Shadows of Reality is a book that not only makes a good first impression but also follows 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 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 19th 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, 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 4D world is to our space as a 3D world is to a Flatlander. While an effective analogy, it largely ignores the use of projective techniques to study 4D 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 space-time. 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 4D geometry, including the work and pioneering drawings of Washington Irving Stringham, Pieter Henderick Schoute and Esprit Jouffret. Integrating a number of case studies, Part One examines past uses of the projective model. Here Robbin walks us through early 20th-century ideas, examples of painting and various constructions of the 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...