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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
160pp., illus: 90 b/w, 8 col. Trade, $ 40.00
ISBN: 0300110391.

Reviewed by Amy Ione
The Diatrope Institute


Shadows of Reality is a book that not only makes a good first impression but also carries through as one becomes better acquainted with it. Written by Tony Robbin, whose innovative art and computer visualizations of hyperspace have contributed to efforts to conceptualize other dimensions, the text provides a revisionist math history as well as a revisionist art history. This work, the latest in a series of works by Robbin probing the fourth dimension, builds on his earlier studies. In it, Robbin investigates with more specificity how dimensional research contributes to our comprehension of different models of the fourth dimension and examines their applications to art and physics. A foundational component of this cross-disciplinary revision is the author’s introduction of the distinction between what he terms the slicing, or Flatland, model as compared to the projection, or shadow, model. On the one hand, the strength of the slicing model is its grounding in calculus, which makes it mathematically self-consistent. Thus, the slicing model is often taken to be an accurate, complete, and exclusive representation of fourth dimensional reality. On the other hand, the projection (or shadow) model is also self-consistent and mathematically true. Yet, it nonetheless offers a parallel approach. What is key here is that this second view enriches geometry by offering a system that makes infinity a part of space. This not only changes the geometry but, more significantly in Robbin’s view, allows for a presentation more like the way space is.

Mathematicians and philosophers first explored and comprehended the two competing models during the nineteenth century. Today, the slicing or Flatland model is best seen as a God’s-Eye-View, popularly presented by E.A. Abbott in his classic 1884 book titled Flatland. Abbott described the experience of seeing a higher dimension as a direct experience, the kind in which the insight is only conceptualized through inference. Essentially, this translates into the idea that a four-dimensional world is to our space as a three-dimensional world is to a Flatlander. While an effective analogy, it largely ignores the use of projective techniques to study four-dimensional figures and spaces. Robbin clarifies that the projective alternative provides a more mathematical orientation, or a shadow model. The value he places on this latter approach comes through in his decision to title this book Shadows of Reality.

In Shadows of Reality, historical sections, analysis of the tensions between the two models, and the examination of the uses and misuses of the two models in popular discussions are presented insightfully. The comprehensive approach to fourth-dimensional thinking is impressive, as is the overview explaining why the powerful role of projective geometry in the development of current mathematical ideas was long overlooked. Particularly thoughtful is Robbin’s review of how projective ideas are the source of some of today’s most exciting developments in art, math, physics, and computer visualization. Perhaps what is most needed in our general popular discussions are the sections in which Robbin proposes that our attachment to the slicing model is essentially a conceptual block that hinders progress in understanding contemporary models of spacetime. Also of note is Robbin’s attention to detail, as evident in the many sidebars and well-chosen visuals. These effectively supplement the text and add to the well-developed arguments, which at times become quite difficult.

Robbin begins with an outline of historical theories of four-dimensional geometry, including the work and pioneering drawings of Washington Irving Stringham, Pieter Henderick Schoute, and Espirit Jouffret. Integrating a number of case studies, Part One examines past uses of the projective model. Here Robbin walks us through early twentieth century ideas, examples of painting and the various constructions of Fourth Dimension, drawing upon his own research and relevant studies by art historians (e.g. Linda Henderson, Pierre Daix, and Josep Palau I Fabre) and historians of science (e.g., Arthur I. Miller). I particularly liked how this author makes the case that Picasso’s Les Demoiselles d’Avignon (1907), frequently described as the first important painting of the twentieth century, should perhaps be characterized as the last important painting of the nineteenth-century because the themes that it attempts to fuse are of the nineteenth century. Placing it next to Picasso’s Portrait d’Ambroise Vollard (1910) we see the latter contains an exaggeration of color and line that breaks from the more generic formalism evident in Les Demoiselles d’Avignon. Evaluating the visual evidence from his perspective as a painter, Robbin concludes that Picasso adopted the methods of the mathematician E. Jouffret in 1910 and essentially used the projection model to invent cubism. Also noteworthy is how well this author fleshes out Minkowski’s work. Briefly, Robbin claims that Minkowski had four-dimensional projective geometry in mind when he structured special relativity.

