Only on the surface, so to speak, does Flatland belong to the arcane subgenre of mathematical fantasy. In fact, this harrowing memoir could be more properly grouped with such fictional representations of totalitarian mind control as Arthur Koestler’s Darkness at Noon, George Orwell’s Nineteen Eighty-Four and Anthony Burgess’s A Clockwork Orange. I’m only exaggerating a little.
In Flatland, the narrator, A. Square, describes a two-dimensional world, as smooth as a piece of paper, upon which dwells a civilization of geometrical shapes. Its citizens range from the triangular workers to the square-like middle class to the aristocratic polygons (who would like to think of themselves as perfect circles). All these are male. The female residents of Flatland, no matter what their class, are simply short straight lines.
In their planar realm, A. Square tells us, the Flatlanders can “move freely about, on or in the surface, but without the power of rising above or sinking below it, very much like shadows—only hard and with luminous edges.” Note that word “hard.” Any sort of sharp point, whether the apex of certain triangles or the end of a straight line, is a potentially deadly weapon, being able to pierce the sides of an innocent square or hexagon.
Even more dangerous, though, is the entire class of irregular figures, which lack proper angles and proportions, and are innately destined for moral delinquency and criminal behavior. As the exalted Pantocyclus affirmed, “Configuration makes the man; that if, for example, you are born an Isoceles with two uneven sides, you will assuredly go wrong unless you have them made even—for which purpose you must go to the Isoceles Hospital; similarly, if you are a Triangle, or Square, or even a Polygon, born with any Irregularity, you must be taken to one of the Regular Hospitals to have your disease cured, otherwise you will end your days in the State Prison or by the angle of the State Executioner.” A. Square himself declares that he has “never known an Irregular who was not also what Nature evidently intended him to be—a hypocrite, a misanthropist, and, up to the limits of his power, a perpetrator of all manner of mischief.”
While Flatland has long enjoyed a placid history, it has nonetheless weathered occasional crises. For instance, during the “Chromatic Sedition” an incendiary passion for color led to domestic violence and social chaos, culminating in a political movement demanding a more democratic government. Fortunately, such misguided idealists, these sponsors of a Universal Colour Bill, were firmly dealt with by the high-ranking Circles and the country quickly returned to its monochromatic serenity, as each citizen was again taught to “attend to your Configuration” and not get too uppity.
Configuration, in fact, is destiny, though some people do speak of right and wrong “as if they believed that these names represented real existences, and that a human Figure is really capable of choosing between them.” With women, that volatile and deadly sex–the needle-like creatures can inflict great injury and damage—there is, of course, a tendency among males “to speak of ‘love,’ ‘duty,’ ‘right,’ ‘wrong,’ ‘pity,’ ‘hope,’ and other irrational and emotional conceptions, which have no existence, and the fiction of which has no object except to control feminine exuberance; but among ourselves, and in our books, we have an entirely different vocabulary and I may almost say idiom. ‘Love’ then becomes ‘the anticipation of benefit’; ‘duty’ becomes ‘necessity’ or ‘fitness’; and other words are correspondingly transmuted.”
Such double-speak, as Orwell would call it, reveals the misogyny and myriad inequities of a ruthlessly hidebound and hierarchical world, one oddly reminiscent of Victorian England. Still, it is the only world A. Square knows—until that fateful evening when he is visited by a being from another dimension.
“It was the last day of the 1999th year of our era. The pattering of the rain had long ago announced nightfall; and I was sitting in the company of my wife, musing on the events of the past and the prospects of the coming year, the coming century, the coming Millennium.” While in this pensive mood, recalls A. Square, “Straightaway I became conscious of a Presence in the room, and a chilling breath thrilled through my very being.” An intruder has simply appeared out of nowhere.
It is, in fact, a Sphere, perceived on Flatland’s two-dimensional surface first as a point and then as a larger and larger circle, until it finally stops growing. (Were the Sphere to continue passing through Flatland’s plane, the horizontal circle–which is all that A. Square can see–would gradually shrink until it disappeared.) Already bewildering, the situation grows frightening when this strange, probably insane creature begins to rave about an incomprehensible “Gospel of Three Dimensions.” There is, in particular, some nonsense about “upward.”
After several unsuccessful attempts to prove that there is more to the universe than just breadth and width, the Sphere furiously picks up our narrator and simply carries him aloft into Space. There, A. Square undergoes a transcendental out-of-body experience: “An unspeakable horror seized me. There was a darkness, then a dizzy, sickening sensation of sight that was not like seeing; I saw a Line that was no Line; Space that was not Space; I was myself, and not myself.” From the vertiginous heights the unmoored Flatlander looks down into the rooms of his own two-dimensional pentagonal house. And slowly at first, but then with the enthusiasm of the newly converted, A. Square believes. He knows, he can actually see with his own eyes that there is a third dimension.
