Hiding in the Mirror (10 page)

All of these diverse notions about a fourth
dimension were widely debated and culminated in a 1909 essay
contest sponsored by
Scientific Ameri-
can
for the best “explanation of the Fourth
Dimension.” Of particular interest today, because of their
prescient resemblance to arguments that would later become part of
modern lore, was the stated possibility, à la Hinton, of multiple
three-dimensional universes existing within a fourdimensional
framework. The development of special relativity ultimately did
provide, in 1908, via Minkowski’s work, a scientific basis for a
fourth dimension, but not the spatial fourth dimension so cherished
by Hinton, Abbott, Wells, and others. Nevertheless, Einstein’s work
did play at least an indirect role in rekindling a surge of
cultural interest in extra spatial dimensions. One of the chief
instigators of this was Henri Poincaré, the French mathematician
whose own work on symmetries of space and time played a role in the
development of relativity. The relativity of length and time
measurements that were a hallmark of the special theory somehow
implied to Poincaré that
all
our sense
perceptions were relative, including even our perception of the
number of dimensions.

In his book
Science et
Méthode
(1908), Poincaré wrote: “So, the characteristic
property of space, that of having three dimensions, is . . . an
internal property of human intelligence, so to speak.” Like Hinton
before him, Poincaré believed that the key to revealing the inner
reality of extra dimensions involved breaking the bonds of our
limited three-dimensional intuition. As he put it: “One who devoted
his life to it could
perhaps
eventually be
able to picture the fourth dimension.” He was a tremendously
influential intellectual figure in early-twentieth-century France,
and his extended notion of what one might call philosophical
relativism and the associated idea that the four-dimensional world
was accessible to us had wide impact.

The ways in which science has had an impact
upon our culture are fascinating, and no doubt deserve more
discussion than I can provide here. Yet what we see in the adoption
of concepts like four dimensions and relativity as a framework for
other philosophical purposes is, I suspect, more universal. People
adapt what they
perceive
are scientific
ideas and apply them with their own particular prejudices. They
pick and choose what resonates, and the results may ultimately bear
little resemblance to the actual underlying science.

Among those who helped further popularize the
French fascination with four dimensions was the journalist, editor,
theater critic, and science fiction writer, Gaston de Pawlowski,
whose
Journey to the Country of Four
Dimensions
(1912) was first serialized in installments on the
front page of the literary journal
Comoedia
.

Pawlowski’s literary effort, like Wells’s
Time Machine,
involved a voyage to the
future. But unlike Wells, he used the fourth dimension as a plot
device to reflect a time when the tyranny of scientists, with their
threedimensional science, would be replaced in a future Utopia,
once the existence of four dimensions was revealed to the world.
Whatever one may think of this premise, Pawlowski helped instill a
notion that would be popular in France and elsewhere for
generations: Namely, that a lack of the proper vocabulary, both
visual and verbal, has hindered our ability to free our minds to
fully appreciate the underlying reality of four dimensions. As he
wrote:

The vocabulary of our language is in fact
conceived according to the given facts of three-dimensional space.
Words do not exist which are capable of defining exactly the
strange, new sensations that are experienced when one raises
himself forever above the vulgar world. The notion of the fourth
dimension opens absolutely new horizons for us. It is precisely
this excitement of freeing our minds, extending the range of our
senses, and opening ourselves to new experiences that is so
seductive.

Ultimately the growing call for a new
vocabulary with which to explore our world resonated most strongly
with visual artists, whose aesthetic is directly tied to pushing
against the limits of our reality. I wrote earlier about Vincent
Van Gogh freeing us in 1882 from the tyranny of color, and
demonstrating exotica otherwise hidden in ordinary objects. But as
strange and hauntingly pleasing as his images are to the modern
eye, they nevertheless preserve the spatial relationships of all
the objects they represent, which remain, in spite of their jarring
colors, more or less instantly recognizable. This was not to be
true of a school of artists that comprised perhaps the most
influential painters and sculptors of the twentieth century who,
starting about 1910, also began to transform the very definition of
art. One merely has to glance at Picasso’s famous
Man with Violin
(1911–12) to realize that a new way
of viewing the world was emerging. Even in his early
Les Demoiselles d’Avignon
(1907), one can see in the
distorted faces the beginnings of what would become a
characteristic trait of presenting different perspectives on parts
of figure in the single plane of a painting.

