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
jferng@mit.edu
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 19^{th}
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.