As the new millennium draws near, we in the United States are suffering from a dearth of millennial ideas. Supply-side economics, multiculturalism, cold fusion, caller I.D.--all these Promethean schemes for transforming the human condition have lately come to naught. In their wake, small ideas and incremental thinking hold sway. Yet the popular appetite for visionary conceits remains strong. Has Hillary Clinton not struck a chord with her "politics of meaning," especially when she calls for the "remaking of the American way of politics, government, indeed life"? These are nebulous words, admittedly, but I think I know what she is groping toward. I should therefore like to make a proposal, or rather a plea: a plea that we open our minds to the existence of higher dimensions. And since (as I shall explain) the beginning of higher dimensional wisdom lies in willfully inducing confusion between what is left and what is right, I can think of no forum in which this plea could be more aptly voiced than The New Republic.
What is a dimension, anyway? This has always been a vexed question. Dr. Johnson, after what he described to Mrs. Thrale as an "infernal struggle," defined it as "that which is mensurable in a direction orthogonal to a similar and coexistent extension." He accompanied this somewhat unhelpful gloss with the literary citation, "Hath not a Jew dimensions? (Shakespeare)." A line, we all know, has one dimension; a plane has two. We are said to live in a three-dimensional world, one structured by height, breadth and depth. For many great thinkers this does not just happen to be true--it is a matter of necessity. Aristotle, for instance, declared at the outset of On the Heavens that "the three dimensions are all that there are." Since everything has a beginning, a middle and an end, he argued, it follows that the number of the world is the triad. A body with more than three dimensions, he maintained, cannot exist even in thought. Ptolemy adduced a simpler demonstration of the intrinsic tri-dimensionality of nature: to wit, that it is impossible for more than three mutually perpendicular rods to meet at a point. This less than overwhelming bit of reasoning was also seized upon by Leibniz and by Galileo, who had the intellectual boldness to throw over Ptolemy's geocentric universe but could not part with the equally parochial 3-D worldview.
It was not until the seventeenth century that the possibility of higher dimensions was seriously considered. The Cambridge Platonists were probably the first to talk of a "fourth dimension"; one among their number, Henry More, suggested in 1671 that spirits were beings from this place. Around the same time Descartes took the seemingly innocuous step of adding an extra algebraic variable to his geometry, which enabled him to describe four-dimensional figures he called sursolides. Timid contemporaries found this intolerable; in 1685 John Wallis denounced the sursolide as a "Monster in Nature, less possible than a Chimera or a Centaur!"
The quest for higher dimensions suffered an apparent setback in the next century with Kant, who declared the three dimensions of space to be a pure form of intuition prior to all experience, and thus "apodictically certain." Yet in making dimensionality phenomenal rather than real--a matter of how our perception is structured, not an objective feature of the external world--he at least seemed to leave things flexible: Could we not somehow remove the 3-d glasses with which we make sense of our experience and replace them with, say, a 5-D pair? And what could the Konigsberg philosopher have had in mind when he wrote that "if it is possible that there could be regions with other dimensions, it is very likely that God has somewhere brought them into being"?
In the early nineteenth century Hegel presented in his Encyclopedia the latest and least intelligible in the long line of "proofs" that space must have precisely three dimensions. But just as Hegel had had the misfortune to demonstrate, by a priori reason alone, the necessity of there being seven planets in our solar system on the eve of the discovery of the eighth, so too was his dimensional proof infelicitously timed. For around the middle of the century, with the pioneering of non-Euclidean and n-dimensional geometries by Lobachevsky, Riemann and Grassmann, the mathematical reality of higher dimensions became undeniable. The great French mathematician Jules Henri Poincare reaffirmed the Kantian theme that space is a mental artifact: "Experience does not prove to us that space has three dimensions," he wrote. "It only proves to us that it is convenient to attribute three to it." (Poincare briefly considered, but ultimately rejected, the hypothesis that we experience space as three-dimensional for no deeper cause than that our sense of balance and orientation is due to three semicircular canals in our ears.)
At first this mathematical breakthrough, this magnificent triumph of the imagination over the senses, was greeted by scientists and philosophers with incredulity and downright enmity. Within a couple of decades, however, the notion that the space around us might possess occult dimensions came to beguile the public imagination much as the Big Bang does today. In England the fascination was especially acute, owing to the 1884 publication there of E.A. Abbott's Flatland: A Romance in Many Dimensions. In this first popular fiction on the subject--which doubled as a satire on Victorian society--a planar world of two-dimensional creatures is visited by a spherical sojourner from the third dimension, with subversive results. In the United States accounts of the fourth dimension abounded in the popular press; between 1902 and 1913 Harper's alone published no fewer than five articles on the subject.
