Flatland

Flatland: A Romance Of Many Dimensions is a 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. Square. 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".


 * 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.
 * Alien Geometries: The third dimension is unfathomably alien to the Flatlanders, and so is the second dimension to the Linelanders.
 * 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.
 * Black and White Insanity: the ruling caste enforce a Black and White Morality worldview to the point where they outlaw color, enforcing the world to literally be black and white. Their excuse for this draconic law is that it's needed for preserving the sexual purity of their women.
 * 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 lines. It is a satire, thinly veiled, of Victorian society.
 * Flat World: Ultimate Example.
 * Fridge Brilliance: The main character is named A. Square, because that's the shape that the character has. However, once one realizes that the author's name is Edwin Abbot Abbot, which for the mathmatical types can be represented as Abbot Squared, then it becomes a sly self-reference.
 * 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.
 * One-Dimensional Thinking: Justified in Lineland.
 * Not So Different: Lord Sphere is no more willing to accept a fourth dimension than the Circles are a third.
 * Paper People
 * Painting The Fourth Dimension: or possibly Leaning on the Fourth Wall.
 * Pals with Jesus: The hero is befriended by a helpful higher-dimensional being.
 * Shout-Out: A number of stories are unofficial sequels to or imitations of Flatland; others were inspired by it, and still others use this story to explain the concept of dimensions beyond our own.
 * Carl Sagan laid it out in the Cosmos episode, The Edge of Forever.
 * The Outer Limits has an episode with an alien freely inspired by Flatland.
 * And in Animorphs, Elfangor explains the abilities of the Ellimist by telling Loren to imagine that they were Flatties, lived on a sheet of paper, and could be imprisoned by the three-dimensional Ellimist drawing a square around them... or freed by being plucked up off the paper. Well, they're Cubies.
 * And then there's the Network TV show The Big Bang Theory. The writers certainly know how to please a demographic.
 * At one point during their interdimensional travels in A Wrinkle in Time, the protagonists are accidentally transported to Flatland, where they almost suffocate.
 * The Hypernaut from Alan Moore's 1963 has a Flatland in his space station.
 * Flatland and its inhabitants end up as the MacGuffin of the backstory for Transformers Alternity which, itself, deals extensively with the concepts of pan-dimensional transcendence. You know, To Sell Toys.
 * The Original Flatland Role Playing Game by Marcus L Rowland. "Action and adventure in a world where characters are supposed to be two-dimensional!"
 * Sinister Geometry: Averted; Flatlanders value perfect geometric symmetry as a sign of high breeding and intelligence.
 * Skepticism Failure: A. Square and the Linelanders are skeptical of higher dimensions, and the plot proves them wrong.
 * Small Secluded World: Pointland.
 * Spiritual Successor: The Planiverse by A.K. Dewdney, a 1984 novel about a college professor and a team of students who create a 2D world simulation which somehow contacts a real 2D world.
 * Stay in the Kitchen: Women are restricted in their social roles because they are dangerously sharp lines. They are restricted because they can kill people by walking into them: it is segregation based on a physical characteristic.
 * Straw Man: The Flatland priests and government officials' view of the a world not being flat.
 * 2-D Space: Flatland.
 * World Building: Over half the book is this.
 * 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.