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Geometrical Creatures


One feature of the world that few people stop to puzzle over is how many dimensions it has. Although it is a little tricky to say just what a dimension is, it does seem fairly obvious that we, the objects that surround us, and the space we move about in are structured by three dimensions, conventionally referred to as length, width, and depth. Even philosophers have tended to take this for granted. Aristotle, at the beginning of On the Heavens, declared that “the three dimensions are all there are.” Why? Because, he argued in a somewhat mystical vein, the number three comprises beginning, middle, and end. Therefore, it is perfect and complete.

The first philosophers to talk about a “fourth dimension” were the seventeenth-century Cambridge Platonists, but they seemed to have something more spiritual than spatial in mind. One of them, Henry More, suggested in 1671 that the fourth dimension was the abode of Platonic ideas and, quite possibly, ghosts. Around the same time, Descartes took the seemingly innocuous step of adding an extra variable to his coordinate geometry, which enabled him to define four-dimensional entities he called sursolides. Timid contemporaries found this intolerable; in 1685, the mathematician John Wallis denounced the sursolide as a “Monster in Nature, less possible than a Chimera or Centaur!” Kant, in his early writings at least, flirted with the idea that three-dimensional space might be contingent; perhaps, he conjectured, God created other worlds with different numbers of dimensions. By the time he wrote the Critique of Pure Reason, however, Kant had decided that space was not an objective feature of reality, but something imposed on it by the mind to give order to experience. Moreover, he held, its character was irrevocably Euclidean and three-dimensional—this we know with “apodictic certainty.”

Meanwhile, in mathematics, a revolution was getting underway. Some mathematicians began to investigate curved geometries, where the shortest distance between two points was no longer a straight line; others extended the Euclidean system to spaces of more than three dimensions. These developments were brought together in a magnificent synthesis by Georg Friedrich Bernhard Riemann (1826– 1866). In an 1854 lecture before the faculty of the University of Göttingen, titled “On the Hypotheses Which Lie at the Foundation of Geometry,” Riemann toppled the Euclidean orthodoxy that had dominated mathematics—and, indeed, Western thought—for two millennia. According to Euclid, a point has zero dimensions; a line, one; a plane, two; and a solid, three. Nothing could have four dimensions. Moreover, Euclidean space is “flat”: parallel lines never meet. Riemann transcended both these assumptions, rewriting the equations of geometry so that they could describe spaces with any number of dimensions, and with any kind of curvature. (In a spherical space, for example, there is no way that parallel lines could not meet.)

The Riemannian revolution destroyed the old notion of geometry as the science of physical space. Clearly, there was nothing metaphysically necessary about three dimensions. An endless variety of other spatial worlds was possible; they could be described by a logically consistent theory, and hence were mathematically real. This led to an interesting line of speculation. Could such worlds ever be visualized by the human imagination? Was it just an accident that, among all the possible spatial architectures, we find ourselves living in a world of three dimensions? Or—an even more heady thought—could it be that we do live in a world with more dimensions, but, like the prisoners in Plato’s cave, we are too benighted to realize it?

A person who devoted his life to it might perhaps eventually be able to picture the fourth dimension,” wrote the great French mathematician Henri Poincaré. Some nonmathematical charlatans went even further. In 1877, the fourth dimension achieved notoriety when an American psychic named Henry Slade was put on trial in London for fraud. In a series of séances with prominent members of London society, Slade had purported to summon spirits from the fourth dimension. Distinguished physicists, including two future Nobel laureates, sprang to Slade’s defense, gulled by his knack for doing parlor tricks that supposedly exploited this unseen extension of space (like miraculously extracting objects from sealed three-dimensional containers). Although Slade was convicted, the mysterious fourth dimension had captured the public imagination. It was in this atmosphere that Edwin A. Abbott (1838–1926) wrote the first, and most enduringly successful, popular fiction on the subject, a little masterpiece entitled Flatland: A Romance of Many Dimensions.

Since its initial publication in 1884, Flatland has been through countless editions, with introductions by Ray Bradbury and Isaac Asimov, among others. The most rewarding of them, to my mind, is this new one annotated by Ian Stewart. Why an annotation? Because, as its subtitle suggests, Flatland has several dimensions: it is a scientific fantasy that coaxes the reader into imagining unseen spatial realms; it is a satire on Victorian attitudes, especially those concerning women and social status; it is an allegory of a spiritual journey. Stewart, a professor of mathematics at the University of Warwick, is himself an able (and prolific) popularizer. Here he does an admirable job of expanding and updating the scientific ideas evoked by Flatland. He also plays the amateur historian and literary sleuth, often letting his fancy take its own sweet way.