It is said that a picture is worth a thousand
words. But what if, as Pawlowski stressed, words fail completely?
Charles Hinton spent much of his life attempting to teach others
how to develop a visual intuition about four-dimensional space, as
he believed he had himself done. Recall that the heart of his
technique, which was reflected in essentially every other subsequent
effort, including A. Square’s, was to display different
threedimensional projections of a four-dimensional object as it is
“rotated” in the fourth dimension. Just as one can color the six
faces of a cube and display the different colors that result when
one rotates it by ninety degrees in order to help visualize both
the nature of the cube and precisely what is meant by the set of
rotations in three dimensions, one might hope to build up a similar
understanding of four-dimensional space by considering the
different three-dimensional projections of a tesseract, for
example. The similarity between Hinton’s approach to the tesseract
and Picasso’s approach to his models is striking. But is there more
to it than a simple spatial operation? Certainly, Picasso never
claimed there was. His famous statement, “I paint objects as I
think them, not as I see them,” was more a reflection of his protest
against the confines of standard perspective than a claim to be
interpreting higher dimensions. Just as Van Gogh fought against the
tyranny of color, one might say that Picasso and his contemporaries
Braque, Gris, Metzinger, Weber, and Duchamp were struggling to free
us from the tyranny of space. Yet, at the same time the ultimate
goals of the mathematicians and the artists were similar: to compel
us to use our minds to liberate ourselves from the confines of our
own experience. Picasso was a product of the intellectual ferment
of those heady times after the turn of the century, and this was
also reflected in the cubist revolution, in which he was a leading
figure. The circle of artists and writers at the Bateau Lavoir in
Montmartre, where cubism had its origins, discussed many of the
exciting ideas of the day, including extra dimensions. While cubism
was born out of a sense of questioning of the traditional views of
the world, if the existence of an extra dimension could provide
validation for its attempt to extract a new, hidden reality in
nature, all the better. Certainly those authors who chose to write
about cubism—notably Jean Metzinger and Guillaume Apollinaire—as
well as related French literary figures like Jarry—and ultimately
the artist perhaps most closely associated in the modern mind with
this aspect of the movement, Marcel Duchamp, all explicitly
described a relationship between cubist art and four dimensions,
with the analogies being alternately poetic and explicit. Witness
Duchamp, in a later interview, discussing his motivation in
creating one of his most famous pieces representing a
higher-dimensional reality,
The Bride Stripped
Bare by Her Bachelors, Even (the Large Glass),
created between
1915 and 1923:

What we were interested in at the time was the
fourth dimension. . . . Do you remember someone called, I think,
Povolowski?

He was a publisher, in the rue Bonaparte. . . .
He had written some articles in a magazine popularizing the fourth
dimension. . . . In any case, at the time I had tried to read
things by Povolowski, who explained measurements, straight lines,
curves, etc. That was working in my head while I worked, although I
almost never put any calculations into the
Large Glass
. Simply, I thought of the idea of a
projection, of an invisible fourth dimension, something you
couldn’t see with your eyes.

Notes, however, for
Large
Glass
do contain substantial references to mathematical
discussions of a fourth dimension, including the writings of
Poincaré. While Duchamp claimed only a passing knowledge of these
ideas, observations he made in these notes, such as, “Poincaré’s
explanation about n-dim’l continuums by means of the Dedekind cut
of the n-1 continuum is not in error,” demonstrate the depth of his
interest in the topic. Interestingly, in spite of his truly
meticulous efforts to methodically attempt to portray projections
of a fourth dimension—efforts that made him more than any other
artist an explicit student of this mathematics—

Duchamp later disavowed them. “It wasn’t for
love of science that I did this,” he said. “On the contrary, it was
rather in order to discredit it, mildly, lightly, unimportantly.
But irony was present.”

