Flatland: A Romance of Many Dimensions is an 1884 satirical novella by the English schoolmaster Edwin Abbott Abbott. Writing pseudonymously as "a square" , Abbott used the fictional two-dimensional world of Flatland to offer pointed observations on the social hierarchy of Victorian culture. However, the novella's more enduring contribution is its examination of dimensions; in a foreword to one of the many publications of the novella, noted science writer Isaac Asimov described Flatland as "The best introduction one can find into the manner of perceiving dimensions". As such, the novella is still popular amongst mathematics, physics, and computer science students.

Several films have been made from the story, including a feature film in 2007 called Flatland. Other efforts have been short or experimental films, including one narrated by Dudley Moore and a short film with Martin Sheen titled Flatland: The Movie.

The story is about a two-dimensional world referred to as Flatland which is occupied by geometric figures, line segments (females) and regular polygons with various numbers of sides. The narrator is a humble Square, a member of the social caste of gentlemen and professionals in a society of geometric figures, who guides us through some of the implications of life in two dimensions. The square has a dream about a visit to a one-dimensional world (Lineland) which is inhabited by "lustrous points." He attempts to convince the realm's ignorant monarch of a second dimension but finds that it is essentially impossible to make him see outside of his eternally straight line.

The narrator is then visited by a three-dimensional sphere, which he cannot comprehend until he sees Spaceland for himself. This sphere, who remains nameless, visits Flatland at the turn of each millennium to introduce a new apostle to the idea of a third dimension in the hopes of eventually educating the population of Flatland of the existence of Spaceland. From the safety of Spaceland, they are able to observe the leaders of Flatland secretly acknowledging the existence of the sphere and prescribing the silencing of anyone found preaching the truth of Spaceland and the third dimension. After this proclamation is made, many witnesses are massacred or imprisoned (according to caste).

After the Square's mind is opened to new dimensions, he tries to convince the Sphere of the theoretical possibility of the existence of a fourth (and fifth, and sixth ...) spatial dimension. Offended by this presumption and incapable of comprehending other dimensions, the Sphere returns his student to Flatland in disgrace.

He then has a dream in which the Sphere visits him again, this time to introduce him to Pointland. The point (sole inhabitant, monarch, and universe in one) perceives any attempt at communicating with him as simply being a thought originating in his own mind (cf. Solipsism):

The Square recognizes the connection between the ignorance of the monarchs of Pointland and Lineland with his own (and the Sphere's) previous ignorance of the existence of other, higher dimensions.

Once returned to Flatland, the Square finds it difficult to convince anyone of Spaceland's existence, especially after official decrees are announced – anyone preaching the lies of three dimensions will be imprisoned (or executed, depending on caste). Eventually the Square himself is imprisoned for just this reason.

In the book, men are portrayed as polygons whose social class is directly proportional to the number of sides they have; therefore, triangles, having only three sides, are at the bottom of the social ladder and are considered generally unintelligent, while the Priests are composed of multi-sided polygons whose shapes approximate a circle, which is considered to be the "perfect" shape. On the other hand, the female population is comprised only of lines, who are required by law to sway back and forth and sound a "peace-cry" as they walk, because when a line is coming towards an observer in a 2-D world, it appears merely as a point. The Square talks of accounts where men have been killed (both by accident and on purpose) by being stabbed by women. This explains the need for separate doors for women and men in buildings. Also, colors in Flatland were banned, when lower classes painted themselves to appear to be higher ordered.

In the world of Flatland, classes are distinguished using the "Art of Hearing", the "Art of Feeling" and the "Art of Sight Recognition". Classes can be distinguished by the sound of one's voice, but the lower classes have more developed vocal organs, enabling them to feign the voice of a polygon or even a circle. Feeling, practised by the lower classes and women, determines the configuration of a person by feeling one of their angles. The "Art of Sight Recognition", practised by the upper classes, is aided by "Fog", which allows an observer to determine the depth of an object. With this, polygons with sharp angles relative to the observer will fade out more rapidly than polygons with more gradual angles. The population of Flatland can "evolve" through the Law of Nature, which states: "a male child shall have one more side than his father, so that each generation shall rise (as a rule) one step in the scale of development and nobility. Thus the son of a Square is a Pentagon; the son of a Pentagon, a Hexagon; and so on."

This rule is not the case when dealing with isosceles triangles (Soldiers and Workmen), for their evolution occurs through eventually achieving the status of an equilateral triangle, removing them from serfdom. The smallest angle of an isosceles triangle gains thirty minutes (half a degree) each generation. Additionally, the rule does not seem to apply to many-sided polygons; the sons of several hundred-sided polygons will often develop fifty or more sides more than their parents.

Regular polygons were considered in isolation until chapter 7 of the book when the issue irregularity, or physical deformity, became considered. In a two dimensional world a regular polygon can be identified by a single angle and/or vertex. In order to maintain social cohesion, irregularity is to be abhorred, with moral irregularity and criminality cited, "by some" (in the book), as inevitable additional deformities, a sentiment concurred by the author. If the error of deviation is above a stated amount the irregular faces euthanasia, if below he becomes the lowest rank of civil servant. In this study of what was to later to become known as eugenics, an irregular polygon should not be destroyed at birth, but rather allowed to develop in order to see if its irregularity could be “cured” or reduced to within society's level of tolerance. If the deformity could not be correct then the irregular should be “painlessly and mercifully consumed.”

In the book, the three-dimensional Sphere has the ability to stand inches away from a Flatlander and observe them without being seen, can remove Flatland objects from closed containers and teleport them via the third dimension without traversing the space in between, and is capable of seeing and touching the inside and outside of everything in the two-dimensional universe; at one instance, the Sphere gently pokes the narrator's intestines and launches him into three dimensions as proof of his powers.

Numerous imitations or sequels to Flatland have been written, including:

Short stories inspired by Flatland include:

Flatland: A Romance of Many Dimensions

Flatterland • Sphereland

The Planiverse • Spaceland • The Dot and the Line: A Romance in Lower Mathematics

Flatland (2007)  • Flatland: The Movie (2007 short film)

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