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Literature / Flatland

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Flatland: A Romance Of Many Dimensions is an 1884 satirical novella by Edwin A. Abbott. The story takes place in a two-dimensional world made up of polygons, and is narrated by a square named A. Squarenote . It's also a scathing dissection of Victorian class structures, of biological racism and eugenics, and of misogyny.

The square dreams one night about visiting Lineland, where there is only one dimension, and he tries and fails to explain his existence and that of a second dimension to its king. Later, a sphere from Spaceland speaks to him, pokes his insides, and appears before him in his home, then carries him in an incomprehensibly new direction called "up", where he is able to look down and see into houses and the insides of the other polygons. Suddenly understanding, he speculates that there may be dimensions beyond Spaceland, but the sphere is discomfited by this and returns him to Flatland, where he seems to just appear. Later, he is imprisoned for this talk of a third dimension, and he dreams of himself and the sphere visiting Pointland, where the Point — monarch, sole inhabitant, and universe in one — is unable to perceive them as anything but his own thoughts. This causes him to connect the uncomprehending ignorance of the Point, the king of Lineland, and the rulers of Flatland together with the sphere's astonishment at the thought of some dimension beyond Up.

It is part sci-fi, part satire, part philosophy, and part mathematics. Isaac Asimov described Flatland as "The best introduction one can find into the manner of perceiving dimensions".

Adaptations include a 1965 animated short (Flatland), a 1982 stop-motion short only available in Italian (Flatlandia), a 2007 animated feature film (Flatland, aka Flatland: The Film), and, confusingly, another 2007 animated film, this one a short (Flatland: The Movie), and its 2012 sequel, Sphereland. The story is also summarized by Carl Sagan as part of Cosmos.


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     Tropes found in the novel Flatland 
  • 1-Dimensional Thinking: Justified in Lineland, in which the second dimension does not exist.
  • Alien Geometries: The third dimension is unfathomably alien to the Flatlanders, and so is the second dimension to the Linelanders. The fourth dimension is briefly discussed.
  • Another Dimension: One of the few times in fiction that this term was used correctly.
  • Bed Trick: An isoceles triangle (lower class, basically serfs) paints himself up as a polygon (aristocracy) in order to bed a woman that had rejected him.
  • Bizarre Alien Biology: Linelanders have eyes and mouths at both ends. Additionally, males have two voices, baritone from one end and Tenor from the other.
  • Bizarre Alien Sexes: The male Flatlanders are polygons of varying angle count (from triangle on up to quasi-circle) in which intelligence is dictated by angle count, while the females are straight lines and nearly mindless.
  • Black-and-White Insanity: The ruling caste enforce a Black-and-White Morality worldview to the point where they outlaw color, forcing the world to literally be black and white. Their excuse for this draconian law is that it's needed to preserve the sexual purity of their women.
  • Call a Smeerp a "Rabbit": The Living Polyhedrons that inhabit Flatland are called "human beings".
  • Cassandra Truth: The square eventually is imprisoned for preaching the Gospel of Three Dimensions. No one will believe him, not even his brother, who witnesses the sphere appearing before the Flatland parliament.
  • Does This Remind You of Anything?: In addition to its frank discussion of Euclidean geometry, many of the book's depictions of different dimensions are thinly-veiled social satire, largely concerned with the evolution of social and political systems.
    • Flatland is Abbott's own Victorian Britain—a world whose inhabitants revel in their supposed freedom, but are actually stifled by a rigid class system, shunning any ideas that might expose their limited worldview for what it is. Because they can only think in two dimensions, Flatlanders are utterly incapable of understanding "upward mobility", which is the only way to reach Spaceland and the freedom that it offers.
    • Lineland is a very primitive monarchy, where all power and influence is determined by proximity to to an autocratic king — who is actually just a humble line on a line. Because he only exists in one dimension, the King of Lineland is incapable of even basic movement, since there's no way for anyone to move aside to permit someone else passage, but he's too drunk on his own ego to realize how little his "authority" really matters.
  • Fantastic Racism: More precisely, fantastic classism and fantastic sexism— polygons with more sides are the higher classes; triangles — especially isosceles ones — are servants or soldiers; circles (technically many-many-many-sided polygons) are priests; women of all classes are just linesnote . It is a satire, thinly veiled, of Victorian society.