The fourth dimension was warmly received in spiritualist circles, where experience with it was apparently quite routine in seances, mainly in the form of parlor tricks such as joining solid wooden rings and extracting objects from sealed boxes. Although Madame Blavatsky was less than enthralled by the idea, many of her fellow theosophists saw the fourth dimension as an antidote to the evils of materialism and strove to cultivate their "astral vision," the better to apprehend this noumenal reality. Writers such as Dostoyevsky, Wells, Proust, Wodehouse and Gertrude Stein sprinkled their books with references to it. In 1901 Joseph Conrad and Ford Madox Ford co-authored a novel called The Inheritors about a ruthless people from the fourth dimension who gradually take over our world. The cubists and the Russian futurists endeavored to paint it. Scriabin and Edgar Varese thought they could express it in their music. Oscar Wilde lampooned it: in his 1891 story "The Canterville Ghost," a wraith who has anciently haunted an English manor is exasperated by his inability to frighten the new tenants, a family of vulgarly unflappable Americans--"so, hastily adopting the Fourth Dimension of Space as a means of escape, he vanished through the wainscoting and the house became quiet."
Alas, just as the gospel of the fourth dimension was on the verge of liberating mankind, of bringing about a Brave New World of rococo geometry and boundless space, along came Einstein with his theory of relativity. By conflating space and time into the seamless four-dimensional manifold "spacetime," Einstein's theory misled the public and the intelligentsia alike into believing that the fourth dimension was merely time--nothing very marvelous or occult about that. Soon after, Freud did his part, submitting that believers in a fourth dimension were merely "projecting." The idea of higher spatial dimensions came to seem passé, and interest in it dried up.
Still, there remain a number of unexplained incidents that can be rendered intelligible only by appeal to the fourth dimension. The case of Gottfried Plattner is typical of these. In 1896 this Alsatian-born science and modern languages master at a West Sussex grammar school vanished without a trace when an explosion occurred during a chemistry experiment as a result of his students mischievously mismeasuring the reagents. Nine days later he reappeared just as suddenly, landing on top of a field worker some miles from the school. Medical examination revealed him to be quite fine, except that his heart now beat on the right side of his chest--that is to say, on the opposite side from everyone else's. His liver, lungs and other internal organs were similarly contraposed, and he had gone from being left-handed to right-handed. The only explanation for this anomaly was that the blast had propelled him into the fourth dimension, where a "rotation" of his body reversed left and right (in the same way a two-dimensional silhouette of a body can be reversed by flipping it over in the third dimension).
An even more disquieting episode apparently involving higher dimensions occurred in Los Angeles in 1940, when an obscure but gifted architect named Quintus Teal set out to design a house in the form of a 4-D hypercube, or "tesseract." Lacking direct access to the fourth dimension, he was forced to "unfold" the object in three dimensions. (Just as a regular 3-D cube can be unfolded and flattened into a cruciform array of its six square facets, so can the eight cubes that make up the facets of a 4-D hypercube be disassembled into a three-dimensional arrangement; Dale depicted Christ crucified on such a structure in his 1954 painting The Crucifixion: Corpus Hypercubicus.) Teal's house was more unstable than he realized, however. Soon after it was built in Laurel Canyon, a mild earthquake apparently caused it to fold up into an actual hypercube, presumably trapping the architect and his two clients, who were inspecting the new abode at the time, in the fourth dimension. None of the three has been heard from since.
In addition to such anecdotal evidence, powerful support for the existence of higher dimensions comes from contemporary particle physics. The only plausible way physicists have found to theoretically reconcile gravity (which governs the cosmos in the large) with quantum mechanics (which governs it in the small) is by supposing that space actually consists of nine dimensions. The six supernumerary ones go unnoticed by us in ordinary experience because of their extreme tininess: rather than being infinitely extended like length, breadth and depth, they are curled up into little circles of infinitesimal radii. According to this story, all nine dimensions were created equal at the Big Bang. For some reason, though, only three of them got inflated in the subsequent expansion; the rest have supposedly atrophied, at least in our region of the universe.