Edwin Abbott was a typically tireless Victorian, rating two double-columned pages in the Dictionary of National Biography. He was the longtime headmaster of the City of London School, where his devoted pupils included the future prime minister H.H. Asquith. He was a Broad Church reformer, celebrated for his preaching at Oxford and Cambridge. An avid Shakespeare scholar, he produced A Shakespearian Grammar (1869), a standard reference, along with some fifty other books, many on ponderous theological matters. What possessed him to write a consciousness-raising mathematical satire like Flatland—his only book still in print—is not really known. As a progressive educator, he may have wished to shake up the English mathematics curriculum, with its dreary emphasis on the memorization of long proofs from Euclid. And, as a modern-minded churchman, he was no doubt attracted to the challenge of reconciling the spiritual and scientific worldviews.

Abbott’s doorway to higher dimensions was the method of analogy. We cannot readily picture to ourselves a space with one more dimension than our three-dimensional world. We can, however, picture a space with one fewer dimension: a plane. Suppose there were a society of two-dimensional creatures confined to a planar world. What would they look like to us? And—a more interesting question—how would three-dimensional beings like us look to them, assuming we could somehow pass through their world or lift them into ours?

Abbott’s Flatland consists of an infinite plane inhabited by a rigidly hierarchical society of geometrical creatures. Social pedigree is determined by the number of sides the creature has: lowly women are mere line segments, whereas men range from working-class triangles to bourgeois squares to aristocratic polygons with five sides and up; the priestly caste consists of polygons with so many sides that they are virtually circular. There is some upward mobility: in a caricature of Victorian notions of progressive evolution, well-behaved members of Flatland’s lower classes occasionally bear children with a greater number of sides than themselves.

The narrator of the book is a conservative lawyer aptly named A. Square. In decorous language, he explains to the reader (presumed to be a three-dimensional “Spacelander” like us) the history and folkways of his world. Flatland abounds in absurdities, which are often imperfectly appreciated by Mr. Square. For example, women, though “devoid of brain-power,” are extremely dangerous because of their needle-like linear form; in a fit of anger (or sneezing) the “Frail Sex” is capable of inflicting instant death by penetrating a polygonal male. Life in a two-dimensional world also poses many practical difficulties. How can Flatlanders recognize one another by sight? Suppose you are looking at a penny on a table. Viewed from above, its round shape is apparent. But if you get down to the level of the tabletop, the penny’s edge appears to be a straight line. In Flatland, everyone and everything looks like a straight line. To overcome the recognition problem, women and tradesmen feel one another in their social intercourse. The upper classes find this beyond vulgar; “to feel a Circle would be considered a most audacious insult,” Mr. Square tells us. Instead, they rely on their cultivation of depth perception and geometry to determine one another’s angles, and hence social rank.

At one point in Flatland’s history, reformers attempted to eliminate the invidious distinction between the feeling and the nonfeeling classes by in-troducing a “Universal Colour Bill.” Henceforth, Flatlanders would have the right to adorn their linear profiles with colors of their choice, thus achieving equality of recognition and, incidentally, relieving the aesthetic dullness of their world. But the popular uprising in favor of this reform was bloodily suppressed, and the forces of reaction banished color from Flatland.

The social satire in Flatland tends to be schematic and a bit labored. Nor is Abbott altogether successful in setting up a consistent scheme for two-dimensional life, as Ian Stewart nicely shows in his notes. Take the question of how sound and hearing work in Flatland. In a space with an odd number of dimensions, like our own three-dimensional world, sound waves move in a single sharp wavefront. If a gun is fired some distance away, you hear first silence, then a bang, then silence again. But in spaces with an even number of dimensions, like a two-dimensional plane, a noise-like disturbance will generate a system of waves that reverberate forever. Living in Flatland would, in Stewart’s words, “be like living inside a tin can.”

Another matter overlooked by Abbott is how a Flatlander’s brain might operate. Imagine the nightmare of having to draw up neural circuitry on a two-dimensional piece of paper, where there is no room for wires to cross each other without intersecting. Stewart cleverly solves this problem for the author by invoking the theory of “cellular automata”: “two-dimensional arrays of cells that, communicating with their neighbors according to simple rules, can carry out any task that a computer could accomplish.” With cellular automata for brains, Flatlanders would be capable of intelligent behavior (although, as philosophers like John Searle have argued, they might fall short of consciousness). As for more intimate details of the Flatlanders’ existence—their method of sexual congress, for example—the author maintains a proper Victorian reticence, which his annotator respects.

But Abbott is not primarily interested in the details of a functioning two-dimensional world. His real theme is higher dimensions, and that is taken up in the second (and more interesting) half of the book. One night, while Mr. Square is at home with his wife, he is visited by a ghostly “Stranger,” a spherical being from Spaceland. How can the three-dimensional sphere enter a two-dimensional world like Flatland? The reader is invited to picture Flatland as something like the surface of a pond. By rising from the deep and breaking this surface, the sphere can manifest himself to the two-dimensional beings who float upon it. At first, they would see nothing at all. Then, as the sphere first made contact with the surface, they would see a single point. As the sphere continued to rise, they would see this point expand into a circle, whose radius would grow until it reached a maximum when the sphere had passed halfway through the surface. Then the Flatlanders would see the circle begin to contract, shrinking to a point again and then disappearing altogether as the sphere rose completely above the surface. (See illustration on this page.)

