from head to hand; and from stencils, tape and airbrush to viscous, semi-translucent colored pigment, at first acrylic and later oils.
Robbin’s geometric paintings between 1974 and 1982 had already seen an evolution toward asymmetry and richly colored backgrounds. The carefully airbrushed minor key blues and oranges of Lobofour, at the end of that progression, seem nevertheless to remain primarily functional, a means of establishing color-coded rules to be embroidered upon by the frontal lines, actual and painted. The saturated pigments of Sol Lewitt’s cleanly abutted illusions of solid geometry (wall drawings begun in the mid-1980s) have, perhaps, a similar informational edge; the more sensuous, the more stand-offish. But by 1999, Robbin’s newly painterly approach had turned the tables on color. In comparison to Lobofour (1982), 1999-4’s chromatic power is tremendously increased, yet the palette has hardly changed. There are additional foreground elements in the more recent painting–– delicate strings of regular polyhedra that dance in space–– but the principle difference is that the background is no longer tessellated, tiled and airtight. Instead, it is thickly gaseous and luminous, with soft, intense spots of color that give off heat as well as light. The painting is impenetrable with colliding incidents and riddles of structure, yet it’s light on its feet, porous: a muted rainbow that fractures into foreground shards plays a dark scherzo all the way back to the farthest cloud of matter. In 1999-4, in a way that is new to Robbin’s paintings, color and space are intertwined–– relativistic, one could say; entangled.
Robbin’s new painterliness has continued to develop alongside mathematical speculations that are by now so far beyond the grasp of most viewers that plain looking is surely what is being called for. Which is not to say Robbin has given up explaining–– as in this technical notation in a peer-reviewed math journal about a computer study of a “a quasicrystal lattice in 5-fold orientation where the acute angles are 72° and 36°; it is a slice through a quasicrystal cloud that was made with the deBruijn algorithm.” There are grids and there are grids.
Later in this article, Robbin explains his artistic method and purpose. For the math-challenged, we may take comfort in Robbin’s assertion here, addressing math-savvy readers, that “Sorting out all these complications is not the point. My paintings are not equations, and it is not possible to read a mathematical resolution in them.” He continues:
Rather they revel in the richness and paradox of higher-dimensional visual phenomena. With a knowing nod to the mathematical possibilities, the paintings encourage an acceptance of such spatial complexity. Further, they encourage a taste for, and even a giddy joy, in spatial complexity.
2006 began a period of tragedy, misfortune and serious illness for Robbin and his family. Giddiness departed; spatial complexity stuck around. Robbin revived the idea of monochrome backgrounds to highlight phosphorescent imagery that recalls the pulsating cathode rays of early hypercube animations. The dark background in 2007-8 (2007 56”x70”), for example, recedes behind polygonal planes nested in blue, green, and orange matrices, a slashing, compressing framework. Unlike the monochromes of the 1970s, where geometry is inscribed on top, here the translucent lines embed themselves into the paint; brushy and stained, the background opens up into inscrutable space and color, a cosmic cave.
Robbin’s backgrounds didn’t stay dark for long, blossoming, for example, into the lustrous reds of 2008-1, the molten golds of 2008-O-6, and the fierce ceruleans of 2010-O-3. These insistent colors are worked and worried into unapologetic expressionism. On the other hand, 2009-7, with its palette of pale pinks and oranges, powdery yellows and blues, and with its feathery precision and buoyancy, is distinctly Impressionist in feeling. One might take it for an homage to Monet–– a unique one, which acknowledges Monet the scientist as much as Monet the painter. In Fourfield, Robbin had written about Monet quitting Paris for “the scientific study of light on haystacks and facades at different times of day and in different atmospheric conditions.” Later in the book, Robbin hypothesized Monet’s water lilies–– in which surface, sub-surface, and reflection are mingled–– as the completion of Cubism: “From this point of view […] it is the spatial properties, not the color and brushwork, that make Monet’s later work so appealing and enduring.” Those words were written in 1992. Robbin’s spatial point of view began to give ground to color and brushwork when he resumed painting a few years later, and with the Monet-like 2009-7, the two viewpoints achieve a kind of stereo integration.
Color and brushwork continue to be on the upswing. Since 2013 or so, Robbin has dispersed his dense, braided matrices more and more, leaving dimensional ghosts in shimmering fields of color and light. In 2013-6, the orange background subtly dominates, like the tarnished gold leaf of a Buddhist screen of fluttering Fall leaves. It has a richly melancholy feel. 2016-4 brings foreground and background into raw, scribbling equilibrium, achieving an almost psychotic gorgeousness reminiscent of Ensor or Nolde. 2019-1 is translucent and provisional, grays floating upon but not quite hiding deeper hues, and above that, light-struck facets like fragments of box kites in vapor.
This viewer has already confessed to being unable to see the higher dimensional spaces where Robbin’s work is embodied, but his most recent Pattern Paintings–– which one could say are less late-Monet than late-Cézanne–– provide guided-tours to the edge of the spatial horizon more expert, and every bit as lyrical, indeed as musical, as any offered before. Robbin wants us to see (as he put it in 2006 to the mathematicians) “all of the spaces in the same space at the same time.” He is speaking the language of Masaccio and Leonardo; of Picasso and Duchamp; of Al Held and Robert Smithson.
Smithson, of course, invoked four-dimensional paradox in his writings and artworks, notably the mirror displacements–– part of a general revival of interest in the spatial fourth dimension in the 1960s. In 1969, Robbin wrote about Smithson for Art News, before Robbin’s own 4-d obsession had taken hold. In that article, his focus was on Smithson’s dismantling of systems, including those of art. “I want to de-mythify things,” says Smithson in an interview which precedes the article proper.
Robbin: “People will be frustrated in their desire for certainty, but maybe they will get something more after the frustration passes.”
Smithson: “Well, it’s a problem all around, and I don’t think we will work our way out of it.”
In the text that follows, Robbin places Smithson in the company of Cage more than Judd, identifying his arrangements of materials as “reconstructions of thought processes” rather than sculptures, per se. He recognizes the originality and power of Smithson’s critique of systems. Yet Robbin doesn’t quite accept the bedeviled state of affairs that Smithson delights in exposing:
Since our perceptions mold us, we ought to be responsible for them. How they mold us, how we can be responsible for them, how we can change what we see … are only implicit in Smithson’s work. For further explorations we must wait for other artists or for other shows by Smithson.
Quite a prediction. The next year, 1970, Spiral Jetty crystallized, figuratively and literally, Smithson’s message about the open-ended nature of mathematics, of systems; in doing so, this celebrated work epitomized the “something more” that Robbin proposed beyond the horizon of certainty.
Robbin continues to believe we ought to be responsible for the way we see, but the extraordinary flowering of his paintings of the last two decades has made it easy on the eyes to do so. Built upon a gamut of restlessly shifting higher-dimensional grids–– not only quasicrystals, but braided lattices, four-dimensional knot diagrams, hyperplanes, and so on–– Robbin’s painterly improvisations constitute their own kind of systemless sytem, an open-ended spiral at whose tip all spaces coalesce. “Something more” is there for the seeing.
