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
ione@diatrope.com
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.