Mr. Square is alarmed by this apparition. How can the Stranger—who must be of the priestly caste, judging from his seeming circularity—appear from nowhere and make himself smaller or larger at will? The Stranger explains that “in reality I am not a Circle, but an infinite number of Circles,” all of varying sizes. Mr. Square, who cannot grasp the idea of a three-dimensional solid, is incredulous. So the Stranger tries to prove his assertion with several tricks, like rising up out of Flatland, hovering invisibly over Mr. Square, and touching the center of his stomach. Finally, the Stranger lifts the enraged Mr. Square out of Flatland into the world of Space. There, floating as insubstantially as a sheet of paper in the breeze, Mr. Square looks down on his two-dimensional world and sees the entire form and interior of every person and building. (To return to the example of the penny on the table, imagine looking at its edge from the level of the tabletop, and then rising up and looking at it from above: suddenly you can see the whole circular form and Lincoln’s head “inside” it.)

Even more awesome is Mr. Square’s vision of the Stranger, who now appears in his full tridimensionality:

What seemed the centre of the Stranger’s form lay open to my view: yet I could see no heart, nor lungs, nor arteries, only a beautiful harmonious Something—for which I had no words; but you, my Readers in Spaceland, would call it the surface of the Sphere.

Once Mr. Square gets habituated to Spaceland, he is quick to grasp the analogy to higher dimensions. If his familiar two-dimensional world is an infinitesimal sliver of three-dimensional space, he reasons, perhaps three-dimensional space is but a sliver of a four-dimensional space, a space harboring hyperspheres that surpass the spherical Stranger the way the Stranger surpasses the circular priests of Flatland. And why stop there?

In that blessed region of Four Dimensions, shall we linger on the threshold of the Fifth, and not enter therein? Ah, no! Let us rather resolve that our ambition shall soar with our corporal ascent. Then, yielding to our intellectual onset, the gates of the Sixth Dimension shall fly open; after that a Seventh, and then an Eighth….

But the Stranger reacts coldly to Mr. Square’s ecstatic aspirations. Ironically, although he has come to preach the gospel of three dimensions, he himself cannot make the imaginative leap into the fourth dimension. “Analogy!” he harrumphs. “Nonsense: what analogy?” In a fit of dudgeon he hurls our narrator back into Flatland, where the authorities promptly clap him into prison for his subversive ravings about a higher reality.

The spiritual aspect of Flatland is patent. It tantalizes the reader with the possibility that there is an unseen world surrounding us, one that lies in an utterly novel direction—“Upward, not Northward!” as Mr. Square tries to express it; a world that contains miraculous beings which can hover about us without actually occupying our own space. Soon English clergymen were talking about the fourth dimension as the abode of God and his angels. The idea cropped up in the works of authors like Proust, Dostoyevsky, Oscar Wilde, and Gertrude Stein. It seduced the artistic avant-garde, who invoked it to justify the overthrow of three-dimensional Renaissance perspective. The Cubists were especially enthralled: viewed from the fourth dimension, after all, a three-dimensional object or person manifests all perspectives at once. (Recall how Mr. Square, on ascending above Flatland, could for the first time see all the edges of the objects in that two-dimensional world.)

Apollinaire wrote in Les Peintres Cubistes (1913) that the fourth dimension “is space itself” and “endows objects with plasticity.” The notion of unseen higher dimensions was seized upon by Theosophists, who saw it as a weapon against the evils of scientific positivism and who cultivated their “astral vision” the better to apprehend it. Vladimir Lenin, alarmed by its seeming spiritualist implications, attacked the notion in his Materialism and Empirio-Criticism (1908). Mathematicians might explore four-dimensional space, Lenin wrote, but the tsar can only be overthrown in the three-dimensional world.

The one part of culture that was largely untouched by the craze over higher dimensions was science. Not only was the idea rendered disreputable by its association with mystics and cranks; it also seemed devoid of testable consequences. Then, during World War I, Einstein framed his theory of general relativity. By uniting the three dimensions of space and the one of time into a four-dimensional manifold, “space-time,” and explaining gravity as a curvature within this manifold, Einstein’s theory gave the impression that the hitherto mysterious fourth dimension was merely time. Interest in higher spatial dimensions began to die out—quite prematurely, as we shall see.


