Offered as a fictional mathematicians memoirs, Edwin A. Abbotts Flatland: A Romance of Many Dimensions depicts a nightmarish dystopia in which living geometrical figures persecute irregular figures (those with unequal sides) and condemn straight lines, or females, to perpetual ignorance and subservience. The novel is divided into two parts: a preface and the “central event,” as Abbott calls it.

In part 1, titled “This World,” the mathematician A. Square describes his two-dimensional world, Flatland, for an audience in Spaceland, a three-dimensional world. The government of Flatland is administered by a cabal of many-sided polygons who promote a societal hierarchy that ascends gradually from straight lines (women) to circles (priests). In between are irregular or isosceles triangles, the soldiers and working class; equilateral triangles, who are the tradesmen; squares and pentagons, who represent the professional classes, such as lawyers and mathematicians; and polygons of more sides, including hexagons, who enjoy the status of no-bility.

In Flatland, evolution is not only a biological fact but also a state policy. “It is a Law of Nature with us,” writes A. Square, “that a male child shall have one more side than his father, so that each generation shall rise (as a rule) one step in the scale of development and nobility.” To assist nature, the ruling circles engage in selective breeding and extermination. Irregular figures are imprisoned, enslaved, or destroyed, and surgery routinely is performed to correct genetic deformities. The political climate in Flatland is ripe for revolution, and in fact A. Square, who often defends the prejudices and policies of his government, says that there have been at least 120 rebellions, including the infamous Color Revolt, and nearly 250 “minor” outbreaks in the recorded history of Flatland.

In part 2, titled “Other Worlds,” which was written for both “Plane and Solid Humanity,” A. Square narrates his imaginary and real visits to one-, three-, and no-dimensional worlds. His stated purpose is to “stir up a race of rebels who shall refuse to be confined to limited Dimensionality.” He begins by recounting his dream of Line-land, a one-dimensional realm in which points (females) and lines (males) travel through life in single file, communicating with one another, and even copulating, by sound.

The next day, a sphere visits A. Square in his house to communicate “the Gospel of the Three Dimensions.” Unable to explain his world, the God-like sphere decides to transport the mathematician into Spaceland and show him the third dimension. The trip and subsequent dialogue convince A. Square that there is a third dimension and even a fourth dimension, which he terms “Thoughtland.” The sphere, however, cannot accept the proposition of a fourth dimension, consequently becomes very angry, and returns his “apostle” to Flatland abruptly.

As he ponders this experience, A. Square falls asleep again and dreams about Pointland, a world of no dimensions where the inhabitants speak of themselves in the third person because they cannot distinguish themselves from the world. To instruct the people of Flatland about other dimensions, the clever mathematician writes a treatise titled Through Flatland to Thoughtland, which safely fictionalizes his subject. His proselytizing, however, becomes more blatant over time, and he winds up in prison, where he spends the next seven years writing his memoirs.