  • Fantastic Caste System: A Flatlander's intellect is dictated by the degrees of their angles (for triangles), or how many angles they have (for polygons). A perfectly equilateral triangle is the equivalent to an Intrepid Merchant in brainpower; a square is smart enough to be a lawyer. People with "irregular" angles (One example given was a parallelogram) are predestined to criminality, similar to how some people thought of XYY-Chromosome Disorder patients back in the 80's. Females, being straight lines and therefore having no angles, are universally dumb as, well, posts. Polygons go up one caste every generation, and triangles have a chance at producing a square son, who in turn have a chance of producing pentagonal or triangular sons.
  • Flat World: Not really the Trope Maker, as various cultures have believed in a flat Earth since prehistoric times. But Flatland definitely applies a new spin to the notion.
  • Gender Equals Breed: As discussed above, a male Flatlander will be of his father's shape or one angle higher, while females are straight lines regardless of parentage.
  • A God Am I: The sole inhabitant of Pointland spends all his time making these speeches to himself, since he has no way of being convinced that anything else even exists.
  • Government Conspiracy: The circles that rule Flatland know that a sphere comes every thousand years to preach the Gospel of Three Dimensions, so they execute any lower-class Flatlander heard spreading the sphere's message and imprison higher-class Flatlanders that do so.
  • Ignorant of Their Own Ignorance: Former Trope Namer. The King of Pointland believes that he is the only being that exists, because he cannot perceive anything outside of Pointland.
  • Interspecies Romance: Romance between the polygonal males and the linear females of Flatland can approach this.
  • In Which a Trope Is Described: All the chapter titles. The second chapter, for instance, is titled "Of the Climate and Houses in Flatland", since the book is supposed to be the Square's description of his country and people.
  • Living Polyhedron: All inhabitants of Flatland are two-dimensional polygons. There's also a Sphere, which claims to inhabit a Sphereland with other 3D shapes.
  • More than Three Dimensions: Discussed. A. Square is a regular guy who happens to be a square, living in a two-dimensional universe. He is visited by a sphere who preaches to him the Gospel of Three Dimensions. The square is scornful of the idea initially, but eventually the sphere convinces him. When the square talks excitedly of the possibility of a fourth dimension, the sphere immediately dismisses the idea as ridiculous.
  • New Year Has Come: The square is seeing in the new year with his wife when the sphere appears to him.
  • Pals with Jesus: The hero is befriended by a helpful higher-dimensional being.
  • Punny Name: A. Square. Take into account that the author's name is Edwin Abbott Abbott. Abbott squared = A, squared.
  • Sinister Geometry: Averted; Flatlanders value perfect geometric symmetry as a sign of high breeding and intelligence. From a spectator point of view, however, Flatlander society in the indie film is shown to be rather cruel and sinister.
  • Skepticism Failure: A. Square and the Linelanders are skeptical of higher dimensions, and the plot proves them wrong.
  • Stay in the Kitchen: Women are restricted in their social roles because they are dangerously sharp lines (well, super narrow rhombuses). They are restricted because they can kill people by walking into them: it is segregation based on a physical characteristic. Lampshaded, in that the Square mentions that it probably seems incredibly misogynistic to the reader, but it beats getting stabbed to death.
  • Third-Person Person: The King of Pointland, who is unable to conceive of anything other than himself.
  • The War on Straw: The Flatland priests and government officials' view of a world not being flat.
  • When Dimensions Collide: The presence of the more transcendent sphere in Flatland almost embodies this trope.
  • Women Are Wiser: Averted. Flatland women (lines) have the memories of goldfish and come off as Cloudcuckoolanders. One anecdote states that a line once murdered her entire family and then wondered what the hell happened to them a minute later.
  • You Cannot Grasp the True Form: When the hero is taken into the third dimension by a sphere, he literally Cannot Grasp Its True Form or the other creatures there at first.

    Tropes found in various adaptations of Flatland 
  • Adaptational Heroism: Abbot Square in Flatland: The Movie is far more accepting of the idea of 3D and assists his brother in helping spread the message.
