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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. When I could find voice, I shrieked aloud in agony. “Either this is madness or it is Hell.” “It is neither,” calmly replied the voice of the Sphere, “it is Knowledge; it is Three Dimensions.”
—A Square, protagonist of Flatland (1884), upon entering Spaceland
If the fourth dimension exists while we possess only three, it means that we have no real existence, that we exist only in somebody’s imagination and that all our thoughts, feelings and experiences take place in the mind of some other higher being. . . . If we do not want to agree with this we must recognize ourselves as beings of four dimensions.
—P.D. Ouspensky, The Fourth Dimension (1908)
It’s supernatural, for lack of a better word—I mean, it raises all sorts of philosophical-type questions, about the nature of self, about the existence of the soul, you know? Am I me? Is Malkovich Malkovich? . . . Do you see what a metaphysical can of worms this portal is?
—Craig Schwartz (John Cusack) in Being John Malkovich (1999)
The possible existence of higher dimensions has long haunted the popular imagination, representing a great beyond where hard science meets woolly mysticism. Spiritualists and sci-fi writers invoke the fourth dimension as, respectively, phantom zone and deus ex machina; various branches of modern science and mathematics engage regularly with dimensions above the three that we appear to live in. (Superstring theory posits a multiverse: 4-D space-time—an idea popularized by H.G. Wells and Einstein—supplemented by at least six more dimensions, each one so infinitely small as to be undetectable.) But a fourth spatial dimension—distinct from (and perpendicular to) height, width, and depth—is still one brain-spraining leap too far for our 3-D faculties. To the extent that an intuitive understanding of extradimensionality is possible, the most practical and user-friendly manual remains Flatland: A Romance of Many Dimensions, a slender educational novel that was published pseudonymously in 1884 and has never been out of print.
This geometrical caprice and surreptitious head trip was written by Edwin Abbott Abbott (1838-1926), a London headmaster, theologian, and classics scholar who published over 50 books, none of them mathematical except Flatland. (His father married a cousin, hence the double-barreled name.) The math concepts it employs are simple enough: Flatland is a plane inhabited by polygons that glide about freely on its surface. The men are triangles, squares, pentagons, hexagons, and so on. Social status increases with number of sides. The monarchs are circular priests. Women are mere one-dimensional creatures: straight lines. As Flatlanders can only see their fellow polygons edge-on, they behold not shapes but line segments. To tell each other apart, the lower classes depend on a highly developed sense of touch; the nobility favor the refined art of Sight Recognition, deducing forms with the help of a ubiquitous fog that causes brightness to vary with proximity.
Flatland‘s chief accomplishment is its uniquely lucid demonstration of the concept of dimensional analogy. Narrated by “A Square,” the story pivots on the momentous arrival of a Sphere. As the 3-D visitor descends through the planar world of Flatland, it appears to our flustered hero first as a point, which turns into a circle (or more precisely, a line segment that A Square apprehends as a circle). The circle grows until the Sphere’s equator is level with the plane, after which it shrinks back to a point, before finally vanishing. Abbott illustrates the challenges that a lower-dimensional being would face in attempting to visualize our world—and in so doing, invites us to imagine the corresponding situation in three dimensions and beyond. By extension, if a 4-D hypersphere were to intersect our space, it would appear to us first as a point, turning into a sphere and increasing in size over time, before contracting and finally disappearing.
The dimensional analogy dates to Plato’s Republic. The Allegory of the Cave describes the blinkered existence of shackled cavemen, cognizant only of the shadows on the wall, oblivious even to their own solidness—though it never occurred to Socrates and Glaucon to venture beyond three dimensions. Modern software can easily generate n-dimensional polytopes, but Flatland remains an essential tool. “Graphics are fine as aids to thinking,” says English mathematician Ian Stewart, annotator of a new edition of Flatland (Perseus), “but when you think about higher dimensions—which is very important in today’s math, and in many sciences, and even in economics—you need to develop your own mental image.” Almost every pop-science journey through higher worlds, from Rudy Rucker’s The Fourth Dimension to Michio Kaku’s Hyperspace, presents itself to some degree as the further adventures of A Square (or a related Flatlander).
For a treatise on a 2-D universe, Flatland is remarkably multifaceted. The first half, “This World,” breezily outlines the lay of the land (such as it is) in pithy chapters with titles out of Fodor’s—e.g., “Of the Climate and Houses in Flatland” and “Concerning the Women” (matters of decorum, alas, preclude the nitty-gritty of polygonal copulation). Embedded in this deadpan chronicle is a withering social critique, and a satisfyingly coherent joke: For Abbott, this shallowest of worlds—literally devoid of depth—was a reductio ad absurdum of the society he lived in, complete with rigid caste system and rampant sexism. “I think satire was what really motivated the book,” says Stewart. “The writing seems most committed when Abbott is attacking the structure of Victorian society.” In Flatland, class relates directly to physical appearance. Isosceles triangles, with only two equal sides, are at the bottom of the hierarchy; irregularly shaped beings are considered deviant and summarily destroyed.
