Flatland: A Romance of Many Dimensions

On the second-to-last day of the 1999th year of “our era,” The Square stays up late studying geometry and has a dream in which he visits a place called Lineland (43). This is a place consisting of small straight lines and points all moving back and forth and chirping chaotically on a longer straight line. He tries to communicate with them, but they ignore him. Finally, one of the small lines introduces himself as “the Monarch of the world” (43). The Square has a hard time learning anything about Lineland, as the king assumes they accept all the same facts to be true. However, the Square finally determines that the king believes Lineland to constitute all of space: he has no conception of anything that exists outside of his straight line. His subjects consist of men (small lines) and women (points), but to any Linelander, all other Linelanders appear as points, and they can only distinguish age and gender by voice.

The Square asks the king how marriage and reproduction is possible if Linelanders cannot see one another or move from their place on the line. The king explains that contact and closeness are not necessary: Instead, once a week, the Linelanders do a “choral dance” that allows them to find their proper mates using vocals (46). All men have two wives, and all consummations produce two girls and one boy. The king says that sometimes it takes multiple choral dances for these shapes to find their mates.

The Square insists to the king that there must be no way for him to distinguish between lines and points using sight, and the king agrees that this is impossible. He says that he communicates with his wives using two voices rather than one. The Square asks whether people feigning each other’s voices is a problem in Lineland and suggests that feeling or touch might be a way around that problem. The king is disgusted by this idea, declaring that his subjects must maintain space between each other or face the death penalty. This is because women are frail enough that contact with a male figure could kill them. He adds that voice, “the essence of one’s Being,” is a perfectly effective and safe way to recognize and communicate with each other (49).

To the Square, life in Lineland is extremely dull, and he argues that it would be better to have no sight at all than to see so little. He brags about his own ability to see the subjects of Lineland, to which the king replies that he does not understand the concepts of right and left. The two men have a brief dialogue in which the Square tries to explain the difference between right-left motion and North-South motion, but the king cannot grasp it. The Square finally leaves Lineland, and the king, watching him disappear, believes he is dead (although he refers to The Square as a woman on the basis that The Square only has one voice). The king insists that the Square’s mode of travel is not legitimate movement but magic. The Square becomes frustrated and berates the king for ignorantly and arrogantly believing that Lineland constitutes all of space. As the king moves angrily toward him, the Square wakes up.

The next day, the Square spends time with his family. He and his wife reflect on the past and talk about what the future might hold, and he tutors his grandson, a Hexagon, in geometry. The Square teaches his grandson to calculate the area of a square by multiplying the length of one side by itself—raising that number to the second power. The grandson—described as “a most promising young hexagon of unusual brilliancy”—deduces that three to the third power must describe some other shape, a shape for which he has no name (the reader of course recognizes that it describes a cube—a three dimensional shape that cannot exist in Flatland). The Square becomes annoyed and sends the boy to bed as punishment for his “nonsense” (53). He later exclaims that the boy is a fool, to which his wife angrily says he is breaking the commandments of Flatland by dishonoring his grandson. His surprising anger here, and his willingness to hurl insults at a boy he previously described as “brilliant” suggests his uncomfortable suspicion that the boy is onto something.

Suddenly, the Square feels a chill and detects a mysterious presence somewhere in the room. A disembodied voice declares that the boy is not a fool and three cubed has “an obvious Geometrical meaning” (54). The Square’s wife, able to hear the words but not understand them, assumes that a strange woman has entered their home. When she feels the strange visitor, she realizes she is wrong and assumes that the stranger is from the circular class. She apologizes profusely and leaves while calling out her mandated “Peace-cry” (55). The voice proclaims that he is not a Flatland circle but is instead composed of many circles at once. The Square notices that the last grains of sand have passed through the hourglass and a new millennium has begun.

When the Square approaches the stranger, he cannot fathom the latter’s shape and briefly worries that he might be a monstrous Irregular who has broken into their home to rob them. Unable to use Sight Recognition without the Fog, he feels the stranger and realizes that his wife was right: The stranger is a more perfect circle than any in Flatland. He apologizes awkwardly, and the two men begin a dialogue. The stranger describes himself as having arrived “from Space,” to which the Square argues that he is familiar with Space, as Space constitutes “height and breadth indefinitely prolonged” (56). The stranger tries to explain the concept of up-down movement, which The Square cannot understand, and adds that he would be able to understand it if he had an eye on his side. The Square interprets this to mean an eye inside his body, which horrifies him. The stranger says he has come from “the Land of Three Dimensions” and claims that everything the Square perceives as solid in Flatland is actually open and exposed to his gaze (56). To prove this, he describes layout and inhabitants of The Square’s home as seen from above, but the Square remains unconvinced.

The two men then argue about women, with the stranger declaring that even though women are straight lines, they still have a little bit of height; otherwise, they would be invisible. The Square says that when Flatlanders see a line, they see “length and brightness,” and if brightness disappears, so does the line (58). The stranger says that in his world, he is called a Sphere, as he is made up of an infinite number of circles. To demonstrate this, he rises into space, diminishing and eventually vanishing, before slowly lowering himself back into Flatland and fully reappearing. The Square says that this probably seems simple and straightforward to readers in Spaceland, but with only his Flatland mathematics to guide him, he assumes the Sphere is an enchanter or magician playing a trick on him. This faulty assumption echoes the earlier scene in which the King of Lineland made the same assumption about the Square.

After he has returned, the Sphere attempts to explain himself using “the method of Analogy” (60). In another dialogue, he has the Square describe what happens when a point moves in order to form a straight line; he then asks him to consider what would happen if a Flatland square—created by all those points and lines—moved upwards, out of Flatland entirely. The Square eventually finds himself describing the existence of a cube, the same shape his grandson had been trying to describe earlier. He becomes upset and screams at the Sphere, calling him an enchanter and a devil and saying that one or the other of them must die.

The Square attacks the Sphere but is unable to injure him because the Sphere’s shape allows him to slip into another dimension. The Sphere is disappointed in the Square’s reaction, saying he had hoped a mathematician would be a good apostle for the Gospel of Three Dimensions. In another attempt to prove the truth of his claims, the Sphere descends from Space into a cupboard belonging to the Square and removes a tablet. When the Square worries that he is losing his mind, the Sphere tells him that all seemingly solid things are “really superficial” and if the Square wanted to, he himself could rise up into Space (63). The Sphere also touches the inside of the Square’s stomach to prove to him that he can penetrate any seemingly solid surface. The Square again attacks the Sphere, but this time the Sphere sinks below the Flatland plane and out of reach. However, he finds it difficult to rise back up and becomes upset, shouting that he must not be stopped from spreading his gospel. When the Square refuses to help him, the Sphere carries both of them out of the Flatland plane and into Space.