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Shadows of Reality: The Fourth Dimension in Relativity, Cubism, and Modern Thought

by Tony Robbin
Yale University Press, New Haven, CT, 2006
137 pp., illus. 90 b/w; 8 col. Trade, $40
ISBN-10: 0300110391.

Reviewed by Jennifer Ferng
Department of Architecture
Massachusetts Institute of Technology


Is the fourth dimension logically considered space or time, physical or temporal? Or perhaps both? Commencing with such immense questions is no easy task. Yet in combining revisionist art history with history of mathematics and present-day physics, artist Tony Robbin guides the reader on a fast-paced romp through the obscure origins and popular fantasies of fourth dimensional geometry and then straight into research applications and explanations of these same geometries in contemporary art and mathematics. As Robbin insinuates, space and time are not merely abstract theories without practical application but are, instead, dynamic and complex concepts, useful for expanding the visualization of space itself in a variety of artistic media. Time is often mistaken for the fourth dimension when in truth, time is also a form of space (and geometry) as well. This public misperception is what Robbin seeks to remedy. Our current understanding of how space and time relate to movement, perception, and scale radically restructures how contemporary art retools mathematical phenomena in the name of visual representation. From his consistent attention to the formal nature of fourth dimensional shapes, Robbin may be acknowledged as a staunchly committed Platonist whose literary approach precludes any potential opportunities for explaining the final ends behind such geometries, thus deterring any alternative Aristotelian readings of these rather astounding mathematical achievements.

Having written two previous books on closely related topics, Fourfield: Computers, Art, and the Fourth Dimension (1992) and Engineering a New Architecture (1996), Robbin is strongly dedicated to the cross-disciplinary exploration of fourth dimension models, and as part of his professional career, he was also one of the artists included among the Pattern and Decoration movement in New York during the 1970s who incorporated ornamental motifs and undervalued textiles into his paintings [1]. Tutored and educated by talented mathematicians such as Thomas Banchoff and H.S.M. Coxeter, Robbin conscientiously learned how to program hypercubes on the computer and explore quasicrystal geometry in digital space. His training and extensive experimentation with tessellations of hypercubes, graphic projections, and the programming of the de Bruijn algorithm that generates quasicrystals logically culminates in his interests in experimental technical drawing, computer modeling, and mathematical software. While the book is marketed as a dialectical engagement between art history and mathematics/physics, Robbin’s propensity for explicit technical explanations and somewhat hagiographic case studies of higher dimensional geometries deliberately favors a complex mathematical approach for a comprehensive understanding of the fourth dimension.

Named in the book’s title, shadows refer to areas of partial darkness or obscurity in space created when an opaque body cuts off rays of light; they consist only of faint representations of real objects. Robbin argues that it is not necessary to have a four-dimensional object in order to see the fourth dimension. In fact, upon seeing an object’s shadow, one may deduce more information from the shadow than from the object itself. Robbin stringently advocates shifting one’s world view from the slicing model, which is defined as a "parallel projection to a plane – a technique that renders exploded or unfolded figures composed of three-dimensional cells" and is based in calculus, to the projection model, which refers to "drawings and models of fully connected cells which place all the cells in the same (three-dimensional) place at the same time" (121) and is grounded in contemporary physics. In moving away from the limitations of the Cartesian grid, Robbin succinctly contends in one of his previously published essays that,

"In fact, the argument can be made that whenever we see any three-dimensional object we are seeing its shadow or projection; by the time the visual information reaches our eyes, it is two-dimensional – wave fronts of light…I have discovered that a two-dimensional element and a three-dimensional element, when taken together as a single object, provide the visual information necessary to see a four-dimensional figure in precisely this way, by the parallax of their projection rotating around a plane [2]."

This is indeed an innovative premise for the making of art and for the rethinking of representation’s production. Robbin proffers that higher dimensional geometries are merely rotated silhouettes, or even implied utopic sketches, which present a tantalizing glimpse into what is visually possible. If we are to take Robbin’s argument at face value, the definition of an object in mathematics directly purposes a new philosophy of aesthetics which will alter our visual cognitive abilities and hence, our perception of space. However, it becomes unclear over the course of the book if aesthetics is entirely beholden to physics and mathematics for its epistemological bearings and if the science of perception will become the ultimate endgame for representation itself.