But have they really? Might we not be surrounded by wondrous caverns of possibility that our eyes do not yet behold? Perhaps we are like the prisoners in Plato's cave: fettered from childhood so that they are allowed to see only the shadows they cast on the cave wall, these benighted creatures mistake the images before them for reality; hearing their own voices echoed off the shadows, they imagine themselves to have only two dimensions. Or perhaps we are more like the bustard, a bird whose nervous system is adapted to the two-dimensional world of the open plains it inhabits; lacking the spatial imagination simply to step over obstacles it encounters, it goes on bumping against them until it tumbles across. For nearly the entire existence of mankind our own movements have been effectively confined by gravity to two dimensions, with the dream of flight expressing our yearning for the third; it was less than three decades ago that, seeing images of the round blue Earth from space for the first time, we came to appreciate fully the tri-dimensionality of our situation.
But how do we, individually and as a nation, make the leap beyond the third dimension? "Could I but rotate my arm out of the limits set to it, I could thrust it into a thousand dimensions," boasted one of the Utopians in H.G. Wells's Men Like Gods. Unfortunately, Wells left unaddressed the question of how one is actually to do this. Mystics meditate upon higher dimensions; physicists calculate with them. How, though, can we discover them in all of their physical reality and exploit them to become ... as gods?
The first step is consciousness-raising. We must learn to see at least one higher dimension--the fourth, let us say--in the mind's eye. The quickest route to such vision is via analogy. Imagine that, like the Flatlanders of Abbott's fiction, we were two-dimensional beings--circles, squares, triangles and so on--sliding around on an infinite plane. How would it look to us if a three-dimensional object, a sphere, for example, made its way through our horizontal world? Suppose its direction to be downward. At first, with the sphere above us, we would see nothing at all. At the moment in its descent when it made initial "contact" with our planar realm, we would observe a single point. As it continued its passage, the point would expand into a circle--a 2-D cross-section of the sphere--whose radius would grow to a maximum when the sphere was halfway through. Then we would see the circle begin to contract, shrinking down to a point again and disappearing as the sphere completed its transmigration. Now, try to fix in mind the higher-dimensional analog of this: a 4-D "hypersphere" passing through our 3-D space. A pointlike thing would suddenly appear, blow up into a beach ball, deflate again to a point, and then vanish. Simple enough. On reflection, it is not hard to see that any moving, wiggling, expanding, shrinking object can be viewed by us "Spacelanders" as a 4-D hyperobject flitting through our little 3-D world. The tricky part is making the intuitive leap from the 3-D apparition to the 4-D reality--the same leap that the Flatlander must make to imagine a full-blooded sphere from its two-dimensional manifestation.
Flatland cuts three-dimensional space into two parts: the region above and the region below; "Upward, yet not Northward!" was the motto of the Flatlander who aspired to move in a genuinely novel direction. Similarly, the 3-D world to which we believe ourselves confined bifurcates the four-dimensional space surrounding it. Christian doctrine acknowledges this in dividing the cosmos into three layers: the temporal world, heaven "above," though in no assignable direction, and hell "below." But we need new words to name these otherworldly directions. One turn-of-the-century enthusiast suggested ana and kata (Greek for "above" and "below"); an even better choice would be apo ("away") and eiso ("within"). And to denote extension in the fourth dimension, eking out the series "length," "breadth" and "depth," how about throughth? Another aid to intuiting the fourth dimension (as I mentioned at the outset) is deliberately confounding left and right--say, by pretending you are simultaneously driving your car in the United States and in the U.K. From the perspective of the fourth dimension, the distinction is meaningless: our 3-D world viewed from the ana direction is the mirror image of it viewed kata-ward. A left shoe is equivalent to the right one; neoliberals are the same as neoconservatives.
Once the fourth dimension is imaginatively visualized--a task that might require the devotion of a lifetime--stepping out into hyperspace should become practicable. Maneuvering oneself into the fourth dimension means moving, in a rather special sense, toward one's inside, in the direction of one's heart. This must be accomplished through a pure act of will, since our muscles are all oriented for 3-D motility and any "hypermuscles" with which we may be endowed have certainly atrophied for want of use.
