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Literature Study GuidesFlatlandPart 2 Chapters 15 16 Summary

Flatland | Study Guide

Edwin Abbott Abbott

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Flatland | Part 2, Chapters 15–16 : Other Worlds | Summary



Chapter 15: Concerning a Stranger from Spaceland

A Square wakes up and finds that it is the last day of the millennium. At the end of the day he sits with his wife, thinking about a math problem he had tried to explain to his youngest hexagonal Grandson. He had been showing his Grandson that you could infer the area of a square with a length of 3 inches to be 9, or 3 to the second power: 3 squared (32), or 3 times 3 (3 x 3). His Grandson then asked if 3 to the third power—3 cubed (33), or 3 times 3 times 3 (3 x 3 x 3)—meant something as well. Square told him that was nonsense and to go to bed, but the idea still perturbed him. He says the boy is a fool. His wife disagrees and points out that he's breaking the commandment to honor his progeny.

Just then, Square becomes aware of a presence in the room. A spectral voice from nowhere tells him that the boy is right and 3 cubed has a geometrical meaning. Both Square and his wife investigate, and Square thinks at first he sees a woman but then decides it must be an odd sort of Circle. His annoyed wife thinks it is a lady visitor. Square says if she's so sure, she should ask to feel the stranger. She does and is horrified that she may have misbehaved in front of a Circle. The stranger declares he is a sort of circle, or rather, many circles in one, and he has a message for Square that he can only share with Square. Square's wife leaves, and the new millennium begins.

Chapter 16: How the Stranger Vainly Endeavored to Reveal to Me in Words the Mysteries of Spaceland

Worried that the stranger might be a burglar and lacking the fog he needs to discern shape, Square asks to feel the stranger and finds the most perfect circle he has ever encountered. He becomes immediately deferential. The stranger tries to explain that he has come in from above, where he could see Square's interior. However, like the King of Lineland, Square cannot make the intellectual leap that he means another dimension.

The stranger asks Square how many dimensions his wife has. Square acknowledges that to be perceptible, she must possess at least two, though she is so tremendously narrow she appears to be a straight line. The stranger asks him if that does not imply that the world is also visible because of some small degree of height. Square asks if by "height" the stranger means brightness.

The frustrated stranger explains that he is a Sphere and exists in a world of three dimensions. When he passes through Flatland, a Flatlander can only see him as a circle, much as the King could only view Square as a point. To demonstrate, Sphere moves through Flatland, and his circular outline seems to grow and shrink. However, Square cannot comprehend what is happening.

Sphere tries to reason by analogy. He describes how a point has 1 terminal point. A line has 2. A square has 4. Then he asks Square what the next number in the progression is, and Square answers 8. Sphere then describes a cube with 8 points at which lines meet and 6 faces, or, as Sphere puts it, "bounded by six sides, that is to say, six of your insides." Square is horrified and tries to attack the Sphere.


It would be logical to think that because Square already has two dimensions, he has a much better chance of grasping a third dimension than the King could possibly have of grasping a second. The conversation with Square's Grandson demonstrates this. Even though he is only a child, because he already has the model of 3 to the second power, he can conceive abstractly of 3 to the third power. Sphere is able to talk to Square about arithmetic progression (2 + 2 + 2 ...) and geometric progression (2 * 2 * 2 ...). Because Square already understands these and has a model in the way a line becomes a square, he is eventually able to grasp how a square becomes a cube. By extension, a person living in the third dimension should have an easier time conceiving of a fourth. Readers will find, however, that this is not necessarily true.

The Sphere presents his mission in an explicitly religious context. He is here to deliver a revelation of the Gospel of the Third Dimension, and in order to convince Square, he effectively performs miracles. Though Abbott was a theologian, he disliked reliance on miraculous language, which he considered illusory unless it was also accompanied by reasoned understanding. That is to say he believed, like Sphere's intrusion on Flatland, that miracles were manifestations of something rational but outside our limited capacity to understand. To believe requires a leap of faith. Square must open himself up to the possibility of three dimensions before he can begin to understand them. Only once he accepts that Sphere has real things to teach him can he begin to logically comprehend the gospel. Once he applies logic to what he has seen from the other side, what first appeared as magic and miracles make sense as rational occurrences viewed from a limited perspective. However, had he refused to imagine or consider any view outside of his own, none of this revelation would be possible. In the novella as in real life, characters' self-worth, security, and sense of right and wrong are deeply tied to their own understanding of how the world works. All of the characters, even the wise Sphere, when confronted with the possibility that the world operates differently than they believe, react with fear, anger, and violence. This is the core of Abbott's admonishment to modesty and humility and the call to explore ideas that challenge the accepted worldview.

To Square, Sphere's description of what a cube is sounds monstrous and awful when extrapolated out to a human figure, with everything inside a person on display outside. It may be hard for readers to empathize with Square's fear and confusion, as readers implicitly understand three dimensions while Square does not. Here Square, though able to grasp more than the King of Lineland, still reacts viscerally and savagely to Sphere's attempt to explain a third dimension.

The choice of shape for Sphere is resonant with a number of theological and mythological sources. Ancient philosophers such as Greek philosopher Aristotle (384–322 BCE) conceived of the Earth as surrounded by rotating "celestial spheres" containing stars and planets. The Egyptian astronomer Ptolemy (c. 100–c. 170 CE) expanded on this model of the cosmos, geometrically describing it with the Earth incorrectly at its center. Ptolemy's model dominated European understanding until the 16th-century work of Polish astronomer Nicolaus Copernicus (1473–1543). These spheres appear frequently throughout ancient texts, as Abbott would have been well aware as a classical scholar. The visitation of the sphere also resonates with the book of Ezekiel in the Christian Bible. The prophet Ezekiel is visited by servants of God, described as "one likeness, as if a wheel had been in the midst of a wheel," and later "the wheels, were full of eyes round about" (Ezekiel 10:10–12). Though not precisely a sphere, the wheels associated with the heavenly chariot have for the reader the same sort of bizarre and incomprehensible geometry that Sphere has to Square.

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