  • Adaptation-Induced Plot Hole:
    • Flatland: The Film has a setting update from Victorian England to modern day U.S. so why does the "President" still wear a crown?
    • Flatland: The Movie has objects that don't make sense in a 2D world, such as pink unicorn dolls, a fishtank, and skateboards. The sequel Sphereland depicts two-dimensional dogs.
  • Adaptational Intelligence: In both 2007 adaptations, females are not nearly as braindead as they are in the book.
  • Adaptation Name Change:
    • Flatland: The Film:
      • Pantocyclus to President Circle.
      • Lord Sphere to CEO A. Sphere.
    • Flatland: The Movie:
      • A. Square to Arthur Square.
      • B. Square to Abbot Square.
      • Lord Sphere to Spherius.
  • Adaptation Species Change:
    • Chromatistes is changed from a pentagon to an "irregular" in Flatland: The Film. A. Square also only has one hexagonal child, with the other being changed to a pentagon.
    • In Flatland: The Movie females are the same shapes as the males.
  • Adaptational Villainy:
    • B. Square in Flatland: The Film is the one responsible for planning the Chromatist massacre (and thus causing the ensuing riot) and later tries to kill his brother in prison.
    • Pantocyclus was originally mostly a Well-Intentioned Extremist, and had legitimate reasons to fear chromatism and 3D. Even in the Darker and Edgier Flatland: The Film he had the occasional Pet the Dog moment. In Flatland: The Movie on the other hand, he has the purely selfish motivation of keeping the circles on the top of the food chain and also has Arthur sentenced to death as opposed to imprisonment.
  • A Kind of One: In Sphereland, the protagonists discover that there are dozens of Pointlands, with each one having (and being) a self-proclaimed King of Pointland all ignorantly unaware of each other.
  • Age Lift: Hex is A. Square's son as opposed to Grandson in Flatland: The Film. In this case the extra side is explained as the result of being born with a deformity.
  • Alternate Universe: Sphereland reveals that the two-dimensional universe of Flatland is the surface of a sphere, and that many other such spheres exist all in the third dimension.
  • Alternate Self: Near the end of Sphereland, Hex sees multiple alternate versions of the ship, including one where her grandfather Arthur Square is still alive. Hex wonders aloud what that version of her did differently that mader her have a happier ending.
  • Animated Adaptation: Oddly, two were released in the same year (2007). Flatland: The Film was a feature-length indie film. Flatland: the Movie was a big-budgeted edutainment short with an All-Star Cast.
  • Ascended Extra: In Flatland: The Movie, the King of Lineland only appears in one scene, but in the sequel Sphereland, he plays a more prominent role in helping uncover the mystery of Flatland's shape.
  • Asteroid Thicket: A two-dimensional variant in Sphereland that the space crew's ship must avoid.
  • Auto Cannibalism: A. Line (a political prisoner) in the movie does this as a form of suicide before she's executed by the state.
  • The Blank: The Oversphere is depicted without a face. Justified, as this is really but a 3D cross-section of her unseeable full 4D form. Spherius is likewise depicted as such when he is seen as a 2D circle in Flatland. And of course the Kings of Pointland and Lineland don't really have enough dimensions to have a face.
  • Breaking Old Trends: Flatland: The Movie had a recurring theme of increasingly-higher-dimensional beings ignorant of a higher dimension than themselves, with the King of Pointland unable to percieve anything outside himself, the King of Lineland being unable to understand Arthur Square's talk of two dimensions, Arthur himself unable to percieve Spherius and even Spherius scoffing at the mere idea of a fourth dimension. The Oversphere breaks this pattern entirely: she fully accepts the potential existence of a fifth, and sixth, and maybe more dimensions, even scolding and punishing Spherius for being so closed-minded and calling the potential of higher dimensions "an infinite set of possibilities."
  • Chekovs Gunman: The King of Lineland in Sphereland. While Hex and Puncto travel alongside Lineland they encounter him again, in which case they ask him if he's been following them. Only when he replies he hasn't the two hexagons realize that they went right back from where they started just by following Lineland: leading to their realization that Lineland is a circle, and Flatland is a sphere.
