Flatland: An Ingenious Mathematical Fiction About Human Folly And Human Short-sightedness

FlatlandIn my previous post, I discussed the literary power of novels to foster our public imagination about what is socially and politically possible. This post is a continuation of that topic and describes a novel that like Charles Dickens’ Hard Times contains the power to incite public imagination.

Flatland (1884) is a mathematical novel written by Edwin Abbott Abbott. When it was first published, it did not obtain much popularity. The novel however, was discovered again after Albert Einstein published his general theory of relativity. William Garnett wrote in a letter to the editor of Nature that

Some thirty or more years ago a little jeu d’esprit was written by Dr. Edwin Abbott entitled Flatland. At the time of its publication it did not attract as much attention as it deserved… If there is motion of our three-dimensional space relative to the fourth dimension, all the changes we experience and assign to the flow of time will be due simply to this movement, the whole of the future as well as the past always existing in the fourth dimension. (Garnett, 1920)

The novel is a first person narration by the main character, ‘A Square’, who lives in the two dimensional world, Flatland. The protagonist, A Square, is a mathematician and a lawyer who next to offering readers some practical lessons in Euclidean geometry also makes an excellent satire of the many controversies within the Victorian social order, religion and science. At its core, the novel encourages contemporary readers to question the status quo and to challenge our intellectual short-sightedness by stimulating us to think about our assumptions about what is real and logical. Although the novel in general can be interpreted as a wider critique of our contemporary existence, I will focus only on one critique which is on the short-sightedness of human beings. This critique, I believe, manifests itself in A Square’s unsuccessful mission to enlighten beings from the first, second, and third dimension about the existence of a higher dimension. Reading about A Square’s travels to the various dimensions makes us question to what extent we differ in our epistemological arrogance from the beings A Square encounters. It also makes us wonder how it is to be a misunderstood unconventional thinker; how it is to know the truth, while being considered mad.

Abbott describes the lives of geometrical organisms in zero, first, second, and third dimensions with witty facetiousness. A Square’s encounter with the inhabitants of the land of zero dimensions, Pointland, and the land of one dimension, Lineland, and the land of three dimensions, Spaceland, illustrates the short-sightedness and the intellectual arrogance they portray about the world despite their inability to comprehend a reality that is vastly different from what they have ever experienced.

Beings in Pointland

In Pointland for example, A Square, meets a soliloquous being (a Point) who believes that he is all and everything in the world. Indulged in philosophical solipsism, the Point finds perfect self-contentment in his ignorance of the world. He,

himself his own World, his own Universe; of any other than himself he can form no conception; he knows not Length, nor Breadth, nor Height, for he has had no experience of them; he has no cognizance even of the number Two; nor has he a thought of Plurality; for he is himself his one and A, being really Nothing (Abbott, 1884, p. 109).

Beings in Lineland

In Lineland, A Square meets a Monarch who is nothing else but a Straight Line and who is “the longest in Lineland, over six inches of Space… Of Length…” (Abbott, 1884, p. 74). This Monarch regards his Straight Line as his Kingdom and has no perception of ‘left’ and ‘right’. His spatial or worldly experience only consists of ‘back’ and ‘forth’. When A Square attempts to speak to him some truth about a higher dimension, the King interrupts him by saying scornfully: “You speak of an impossibility” (Abbott, 1884, p. 74). When A Square tells the Monarch that his life must be deplorably dull for not being able to infer the existence of Angles, Triangles, Squares, Pentagons, Hexagons, and even Circles, he loses his patience and demands silence. A Square angrily responds

Why waste more words? Suffice it that I am the completion of your incomplete self. You are a Line, but I am a Line of Lines, called in my country a Square: and even I, infinitely superior though I am to you, am of little account among the great Nobles of Flatland, whence I have come to visit you, in the hope of enlightening your ignorance. (Abbott, 1884, p. 76)

Beings in Flatland

Of all lands, Flatland is described in most detail. Flatland’s society is strictly hierarchical with a clear division of people’s statuses. The status and therefore privilege of a person is directly determined by the number of angles or sides he possesses. The lowest class is made up of women who are merely Lines. The female Flatlanders are described as dull, emotional, unimaginative and lacking any form of intelligence.[1] The class that is little higher than the female class is the class of Isosceles who are the Soldiers and the Workmen of society and “who indeed can hardly be said to deserve the name of human Figures, since they have not all their sides equal” (Abbott, 1884, p. 22). Above the Soldiers and Working class are the tradesmen class that only exists of Equilateral Triangles. The respectable classes are those of Squares and Pentagons that fulfil such professional occupations as lawyers and doctors. Above these respectable classes, one finds the lowest noble class and the higher noble class which consists of Hexagons and Polygons respectively. The highest of all is the King who has so many angles that he resembles a Circle.

With so many unfair privileges based on the class distinctions by sides, how does Flatland’s society remain relatively stable where revolutions rarely occur? The political answer is that female Flatlanders have no hopes for the improvement of their fate, and so they also “have no memory to recall, and no forethought to anticipate, the miseries and humiliations which are at once a necessity of their existence and the basis of the constitution of Flatland” (Abbott, 1884, p. 30). In case of the other lower classes, they still have hope that through good conduct (which means working in the interest of the higher classes) and through bringing forth descendants that by the Law of Compensation can gain an extra side, they can improve the lot of their kin. This hope encourages the lower classes to accept their deplorable situations.

After A Square’s narration of the life in Flatland, he tells how a Sphere from the land of third dimensions, Spaceland, came to him to bring the truth about a higher dimension. The Sphere told him that next to the dimensions of length and width, there is also height. Unable to comprehend this concept, the Sphere decides to pick him up and show him Flatland from above. Amazed by the new insights about this higher dimension, A Square then asks the Sphere about the theoretical possibility of the existence of a fourth, a fifth, a sixth… spatial dimension. The Sphere however, who believes that there are only three dimensions and who cannot grasp the possibility of even a higher one became annoyed by A Square’s suggestion of higher dimensions. Thereafter, disgraced by A Square’s suggestions, he returns A Square to Flatland. Once returned into Flatland, A Square attempts to spread his newly found wisdom. However, he miserably fails. Not only does he find it impossible to make other Flatlanders grasp the concept of ‘height’, but the King also ordered that anyone with such heretical views of higher dimensions would be sentenced to jail.

Lesson learned

A Square’s inability to spread his insights about reality teaches us the following, “prophets and inspired people are always considered by the majority to be mad” (Abbott, 1884, p. 115). The story ends with a saddened A Square who declares that “Prometheus up in Spaceland was bound for bringing down fire for mortals, but I – poor Flatland Prometheus – lie here in prison for bringing down nothing to my countrymen” (Abbott, 1884, p. 117).

See here a video of a small summary of Flatland as told by Carl Sagan.[2]


[1] At the time when Flatland was published, women’s role in society was heavily discussed. Many Europeans at the time believed that women were inferior in strength and intelligence due to their inherent biological nature. John Stuart Mill argued in The Subjection of Women (1869), 15 years before Flatland was published, that the female mental development was the result of culturally restricted roles.

[2] In the real story, a Sphere enters Flatland and not an apple.


Abbott, E.A. (1884). Flatland. Oxford: Oxford University Press.
Garnett, W. (1920). Letter to the Editor. Nature, February 12.