And what vistas will be open to us, what new powers will we possess, when this feat is accomplished! To grasp the possibilities, try to conceive of what the Flatlander would experience were he to be lifted above his planar world: since all enclosures in Flatland are merely two-dimensional, he would be able to peer inside of every chamber; the entrails of his fellow Flatlanders would be open to his view; he could effortlessly escape any cocktail party or reach into any safe; and so on. Now, imagine the analogous clairvoyance and omnipotence we would enjoy were we capable of moving freely into the fourth dimension. Opportunities for plunder would be abundant. The greedy hyperspatialist could enter any wine cellar and drink up all the 1961 Lafite-Rothschild without even having to uncork it, or reach "around" the window at Tiffany's and snatch the biggest diamond on display. A surgeon with the 4-D knack could remove your appendix without cutting into your skin, just like a Filipino healer. Iraqi nuclear facilities would be immediately visible to satellites launched into the fourth dimension. The federal deficit could be rotated in hyperspace to become a surplus. The pious would have the option of moving heavenward without all the bother and expense of dying. The Gordian knot would unravel in a trice. An enormous number of parking spaces would suddenly open up; missing socks and tools would be gratefully recovered.
The intellectual and philosophical gains are even more exhilarating to contemplate. As the Russian mystic P.D. Ouspensky noted in his Tertium Organum (1912), the fourth dimension offers us an explanation of all the "enigmas of the world," the riddles of life and death, of matter and mind. How is it, for example, that spirits seem to hover so close to us without actually occupying physical space? Simple: they are but a few feet away in the fourth dimension--guardian angels ana and malevolent demons kata, with poltergeists moving back and forth. Miracles become explicable without ceasing to be miraculous: Jesus fed the 5,000 not by shaming them into bringing out from under their cloaks the box lunches they had been selfishly concealing--as some naturalistically inclined theologians have supposed--but by producing loaves and fishes from the fourth dimension. (The Red Sea crossing is equally easy to account for, but I am still struggling to find a 4-D explanation for how the walls of Thebes were built by a flute alone.) Just as seemingly incompatible scientific theories such as relativity and quantum mechanics can be reconciled in higher dimensions, so can the array of conflicting theories about JFK's assassination be hyperspatially "rotated" to fit together harmoniously. The fourth dimension will permit all contradictions to be resolved, and will usher in the badly needed trope of hyperirony. Metaphysically, it will bring about a great leap forward; as Ouspensky wrote, "When we shall see or feel ourselves in the world of four dimensions we shall see that the world of three dimensions does not really exist and has never existed; that it was the creation of our own fantasy, a phantom host, an optical illusion, a delusion--anything one pleases excepting only reality."
Most important, the apprehension of higher dimensions will lead to a new image of man. Instead of viewing ourselves as finite creatures, born to rut and flourish for a season only to wither and die, we would see that we are truly timeless hyperbeings capable of endless dimensional ascent. Passing through this 3-D vale of tears--manifesting our younger, more comely facets first, as it happens, and our older, uglier ones later--we are destined eventually to make our way into the fourth dimension, and thence to the fifth, the sixth and so on, asymptomatically approaching our Creator, who dwells in a space of infinite dimension.
One morning not long ago I was idly fooling around with some chopsticks left over from the previous evening's takeout Chinese. Suddenly I realized I had done a prodigious thing: I had arranged four of the chopsticks so that they were all perpendicular to one another! In other words, I had stumbled onto an empirical rebuttal of the strongest argument Ptolemy, Galileo and Leibniz could muster to ground their conviction that space is inherently tri-dimensional. Unfortunately, at that moment a fact checker from a ladies' magazine rang up with a few annoying queries, and I had to put the sticks down. Returning to them a few minutes later, I found to my immense frustration that I could not get them back into the 4-D configuration again.
A week or so later I was strolling through Chinatown and, on a whim, dropped into a curio shop. As I browsed in the dusty, gloomy, cluttered interior, a peculiar item composed of delicately painted matchsticks caught my eye. On closer examination I discovered that its intricate geometry included a vertex at which four of the sticks met orthogonally. My heart leapt: a four-dimensional bibelot! When I asked the old Chinese man who ran the place how much he wanted for it, he quoted a price of $12, which struck me as rather dear. After arguing with him for a while, I exited the shop in dudgeon, figuring I could find the same thing cheaper elsewhere.
I am sorry to report that I have not yet succeeded in doing so; nor have I been able to find the shop with the original object again. When I do, I shall certainly buy the thing and bring it to Washington. There I'll make an offering of it to Mrs. Clinton as a sort of seed or crystal around which American society, politics--and indeed life--can be reconstituted along four-dimensional lines, for a truly millennial millennium. We had better hurry: I hear the Japanese have already got something quite similar in the works.
This article originally ran in the July 12, 1993, issue of the magazine.