Although a great deal has been read into Carroll’s two Alice books, any didacticism, if it exists at all, is well hidden. Perhaps a closer comparison to Flatland would be the Russian-born physicist George Gamow’s Mr. Tompkins books: Mr. Tompkins in Wonderland (1940) and Mr. Tompkins Explores the Atom (1945). In these books, Gamow creates fantastic worlds in which his hero Mr. C.G.H. Tompkins, “a little clerk of a big city bank,” travels at speeds close to the speed of light, or shrinks to the size of subatomic particles. Gamow’s explicit purpose is to popularize relativity and quantum theory; he succeeds also in demonstrating the extreme limitation of human sensory perceptions and all intuitive notions of the natural world based on those perceptions. All of these books, including my own Einstein’s Dreams, constitute a literary tradition in which a fantasy world created by some physical or mathematical conceit invites the reader to ponder philosophical questions in the actual world of human existence.

Flatland is divided into two halves. The first half bristles with the portrait of what today can only be described as a repulsively elitist and sexist society. Priests, who are circles, are the highest class and administrate the two-dimensional realm. Everyone aspires to having the highest number of sides, approaching the exalted circles, and the children of aristocrats are actually fractured to increase their number of corners. Tradesmen and soldiers, among the lowest classes, with a scant three or four sides to themselves, are barely human. Women are not even worthy of three sides. They are one-sided figures—straight lines, in other words—and they must be constantly avoided or handled gingerly so that their two extremely sharp ends will not puncture incautious males in the vicinity. Women talk too much and are so dumb that they aren’t even aware of their wretched status at the bottom of the social hierarchy. Occasionally, they go on a berserk rampage and massacre hundreds of males, a practice that is accepted as keeping the population of the lower classes in check (and might have been applauded by a two-dimensional Thomas Malthus).

This description of society is conveyed earnestly by the narrator of the fable, a two-dimensional inhabitant of Flatland. At first, one has the uneasy feeling that the narrator stands for Abbott and his own views of society, but the fictional system is so caricatured and ridiculous and witty that it does not seem possible that Abbott could have shared these views, even in his day. I read this first half of Flatland as social satire, in the tradition of Jonathan Swift.

Also raised in the beginning section are the venerable philosophical controversies over nature versus nurture, free will versus determinism. Among the lowest of the low in Flatland are the “Irregulars,” miserable creatures whose sides are of unequal lengths (as opposed to the equilateral triangles, the squares, the regular pentagons, and so on) and who behave accordingly. Debate rages over whether the Irregulars are natural monsters, born with defective moral constitution, or instead acquire their bitterness and perversion only after being outcast and mistreated by society. The narrator openly confesses that he favors operating on the Irregulars to make them Regulars, and killing them if that process doesn’t work. The Priests claim that personal conduct depends completely on immutable Configuration. If one is born bad (irregular), one remains bad. Will, training, and encouragement are useless. Then again, the Priests have a great deal at stake in protecting their exalted position. The practice of operating on Irregulars to make them Regular or of fracturing the children of aristocrats to give them more sides also echoes the controversial idea of eugenics, in which the “best” human beings are segregated and bred together to improve the species. This notion, discussed today in the context of genetic engineering, was put forth most forcefully by Francis Galton in his book Hereditary Genius (1869), a book that well could have been read by Abbott.

The second half of Abbott’s book takes a sharp turn when the narrator becomes aware of the limited scope of his world and begins conjecturing on three, four, and higher dimensional possibilities. And here begins to emerge the meaning of the book for science.