Being Edwin Abbott

••• How a 19th-century Mathematical Fiction Gave Us Our Point of View

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.

illustration: Anthony Freda

Other Worlds

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.

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