While Robbin’s claim is provocative, it is based upon art historian Linda Henderson’s The Fourth Dimension and Non-Euclidean Geometry in Modern Art (1983), which argued that Picasso’s discovery of cubism was based upon four-dimensional geometry. Henderson compared Picasso’s Portrait of Ambroise Vollard (1910) to a diagram in Esprit Jouffret’s Traité élémentaire de géométrie à quatre dimensions (1903). Robbin extends Henderson’s argument by positing that Picasso’s Portrait of Henry Kahnweiler (1910), composed of multiple, interpenetrating cubes, is a more advanced example of applied four-dimensional geometry than the forms exemplified in Vollard and is also similar to Jouffret’s drawings of the cells of a hypercube. Consequently, the transition from the head of Vollard to the Kahnweiler head marks the paradigmatic move from the slicing model to the projection model, where all of the cells are projected into the same space at the same time without interference (33). Robbin desires to legitimate fourth dimensional geometry by accentuating its legacy in mathematics and physics, rather than by privileging the ambiguous philosophical mysticism that pervaded many painters’ generalized approaches towards non-Euclidean geometry in the first decades of the twentieth century [3].

According to Robbin, the theorization of the complex relationships between time and space originates with the 19th century German geometer Ludwig Schläfli’s contested discovery of six platonic solids, called polytopes, based in the fourth dimension. This is soon followed by Washington Irving Stringham’s 1880 paper "Regular Figures in n-Dimensional Space" and German mathematician Victor Schlegel’s 1882 illustration of a hypercube in perspective [4]. Projective geometry derives many of its principal definitions and concepts from the mathematical work of Gérard Desargues, Karl George Christian von Staudt, and Felix Klein. Its fundamental spirit may be recapitulated as: "…placing infinite points inside the reach of axioms and theorems changes geometry from a system of measurement to a system of incidence alone…" (53). In comprehending the principles behind projective geometry, conventional intuitive relationships between space and time are often reversed and sometimes completely transformed. For instance, one must think of local interactions in relation to global actions as being mutually dependent. Being in two places at the same time, in the case of quantum relativity, is quite plausible. Similarly, particles are equally reliant on each other’s location, magnetic pull, and numeric coordinates. Points intentionally behave as lines, and reciprocally, lines simultaneously act as points. Even the future and the past become formally connected when Robbin elucidates, "When applied to spacetime, the projective model holds futures and pasts in the same shape and shows the continuity between the two. The emphasis on the homotopy of the shapes of the future with the shapes of the past is something supported, if not started, by projective geometry" (115) [5]. Much of the complex phenomena discussed by Robbin, such as Hermann Minkowski’s 1908 merger of Henri Poincaré’s four-dimensional geometry with Albert Einstein’s clocks and measuring rods, are singularly fascinating in their minutiae. Minkowski demonstrated that the geometry of space itself was changed by motion, thus integrating together the once disparate languages of Poincaré’s mathematics and Einstein’s physics (45).

Varied and helpful analogies, involving concrete examples like subways, thoroughly explain esoteric concepts in the book that might otherwise overwhelm or confuse the reader. For example, in box 1.3 on mechanical drawing, Thomas Ewing French (1888) rendered geometric objects in a "glass box" technique, where the object is imagined inside a glass box with hinged faces. In the United States, the viewer is conventionally assumed to be positioned outside the box looking down on it (10). While Edwin Abbott’s classic Flatland is aptly characterized by the chance meeting of the two-dimensional A. Square and the three-dimensional Sphere, sphere eversion, from Simon Newcomb’s 1906 essay "The Fairyland of Geometry," is described as a sinuous rubber band lying on a flat sheet of paper. Similarly, Charles Howard Hinton‘s 1904 drawing of a particle moving in three dimensions illustrates the sequence of slices of a rigid spiral as it moves through a plane. Lastly, a three-sphere, in the field of topology, may be equally envisioned as a sliced egg without the yellow yolk. The resulting concentric rings transform into latitude lines on the earth or nested Russian dolls (78).