As did many Victorians, Flatlanders subscribe to the dubious theories of social Darwinism and phrenology: Intelligence increases with each step up the social ladder, i.e., with the size of brain angle. (A regular n-sided polygon has n angles of 180 – 360/n degrees each, so the greater the number of sides, the bigger the angle.) Women, who have no angles to speak of, are “wholly devoid of brain-power,” though the lethal pairing of volatile temper and needle-like form means they are widely feared and despised. Flatland was evidently perceived in some quarters as a misogynist tract. In a preface to the second edition (it was reprinted within a month), Abbott kept a straight face, responding to critics on behalf of his chastened friend, A Square: “I gather that . . . he has himself modified his own personal views . . . as regards Women.” The confusion persists: Isaac Asimov, introducing a 1983 edition, expresses regret that the book bears the taint of “those now-antiquated social views.”
Research into Abbott’s lifework would have clarified the author’s position. Thomas Banchoff, a professor of mathematics at Brown currently working on an Abbott biography, points out that he was an outspoken advocate of equal schooling opportunities for women. A Square at one point calls for females to be better educated, but this glimmer of enlightenment is couched within a solipsistic harangue. As men are supremely rational and women understand only “emotional conceptions, which have no existence,” male Flatlanders require two sets of vocabularies—one for conversation among themselves, another for their dealings with women; female education would mitigate the strains of this doublespeak. “Abbott recognized the implications of a ‘two cultures’ society a hundred years before C.P. Snow,” says Banchoff. “This was well before anyone was saying, Men Are From Mars, Women Are From Venus.”
Flatland turns downright lysergic in its second half, “Other Worlds,” which flip-flops between fantasy and reality as violently as anything in Borges or Lynch. A Square’s mind-expanding exploits begin with a dream of one-dimensional Lineland. The Sphere arrives to preach the Gospel of the Three Dimensions and catapults the disbelieving Square into Spaceland. But the Sphere bristles when his charge, clearly a quick study, starts inquiring about a 4-D “Thoughtland.” Sent back to Flatland (where he dreams of a zero-D Pointland), A Square attempts to convince the High Council of Circles of his newly discovered direction—”Upward, not northward!”—but (like Galileo) is jailed for heresy.
Stewart, whose copious footnotes in The Annotated Flatland range freely from an explication of the calendrical blunder that caused the millennium to be celebrated a year prematurely to speculations on the wiring of a 2-D brain, places Abbott’s book in science fiction’s prehistory. Flatland shares the math-fiction shelf with geometer-clergyman Lewis Carroll’s Alice novels and theologian Jonathan Swift’s Gulliver’s Travels, especially the “Laputa” section. (“A Modest Proposal” seems to inform Abbott’s chapter on Irregular Figures.) As literature, Flatland pioneered two tropes that would go on to dominate 20th-century SF: the extradimensional visitor and the paranoid conspiracy. Long aware of the third dimension, the high priests of Flatland keep the public in the dark. When the Sphere literally crashes a High Council meeting, those present are ordered destroyed or imprisoned.
Flatland‘s central analogy still provides a sequel-friendly framework for abstract concepts. Dionys Burger’s Sphereland (1957), narrated by A Square’s grandson, A Hexagon, applies Einstein’s relativity theory to Flatland. A.K. Dewdney’s The Planiverse (1984) details the physics and biology of two-dimensionality. Most recently, Stewart’s Flatterland (2001) takes A Square’s great-great-granddaughter, Victoria Line, on a pun-happy romp through higher-dimensional geometry and quantum physics (she meets the Space Girls and a cow named Moobius). Even more whimsical updates abound: Rucker’s “Message Found in a Copy of Flatland” situates the plane in the basement of a London restaurant. A 1965 animated short film features Dudley Moore (!) as A Square. (The primitive but trippy color-table-animation technique is ideally suited to the subject.) Flatland has also inspired an avant-garde performance piece and a computer-generated opera, and pops up in fiction from Infinite Jest to The Cheese Monkeys. In an alarming development, a new TV series set in 2010 Shanghai called Flatland stars Dennis Hopper—no relation, one imagines, to the Space Hopper that guides Vikki Line in Flatterland.
In a 1990 article, Banchoff links Flatland with Abbott’s theological novels, Philochristus and Onesimus, which both build toward epiphanous encounters (with Christ and Saint Paul, respectively) and end with the heroes’ “frustration and eventual persecution.” Abbott’s interdimensional jeu d’esprit is both spiritual allegory and epistemological inquiry. Banchoff says, “He was concerned with our imperfect understanding of the transcendental world—what we do when we come face to face with something we can’t understand, and how we’re easily doomed to failure when we try to explain it to others.”
Abbott had little in common with Victorian spiritualism, a shared interest in the fourth dimension notwithstanding. Banchoff cites an 1897 book, The Spirit on the Water, in which Abbott, referring back to Flatland‘s climax, stresses that the ability to perceive higher dimensions “cannot, by any direct path, bring us closer to God.” Flatland‘s dedication to the inhabitants of Spaceland (i.e., us) wishes for “the possible Development of that most rare and excellent Gift of MODESTY among the Superior Races of SOLID HUMANITY.” The book ends with A Square in prison, “absolutely destitute of converts,” but Abbott’s perspective, at once unassuming and visionary, has found adherents in three dimensions.