In Edwin A. Abbott’s ‘Flatland,’ a 2-D square discovers there’s more to the universe than up, down, left and right.
‘Place a penny on the middle of one of your tables,” says the narrator of an odd little novel. “Look down upon it. It will appear a circle.” As you lower your eye to the edge of the table, the penny becomes an oval. Finally, when your eye is level with the table, right on its rim, the penny is but a line.
Welcome to the two-dimensional world of “Flatland,” by Edwin A. Abbott, an 1884 book that is at once a classic of science fiction, a playful brainteaser about geometry, a pointed satire of Victorian manners—and, finally, a strangely compelling argument about reason, faith, and the greatest mysteries of the universe.
Because Flatland exists only in two dimensions, its inhabitants are like coins on a counter. They can move in the cardinal directions of north, south, east and west but not up or down. They have no notion of what “up” and “down” even mean. The men are shapes, such as triangles, and the women are straight lines, like needles. Flatlanders rely on sense of touch—they’re always feeling each other’s angles—and they live in rigid hierarchies. Sides denote privilege, with hexagons trumping pentagons, for instance. Women have almost no social standing. Circles serve as priests who lord over everyone.
“Flatland” presents itself as the work of “A Square”—a pun on the true author’s peculiar full name, which derives from parents who shared a surname because they were cousins. Thus Edwin Abbott Abbott—Abbott squared, or A 2 —becomes “A Square.”
Abbott (1838-1926) was a clergyman and teacher who ran the City of London School for many years and wrote books on grammar and theology. Along the way, he appears to have come across Charles Howard Hinton, a mathematician who wrote popular articles on the fourth dimension. This gave Abbott the idea for “Flatland.”
In the novel’s first half, A Square explains the way his world works—and, by implication, the way our own 3-D world doesn’t. When he describes Flatland’s women as “wholly devoid of brain-power,” for example, readers are supposed to recognize that this isn’t quite true, and that Flatland’s injustice of denying women an education parallels a similar problem that Abbott knew well in 19th-century England. Abbott also anticipates the rise of totalitarianism in the 20th century, lampooning how Flatlanders enforce conformity: “the toleration of Irregularity is incompatible with the safety of the State.”
This sets up the second half of “Flatland,” in which A Square learns that there’s more to life than two dimensions. First he encounters Lineland, a one-dimensional world whose residents have “no conception of anything out of it.” Later, he chances upon Pointland, “the Abyss of No Dimensions,” whose single, infinitesimal occupant is a hilarious solipsist, unable to contemplate the existence of anything other than itself.
Well before any of this, Abbott’s readers will have guessed that they’re headed toward a fateful meeting between A Square and an interloper from Spaceland, the third dimension. At first, it feels mystical: “I became conscious of a Presence in the room,” reports A Square. He spots a circle but sees that it can change size. That’s because it’s really a sphere, bisecting Flatland—or, as it tells the puzzled protagonist, “I am many Circles in one.”
The sphere dislodges A Square from Flatland and shows him the 3-D reality of “Upward, and yet not Northward.” After this revelation, which involves “sight that was not like seeing,” A Square raises a big question: Is there also a fourth dimension beyond Spaceland? The sphere is scornful: “The very idea of it is utterly inconceivable.” He’s not merely an agnostic but a 4-D denier. The imaginative A Square, however, contemplates the radical possibility of many dimensions.
Abbott devoted much of his life to advancing Christian belief, and his claim here is simple: God lives in a place so far beyond human understanding that we’re figurative Flatlanders, trying to grapple with the incomprehensibility of Spaceland—or literal Spacelanders, struggling to make sense of what lies outside the limits of our perceptions. This becomes a striking metaphor for faith, but it fails to persuade everyone. When A Square returns to Flatland, he faces persecution for preaching “the Gospel of Three Dimensions” and suffers martyrdom, like a Christian in ancient Rome. “Flatland” ends were it begins, with a satirical jab at a Victorian society whose religious habits, following the emergence of Darwinism, had started to fray.
Abbott’s book inspired mathematicians—usually better with numbers than with words—to produce a minor literary subgenre. There are geometry-heavy annotations of “Flatland” as well as pastiches with titles such as “Flatterland,” “Sphereland” and “The Planiverse.” In nonfiction, Rudy Rucker has used “Flatland” as a springboard to discuss curved space, time travel, and other difficult concepts.
Many of these efforts are worthwhile, but none quite match the simple charm or vaulting ambition of Abbott’s original—the greatest math-lit book ever written.
Originally featured here on The Wall Street Journal