For Duchamp, then, as well as for his cubist,
and literary contemporanes, reacting against a three-dimensional
Euclidean world was subversive and thus attractive. I use the terms
three-dimensional
and
Euclidean
here in spite of the fact that there is
nothing about the four-dimensional space-time of Minkowski or, for
that matter, the four-dimensional projections of Hinton and others
that is remotely non-Euclidean. These spaces are quite flat. Having
to go beyond Euclid to consider a possible curvature of space is
essentially never explicit, except perhaps in Duchamp’s piece,
Stoppages,
and in the later distorted
landscapes of Salvador Dali. Yet, in the literature of cubism
non-Euclideanism was rampant. Indeed, in one of the first essays on
cubism, “Du Cubisme” (1912), by Albert Gleizes and Jean Metzinger,
the authors state explicitly: “If we wished to tie the painters’
space to a particular geometry, we should have to refer it to the
non-Euclidean scholars.”

Somehow what was occurring, one might argue,
was a rebellion against perspective, one of the hallmarks of our
three-dimensional world. Certainly a curvature of space, causing
light rays to travel on curved paths, is one way to distort
perspectives, but another is to imagine viewing many different
three-dimensional perspectives simultaneously, which was the
preferred method of the cubists. Duchamp, one of the most
mathematically literate of the emerging school, employed both
non-Euclidean themes and multiple perspectives. This ultimately
allowed him to go even further in his art, becoming perhaps the
first of the modern conceptual artists. While the liberation
achieved by abandoning three-dimensional perspective was
intoxicating, it may have been inspired, at least for some, by an
incorrect understanding of the developments in science at that
time. I have no idea if Einstein, a notorious antiauthoritarian,
coined the word
rel-
ativity
with malice aforethought, but the term
carries a great deal of intellectual baggage, and has encouraged,
and continues to encourage, the incorrect notion that it somehow
does away with all absolutes, making truth itself relative and
observer dependent. And if special relativity, which demonstrated
that space and time are tied together into a four-dimensional
space-time, had everything to do with absolutes, it also has
virtually nothing to do with the non-Euclidean ideas that so
fascinated many of the writers and artists of the time, who may
have seemed in retrospect to have been inspired by it.

One must remember also that Einstein was not
yet the household name he would become in 1919, following the
observations of the bending of light from distant stars which
confirmed the predictions of general relativity. There is no doubt
that with the passing of time his perceived impact on his cultural
contemporaries may be viewed as being more significant than it
actually was. In any case, as I have argued, the facets of a fourth
dimension that most fascinated artists and writers alike actually
had little to do with the actual ideas contained in special
relativity, but were at best culled and adapted from what they
perceived the theory might contain, based on preexisting cultural
fascinations. In spite of the confusions regarding the nature of
the four-dimensional universe implied by relativity, and about the
relations between nonEuclidean geometry and the geometry of extra
dimensions, the almost accidental prescience about these concepts
in the literary and artistic worlds at the beginning of the
twentieth century was remarkable. I have often found (for example,
when I have in other books compared science fiction and science)
that the confluence of ideas and language among different
disciplines is simply due to the fact that when creative people
think about similar problems, even from totally different vantage
points, they sometimes come up with similar ways of approaching
them.

An even more remarkable coincidence, perhaps,
lies in the fact, as I shall next describe, that the first concrete
scientific proposal for the existence of extra spatial dimensions
arose not by generalizing the notions of the space-time of
Minkowski, but rather by attempting to extend Einstein’s general
theory of relativity, building, as fortuitously envisaged by many
of the cubist artists, a bridge between curved space and extra
dimensions that has been central to the scientific pursuit of extra
dimensions into the twenty-first century.