Flatland: A Romance of Many Dimensions is a satirical novella by the English schoolmaster Edwin Abbott Abbott, first published in 1884 by Seeley & Co. of London. Written pseudonymously by "A Square", the book used the fictional two-dimensional world of Flatland to comment on the hierarchy of Victorian culture, but the novella's more enduring contribution is its examination of dimensions.
Several films have been made from the story, including the feature film Flatland (2007). Other efforts have been short or experimental films, including one narrated by Dudley Moore and the short films Flatland: The Movie (2007) and Flatland 2: Sphereland (2012).
Plot
The story describes a two-dimensional world inhabited by geometric figures (flatlanders); women are line segments, while men are polygons with various numbers of sides. The narrator is a square, a member of the caste of gentlemen and professionals, who guides the readers through some of the implications of life in two dimensions. The first half of the story goes through the practicalities of existing in a two-dimensional universe, as well as a history leading up to the year 1999 on the eve of the 3rd Millennium.
On New Year's Eve, the Square dreams of a visit to a one-dimensional world, "Lineland", inhabited by men, consisting of lines, while the women consisted of "lustrous points". These points and lines are unable to see the Square as anything other than a set of points on a line. Thus, the Square attempts to convince the realm's monarch of a second dimension but cannot do so. In the end, the monarch of Lineland tries to kill the Square rather than tolerate him any further.
Following this vision, the Square is visited by a sphere. Similar to the "points" in Lineland, he is unable to see the three-dimensional object as anything other than a circle (more precisely, a disk). The Sphere then levitates up and down through Flatland, allowing the Square to see the circle expand and contract between great circle and small circles. The Sphere then tries further to convince the Square of the third dimension by dimensional analogies (a point becomes a line, a line becomes a square). The Square is still unable to comprehend the third dimension, so the Sphere resorts to deeds: he gives info about the "insides" of the house, moves a cup through the third dimension, and even goes inside the Square for a bit. Still unable to comprehend 3D, the Sphere takes the Square to the third dimension, Spaceland. This Sphere visits Flatland at the turn of each millennium to introduce a new apostle to the idea of a third dimension in the hope of eventually educating the population of Flatland. From the safety of Spaceland, they can oversee the leaders of Flatland, acknowledging the Sphere's existence and prescribing the silencing. After this proclamation is made, many witnesses are massacred or imprisoned (according to caste), including the Square's brother.
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 dimension and higher spatial dimensions. Still, the Sphere returns his student to Flatland in disgrace.
The Square then has a dream in which the Sphere revisits him, this time to introduce him to a zero-dimensional space, Pointland, of whom the Point (sole inhabitant, monarch, and universe in one) perceives any communication as a thought originating in his own mind (cf. Solipsism):
The Square recognises the identity of the ignorance of the monarchs of Pointland and Lineland with his own (and the Sphere's) previous ignorance of the existence of higher dimensions. Once returned to Flatland, the Square cannot convince anyone of Spaceland's existence, especially after official decrees are announced that anyone preaching the existence of three dimensions will be imprisoned (or executed, depending on caste). For example, he tries to convince his relative of the third dimension but cannot move a square "upward," as opposed to forward or sideways. Eventually, the Square himself is imprisoned for just this reason, with only occasional contact with his brother, who is imprisoned in the same facility. He cannot convince his brother, even after all they have both seen. Seven years after being imprisoned, A Square writes out the book Flatland as a memoir, hoping to keep it as posterity for a future generation that can see beyond their two-dimensional existence.