A short course in projective geometry, the Entr’acte, separates the historical studies from the presentation of contemporary uses of the projective model (Part Two). Turning to the projection model, the latter part of the book examines creative ideas about space in contemporary mathematics such as twisters, quasicrystals, and quantum topology. Given the complexity of this material, however, I am not prepared to evaluate his research in detail. Suffice to say, that the concepts would have certainly translated better to a mathematician. As a layperson, I found the scope and intelligence within this part (and the book as a whole) breathtaking as well as challenging. This author goes far beyond confronting how a two-dimension model can be an accurate representation of 4D reality (if there is one). He offers examples that point to ways of viewing dimensions and grapples with how our models might co-exist and intersect. He reminds the reader that a key to the projection model is building an understanding of how the tesseract is a three-dimensional perspective projection of a four-dimensional hypercube and reaches beyond this as well. After an initial review of the ideas, I am not certain I integrally grasped the possibilities in their entirety.

The translation problem I encountered was not only within the specialized sections and the density of the material. In some cases I found myself continually re-thinking my conclusions as to how he was defining the fourth dimension. It was clear that some things that happen in space couldn’t be reduced to the slicing model. I also had no problem with his proposal that the projective model is not merely an alternative but often a requirement in representing the fourth-dimension. Less clear are his definitions of large concepts like time and spacetime, which seemed to be viewed through several lenses. The difficulty in comprehending the technical meaning(s) he wished to present led me to ask why we lack firm definitions and how this ambivalence has muddled our thought. These are not questions I feel qualified to answer. I do, however, feel comfortable stating that the definitional anomalies that result from the current state of science seem to put some arguments at cross-purposes in the ongoing discussions of other dimensions. In terms of Shadows of Reality, the terminological challenge was coupled with the thought that more and more multi-dimensional reality theories keep cropping up these days and are now being debated (e.g., string theory). With this in mind, I wish there had been additional material on the range of competing multi-dimensional proposals. This is not to negate the study’s contributions. Rather, with a range of dimensional variations now emerging to address the unknowns and theoretical challenges, I think we are remiss if we do not also ask ourselves if (whether) we are making real progress. That said, the book offers much to ponder and is definitely a contribution to the "dimensional" dialogue.

From another perspective, as a visual artist, I found the strength of the book within its visual component. Since the author’s effort to clarify many complicated concepts is not always as accessible, I appreciated the numerous drawings and diagrams that simplify concepts. These are stimulating and immensely pleasing to the eye. In a publication that hovers between the easily grasped and the dense, harder-to-decipher the illustrations aid the reader immensely. They also seemed to acknowledge that Robbin’s eclectic audience includes many disparate communities: artists, scientists, mathematicians, art historians, etc. That said, one could argue that Robbin’s addressed the range of these groups skillfully, giving all readers something to work with in ferreting out his views. In my case, when I found myself thinking I was not following the mathematical point, the drawings drew me in and offered another perspective through which I could grapple with the concepts. Given the importance of the visual element, I applaud Yale for a fine design.

In summary, Shadows of Reality is a book that takes the reader in and makes her welcome. It compares the slicing, or Flatland model, the God’s-Eye-View, with the projective spatial models that are only faulty approximations of physical events. It offers entry to the projective modality on its own terms as well. Groundbreaking and thought provoking, Robbin’s insights have an appeal that is well matched by its visual examples. Although the projects presented and the book’s many exciting ideas will require some time to sift through, they linger in my mind as I write. For this reason, I am sure I will return to Shadows of Reality on many occasions. I recommend it highly and think it would be a great addition to the libraries of all in our society seeking new understandings in art, science, mathematics, and visualization.



Updated 1st December 2006

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