Being Edwin Abbott

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

 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)

illustration: Anthony Freda

This World

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.

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