Adventures in hyperspace

Shadows of Reality: The Fourth Dimension in Relativity, Cubism, and Modern Thought, by Tony Robbin, New Haven and London, Yale University Press, 2006; 160 pages, $40.

Not long ago, the problem of space was central to artistic discourse. Any serious discussion of works by artists ranging from Picasso to Mark Rothko, Yves Klein to Al Held, would require one to address the issue of spatial relationships and the means to convey them in artistic terms. While the art world's focus may have shifted away from this area, overcoming the limitations of human perception of space remains a central preoccupation for some artists, as it does for many scientists and mathematicians. Direct observation and conventional mathematical and scientific analysis suggest that the world we take for granted is composed of three-dimensional space in which objects are defined by height, width and depth. More difficult to define or to prove, though far more accurate in explaining the nature of the universe's varied spatial phenomena, is the notion of hyperspace. In this theoretical but wholly credible realm of four or more dimensions inserted into three-dimensional space, data may be gauged by means of a geometry that is seemingly in perpetual flux.

How can one describe the four-dimensional space that we all experience as forms in motion but lack the capability to illustrate convincingly and in a consistent manner? Since the mid 19th century, mathematicians have attempted to illuminate the principles of hyperspace by means of elaborate equations. But visualizing this multidimensional domain and making it intelligible in a two-dimensional illustration require the skills and imagination of an artist. Shadows of Reality: The Fourth Dimension in Relativity, Cubism and Modern Thought by New York-based painter and theorist Tony Robbin traces the development of fourth-dimension imagery, from its origins in mathematical diagrams through Cubist painting and recent computer-generated representations.

Robbin, who has been engaged with this topic for over 30 years, is a well-known lecturer on the fourth dimension in art and architecture. One of my more inspiring undergraduate art history professors in the mid-1970s, he discussed the topic at length in class. Robbin pointed out, for example, how Al Held's hard-edge abstractions from the early 1970s on suggest a four-dimensional space. Robbin does not consider time as the fourth dimension, but here, on perhaps the most rudimentary level, one can recognize that the element of time is intrinsic to Held's work. The intricacies of the geometric forms cannot be taken in at a single glance, and the time element required to fully grasp the overall design will vary with individual perception. While Held's compositions are not based on mathematical formulae and the artist discounted the influence of experimental geometry in his work in a conversation I had with him some years ago, the paintings nevertheless served Robbin well as examples of how one might visualize four-dimensional space.

In the 1980s, Robbin became a pioneer in computer visualization of four-dimensional geometry; he holds a number of patents for its application in architecture. He touches upon some of these building concepts in Engineering a New Architecture, a 1996 book focused on an examination of the experimental geometry that inspired key structures by R. Buckminster Fuller, Frei Otto, Emilio Pinero and others.

In his current book, Robbin discusses a number of the best-known visual representations of hyperspace while exploring the field's fascinating and complex history beginning in the early 1800s. He highlights the resurgence of interest in the topic in the wake of Einstein's theory of relativity in the early 20th century and Einstein's subsequent impact on developments in four-dimensional illustration, including recent computer imaging. It is a story filled with intrigues, breakthroughs, setbacks, dead ends and revelations. But readers ought not to expect a nail-biting suspense tale. The text is sprinkled with technical terms and mathematical equations that might put off those of us who are mathematically challenged. A glossary would certainly have helped greatly. In addition, Robbin's overreliance on numerous sidebars to elaborate upon issues briefly addressed in the main narrative causes the book to seem cumbersome at times. Yet his historical approach and the generous supply of visual information accompanying the text, including rare archival material and numerous reproductions of artworks, help to move the story along.

The author begins by highlighting the two most basic and widely accepted metaphors for four-dimensional space, the Flatland or slicing model and the shadow or projection model. The first, with its grounding in calculus, describes, for instance, a geometric form in motion as a progression of cutaway images, or slices. The projection model, which Robbins believes is the more accurate of the two, also has a firm mathematical foundation. It could be imagined as the shadows cast by the object as the sun moves overhead. While lengths and angles of the shadows might be distorted, the corresponding relationships among the object's parts remain consistent and thus better preserve the object's integrity as a whole. As Robbin says later in the book, "A projective model that rids us of the false notion of spaces stacked on spaces puts us on the road to reality."