Each chapter is self-contained, but every author must strike a balance between the elaboration of details and the greater general picture. This is true for Robbin who must deal with extremely complex concepts. He, however, errs on the side of detail. As a result, his arguments become misplaced in a morass of diagrams and theories, an organizational shortcoming that does not do any viable justice to his obvious breadth of knowledge. Secondly, his intellectual history of the fourth dimension is subtly implied, through his choice of case studies and historical actors, but it could be made much more explicit for the sake of his readers. In addition, Robbin is often unclear about the terminology he introduces. Despite mentioning perspective, projection, projectivity, and projective in the preface and in other given places, the reader is given very little direction to trace how each expression differs from sub-field to sub-field, discipline to discipline. For instance, perspective is a "specific scene" while projective is considered "systems defined by homogenous coordinates where concepts like metric dimension and direction lose all traditional meaning, but gain a richness relevant to modern understanding" (Preface, x). We are then later introduced to the theory of perspectivity, which relates to a range or an arbitrary selection of points on one line to a range of points on another (54). Again, it is difficult to pinpoint accurately the type of "modern understanding" towards which Robbin intends to lead us, and more importantly, if this new mode of knowledge is strictly cultural, logical, or philosophical in practice.

The most intriguing portions of the book are the work of contemporary artists and scientists who are directly exploring the inventive potential of fourth dimensional geometry. His footnotes do provide a wealth of valuable information. An interesting tangent, buried in a footnote three of chapter six, leads us to Haresh Lalvani, an architect and professor at Pratt Institute in New York. Lalvani’s lectures "Hyperstructures" and "The Morphological Universe: Expanding the Design Possibilities in Nature and Architecture" discuss some of the tectonic possibilities of fourth dimensional geometry, and he has analyzed the relationship between levels in a projection, where the angles in the unit cells of flat patterns are the dihedral angles of the units of the next higher-dimensional unit cells (122). Lalvani’s research, in collaboration with Bruce Gitlin, formed the basis for a current commercial venture called AlgoRhythm Technologies that specializes in curvilinear surfaces for architectural environments. Complexity in design is a suitably debatable extension of Robbin’s ambitious interests.

As Minkowski once poetically articulated, "henceforth space by itself and time by itself are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality" (47). It is this type of conjoined awareness that gives birth to the possibility of new spaces to which Robbin calls our attention. He assists us in expanding our notion of what is considered space (or spatial), and as a result, we are given multiple avenues of intellectual and creative engagement in further developing technical tools for spatial visualization. It is an admirable but difficult search for the ideal geometric object that will define the totalizing kinesthetic experience of space and accurately predict our mediated reactions in kind. This book provides valuable explanations of the contemporary research of mathematicians and physicists whose models of spacetime, when appropriately considered together and not carelessly apart, offer extremely constructive directions for the future production of art and our own human evolution to perceive visual connections in the next unexplored dimension.

[1] For more about the contemporary work of other painters associated with the Pattern movement, see Christopher Miles, ‘Tracking Patterns’, Art in America, vol. 92, no. 2, February 2004, pp.76-81.

[2] Tony Robbin, ‘Painting and Physics: Modeling Artistic and Scientific Experience in Four Spatial Dimensions’, Leonardo, vol. 17, no. 4, 1984, pp.227-233.

[3] A non-Euclidean approach to mathematics is based upon the geometric property that objects moved around in a space may change their shape (47). But as Henderson and Robbin both concur, most painters conflated various types of geometries together without any attempt to differentiate them in their artworks.
[4] A hypercube is a four-dimensional cube. Schläfli expanded Swiss mathematician Leonhard Euler’s formula in order to compute the volumes of higher-dimensional regular figures and to determine which polytopes would fit inside which polyspheres.
[5] Two mathematical objects are considered homotopic if one can be continuously deformed into the other. Henri Poincaré first formulated homotopy as a concept around 1900. See http://mathworld.wolfram.com/homotopy.html.



Updated 1st March 2008

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