To mathematicians, of course, it is irrelevant how many dimensions physical space happens to have. The non-Euclidean revolution freed them to theorize about spaces of all conceivable types, and this they continued to do even after the passing of the fourth-dimension vogue, inventing ever more exotic varieties: “Hilbert space,” with an infinite number of dimensions; “fractal” spaces, which might have, say, two and a half dimensions; rubbery topological spaces; and on and on. Geometry has come a long way since Flatland. To bring the popular imagination up to date, Ian Stewart has written a sequel, inexcusably titled Flatterland. Its main character is “Victoria Line,” the great-great-granddaughter, we are told, of the narrator of Flatland. On discovering the manuscript of her ancestor’s banned narrative of his trip into Spaceland, young Vikki beckons a higher-dimensional being to take her on a similarly enlightening excursion. She is rewarded when a creature called the Space Hopper guides her not only through the third dimension, but through all sorts of wild geometries lurking in various regions of the “Mathiverse.”

Stewart’s sequel has little of the literary charm of Flatland. Instead of Abbott’s stately Victorian cadences, we get a slangy vernacular packed with would-be drolleries and execrable puns. But Flatterland is still valuable both for the exhilarating range of ideas it plays with and for the deep questions it poses. What is more real: the way a space looks from the inside or from the outside? Does anything unite the bestiary of abstractions that mathematicians call spaces? What is geometry? (All the geometries ever imagined turn out to be variations on a single theme: how certain properties of a space remain unchanged when its points get rearranged.)

The space we actually live in might seem boring in comparison with the rococo spaces of higher mathematics. In the last few decades, however, physicists have been forced to consider the possibility that there might be more to our spatial world than meets the eye. To understand why, consider that physics currently has two sets of laws: general relativity, which describes how things behave on a very massive scale (stars on up); and quantum theory, which describes how things behave on a very small scale (atoms on down). That may seem like a nice division of labor. But what happens when you want to describe something that is both very massive and very small—like the universe a split second after the Big Bang? Somehow general relativity and quantum theory must be fitted together into a Theory of Everything. However, it seems impossible to do this for a world that has only three spatial dimensions. The only known way to make relativity theory jibe with quantum theory is by supposing that the basic objects that furnish our universe are not pointlike particles, but two-dimensional strings and still higher-dimensional branes. Moreover, if the unified theory—called “string theory” (or sometimes “M-theory”)—is to be mathematically consistent, these strings and branes must be vibrating in a space that has no fewer than nine dimensions.

Arriving at a complete understanding of the universe may thus entail the acceptance of higher dimensions. Then why don’t we see them? One hypothesis, long favored by string theorists, is that the six surplus dimensions are “compactified”—that is, curled up into circles of vanishingly small radius. (Think of a garden hose: from a distance it looks like a one-dimensional line, but on closer inspection it has a small circular dimension too.) More recently, though, physicists have speculated that the additional dimensions might be vast or even infinite in extent. Yet we fail to notice them because the particles that make up our familiar world are stuck to a three-dimensional membrane that is adrift in the higher-dimensional world. If this proves true, we are in a situation very similar to that of the Flatlanders.

There remains one important question unaddressed by Edwin Abbott in Flatland and Ian Stewart in Flatterland. It is the question implicitly posed (if not satisfactorily answered) by Aristotle: Why does our everyday world have three dimensions? String theorists have come up with some extremely subtle conjectures about how, of the nine spatial dimensions they posit, exactly three expanded to enormous size after the Big Bang, while the remaining six got choked off and remained unobservably tiny. But there may be a simpler explanation. In a space of more than three dimensions, it can be shown, there are no stable orbits, either for planets or for electrons. Therefore, there could be no chemistry, and hence no chemically based life forms. Well, then, what about a world of fewer than three dimensions? As Stewart has noted, sound waves would not propagate cleanly in two-dimensional Flatland. But the difficulties are not limited to sound. It is impossible to send well-defined signals of any type in a space with an even number of dimensions—and that means no transmission and processing of information. So, by a process of elimination, the only world that is congenial to chemically based, information-processing beings like us is one with precisely three spatial dimensions. No wonder that’s the kind of world we happen to find ourselves in. (Physicists call this “anthropic” reasoning.)

But we should not complain. In a fundamental sense, three-dimensional space is the richest space of all. Clearly it beats a space of one or two dimensions, where there is no room for any interesting complexity. As for spaces of four dimensions and up, they are too “easy”: there are so many degrees of freedom, so many options for moving things around, that complexities are readily rearranged and dissolved. Only in three-dimensional space do you get the right creative tension, which is why mathematicians find it so challenging. Training ourselves to conceive of “a more spacious Space” may enlarge our imagination and contribute to the progress of science. And we can surely sympathize with the aspirations of A. Square—not to mention assorted Theosophists, Platonists, and Cubists—to rise up into the splendor of the fourth dimension and beyond. But we need not follow them. For intellectual richness and aesthetic variety, a world of three dimensions is world enough.

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