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Literature Study GuidesFlatlandPart 2 Chapters 19 20 Summary

Flatland | Study Guide

Edwin Abbott Abbott

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Flatland | Part 2, Chapters 19–20 : Other Worlds | Summary



Chapter 19: How, Though the Sphere Shewed Me Other Mysteries of Spaceland, I Still Desired More; and What Came of It

A Square tries to rescue his brother but cannot move in three-dimensional space. Sphere tells him not to worry and introduces him to the solids. He builds a cube by stacking square cards. However, Square is accustomed to thinking in two-dimensional shapes and focuses on the outline of the cube, which he perceives as an irregular figure. He does not understand shading and perspective, and Sphere encourages him to feel the cube. Square marvels at the cube and feels awe that they are geometrically related. Sphere moves around the light and changes Square's position relative to the cube to help him understand how light and perspective work.

Square extrapolates that there must also be higher dimensions than the third and asks Sphere to take him to fourth-dimensional space so that he may see what Sphere's interior looks like. He reasons that just as Sphere is composed of many circles, there must also be beings composed of many spheres. Sphere tells him to stop talking nonsense, but Square persists, believing someone as wise as his teacher must know. Square tries the same geometric reasoning that Sphere used, saying that there surely must be some sort of divine fourth-dimensional cube with 16 points of intersecting planes. He asks if the inhabitants of Spaceland have ever seen manifestations of strange beings in the same way Sphere passed through the council chamber. Sphere grumpily answers that there are stories of such, but wise men say it was probably just hallucination.

Excited, Square wonders about fifth, sixth, seventh, and even eighth dimensions before Sphere angrily silences him. Sphere returns Square to his home.

Chapter 20: How the Sphere Encouraged Me in a Vision

Square decides not to tell his wife what has happened and instead lies about having fallen down. His wife doesn't believe him but doesn't argue. She thinks he's ill and tells him to lie down. He tries to recall Spaceland, but it's not as clear as he'd like. He falls asleep and dreams that he is beside Sphere, who is no longer angry.

Sphere shows Square a single point, saying he is taking Square to the "lowest depth of existence ... the realm of Pointland, the Abyss of No dimensions." Pointland is occupied completely by a single being that cannot conceive of anything outside itself because it has no conception of length, breadth, height, or anything beyond its point. The Point congratulates itself on the expansiveness of its being. Sphere tells Square to try to explain higher dimensions to the Point. However, when Square tries, the Point assumes that Square's voice is coming from inside itself and congratulates itself on its capacity for novel thought without reflecting on the message. Sphere delivers the moral of Square's journey to Pointland: "To be self-contented is to be vile and ignorant, and that to aspire is better than to be blindly and impotently happy."

Sphere returns Square to his home, saying that nothing they could say would reach the Point. Sphere apologizes for getting angry and concedes that Square is right. He urges Square not to be complacent, but to aspire and teach others to aspire.


Though Sphere is arguably a higher order of being than Square, he still reacts angrily to the idea that his concept of the world is incomplete. He is so focused on making sure Square understands the facts of the third dimension that he does not recognize the validity of Square's question about a fourth. This is an almost exact mirror of the conversation between Square and his Grandson, and it goes back to the hope in Abbott's dedication that this book will help readers cultivate modesty. Both Sphere and Square, as teachers, discount the possibility that their students may have something worth teaching them. In reality, both Square and his Grandson are capable of making leaps that would be hugely valuable to their older tutors. "Why," Square asks, "should the thirst for knowledge be aroused, only to be disappointed and punished?"

Square cannot see in the third dimension, in part because he has no idea how light and shadow work. Even in Spaceland, visual perspective is both limited and limiting. Represented as an expression, in each of the countries presented in Flatland, a person operating in n dimension always views other people as figures of n – 1 dimensions. This is to say the King of Lineland (one dimension) views others as points and navigates by sound. The people of Flatland (two dimensions) perceive others as lines and use fog to differentiate angles. The people of Spaceland (three dimensions) perceive others as shapes modified by light and shadow. Presumably the inhabitants of a fourth dimension would be able, in this progression, to view the entirety of three dimensions at once, inside and out, but use some other means to presume another person's true dimensions, and so on.

As a corollary, the Point cannot view other people. There is nothing smaller than it, but it perceives itself as so vast that it occupies the entire universe. The Point has the most limited perspective of anyone in the book. It is at once the most ignorant and happy creature in the book, congratulating itself on its every thought. The Point, like the King of Lineland, acts as a criticism of the self-satisfaction and firm belief in their own superiority exhibited by so many in Britain during the Victorian era. Visiting Pointland last acts as a cautionary counterpoint. Though the path of searching for truth may be hard, and indeed Square will be punished for it, striving is always superior to ignorance.

The unfortunate A Square's return home is met with the same kinds of problems as Gulliver experiences when he returns from his travels to strange lands in Swift's Gulliver's Travels. Specifically, during his travels, A Square has come to a new understanding of the world and returned a changed person. He can no longer participate fully in the society in which he was raised. Gulliver, by the end of his own travels, has come to view human beings as an inferior sort of animal, based on the time he spent with the horselike Houyhnhnms. Though he is treated kindly, he cannot return to the life he had. Likewise, Square can no longer revere the Circles, and he cannot pretend convincingly that there are only two dimensions. This common element of character growth and rejection of what they once believed serves to highlight the problems with criticizing society. It shows that the flaws of the world can be recognized and rejected, though doing so often comes with high social and personal costs.

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