But, as in the way of disciples, A. Square also starts to grow a bit fanatical. Could there not be, in fact, a fourth dimension? he asks the Sphere. “As you yourself, superior to all Flatland forms, combine many Circles in One, so doubtless there is One above you who combines many Spheres in One Supreme Existence, surpassing even the Solids of Spaceland. And, even as we, who now in Space, look down on Flatland and see the insides of all things, so of a certainty there is yet above us some higher, purer region, …some yet more spacious Space, some more dimensionable Dimensionality, from the vantage ground of which we shall look down together upon the revealed insides of Solid things.”
The Sphere, as limited in his way as A. Square had once been, refuses to give credence to any dimensions beyond three. But, as our increasingly mystical narrator insists, “of a surety there is a Fourth Dimension,” even if it is only one that can be perceived by “the inner eye of thought.” At which point, the now thoroughly irate Sphere releases the Flatlander who falls back into his old world. Home again, A. Square attempts to explain his new understanding of three dimensions to a bright grandson, who thinks he must be joking. Meanwhile, the government—aware that the Millennium brings out dangerous quackery—has promulgated strict prohibitions against any theories or beliefs that might upset the complacency of the populace. So, one day without having done anything truly wrong, Josef K.—I mean A. Square–is arrested and sentenced to life in prison.
There, as the lonely years grind on, the once bright memory of a three-dimensional universe begins to fade. Hoping to preserve something of his revolutionary vision, A. Square sets down the memoirs we have just read, trusting that they “may find their way to the mind of humanity in Some Dimension, and may stir up a race of rebels who shall refuse to be confined to limited Dimensionality.”
Since Flatland first appeared in 1884, it has gradually come to be seen as a minor masterpiece. Be assured: My summary describes just the major arc of the book and only hints at its ingenious detail, charm and, not least, sly satire. In these pages Edwin A. Abbott (1838-1926) carries on an unrelenting critique of class barriers, an implicit plea for women’s rights modeled after Swift’s ironic “Modest Proposal,” and an impassioned defense of freedom of thought. Yet could Abbott also be hinting, as A. Square’s conjectures seem to imply, that God and a spirit world exist in a “Fourth Dimension” beyond our limited senses?
Both a liberal clergyman and a distinguished teacher, Abbott — according to Ian Stewart in The Annotated Flatland — doesn’t go that far. From all accounts, the author of Flatland was a very down-to-earth sort and typically viewed religion as a source of moral values in this world, while shying away from any otherworldly speculations. Still, his book—written in the early days of 19th-century Spiritualism and rich with mystical language—does seem to point to a metaphysical, as well as mathematical, aspect to multiple dimensions.
That said, Flatland has always been regarded as an early attempt, a baby step, toward our current theorizing about the realm of hypercubes, space-time and superstrings. As A. Square exclaims: “In that blessed region of Four Dimensions, shall we linger at 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.” And why stop there?
At least some of Abbott’s ideas almost certainly derive from Charles H. Hinton (1853-1907), a fellow Brit who ended his life working at the Naval Observatory in Washington D.C. Hinton’s major works—including What is the Fourth Dimension?, A Plane World, “Many Dimensions” and, late in his life, An Episode of Flatland—were first gathered together in two volumes titled Scientific Romances, a term that H.G. Wells and others would soon adopt. (The essays and stories may also be found in the Dover anthology, Speculations on the Fourth Dimension: Selected Writings of Charles H. Hinton, edited by Rudolf V.B. Rucker.) Since Abbott and Hinton, writer-mathematicians have regularly revisited Flatlandor imagined similarly constrained worlds. Outstanding examples include Dionys Burger’s Sphereland, A.K. Dewdney’s The Planiverse, Rudy Rucker’s Spaceland and Ian Stewart’s Flatterland.
In The Time Machine, H.G. Wells hypothesized that duration was the fourth dimension, complementing height, width and breadth. I certainly grew up believing that. But today’s theorists tend to imagine instead the never-ending implications of non-Euclidean space. In our digital age, we can sometimes even advance beyond what A. Square calls “the inner eye of thought” and actually model a tesseract rotating through four dimensions. It looks like a cube designed by M.C. Escher sliding around, in and through itself.
Fortunately, at least for those of us who never went beyond 10th-grade math, Edwin A. Abbott’s Flatland is primarily a literary work, a critique of 19th-century mores coupled with a philosophical plea to think outside the box or, in this particular case, above the plane. As A. Square tells us, “To be self-contented is to be vile and ignorant, and . . . to aspire is better than to be blindly and impotently happy.” Sapere aude!