  • Dimensional Traveler: The Oversphere from Sphereland moves through the fourth dimension and visits Spherius just as he visited Arthur Square in the first film.
  • Happy Ending Override: Flatland: The Movie ends on a high note, with Arthur Square and his proof of the third dimension seemingly having revolutionized Flatland. Sphereland reveals however that he was eventually ostracized for his inability to further explain his discoveries, and he later dies of grief.
  • Fan Sequel: Given that Abbott's book is now in the public domain, several authors have tried their hand at this:
    • An Episode on Flatland: Or How a Plain Folk Discovered the Third Dimension by Charles Howard Hinton. Published way back in 1907, it features a slightly more realistic take on a 2D universe (the Flatlanders in this version exist on the surface of a circular planet rather than just moving through empty space, for instance).
    • "Sphereland: A Fantasy About Curved Spaces and an Expanding Universe" by Dionys Burger. It features A. Square's grandson and tries to explain curved spaces and expanding universes in the same way Flatland tried to explain dimensions. It later gets an animated adaptation as a direct sequel to Flatland: The Movie.
    • The Planiverse: Computer Contact with a Two-Dimensional World by A. K. Dewdney. A Spiritual Sequel that focuses on creating a believable 2-D world with its own physics, biology and culture.
    • Flatterland: Like Flatland, Only More So by Ian Stewart. Follows another one of A Square's descendents and her ventures with an interdimensional Space Hopper. Sacrifices plot and social commentary for higher math theories and puns.
    • "Spaceland: A Novel of the Fourth Dimension" by Rudy Rucker. A Non-Linear Sequel / Spiritual Sequel with its focus on the plot of Silicon Valley hotshot Joe Cube and his encounter with Momo, a woman from the fourth dimension.
    • The iOS game A Noble Circle is based on the novella.
  • Gainax Ending: Flatland: The Film starts to go off the rails in its third act: After hallucinating the King of Pointland, escaping prison and bumping into an both an Identical Stranger and the trapezoid he saw die earlier, A. Square's pursuers are slain by the North Kingdom army. One of the Northern officers begins to chase him, and then everything suddenly goes black. A. Square speaks a single line pleading to see his family, and is answered by an unknown voice. Roll Credits.
  • Gender Flip: Hex is female in Flatland: The Movie. It and its sequel Sphereland potray more female characters due to them being the same shape as the male ones.
  • Ignorant of Their Own Ignorance: Aero the isoceles triangle in Sphereland vehemently denies the existence of the third dimension and is very dismissive of Hex and Puncto's claims.
  • The Movie: You gotta believe them.
  • Reverse Polarity: Happens in all three dimensions in Sphereland:
    • Spherius is punished by the Oversphere for abandoning the Flatlanders by flipping his directions in 3D, forcing him to move in the opposite direction he wants to go.
    • The ship is accidentally flipped over in the second dimension by Spherius, causing their direction to reverse.
    • Hex and Puncto accidentally pop the King of Lineland out of his line, and put him back backwards. The Queen of the Left and the Queen of the Right are thus offended and confused when he refers to them by the wrong names.
  • Nice Job Breaking It, Hero: Spherius is punished by the Oversphere for abandoning the Flatlanders and being himself closed-minded, with her forcing him to return to Flatland by reversing his directions and making the only remedy to find the center of Flatland where she awaits.
  • Not His Sled: In Flatland: The Movie, Spherius rescues Arthur Square from his death sentence and reveals that he intended his granddaughter Hex to be his prophet.
  • Sudden Sequel Death Syndrome: Sphereland takes place 20 years after Flatland: The Movie. By then, Arthur Square has since passed away.
  • Ugly Cute: The guard dogs of the ruins site in Sphereland. Being two-dimensional, we can see their innards. They're still kind of adorable though, as they play chase with Puncto.
  • Values Resonance: Sphereland depicts equality among the Flatlanders after the fall of the Circle Priests, with circles, squares, triangles and all other shapes in between now working together as the scientists of the space exploration base.
  • Wham Line: "The center of Flatland", as revealed by Sphereland. It reveals that the vast 2D plane of Flatland was actually part of the surface of an immense sphere, and that many other Spherelands exist in the 3D plane.