The story describes a two-dimensional world occupied by geometric figures, whereof women are simple 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 Square dreams about a visit to a one-dimensional world (Lineland) inhabited by “lustrous points”, and attempts to convince the realm’s monarch of a second dimension; but is unable to do so. He is himself visited by a three-dimensional sphere, which he cannot comprehend until he sees Spaceland (a tridimensional world) for himself. This Sphere 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. 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; but the Sphere returns his student to Flatland in disgrace.
– Wikipedia plot summary to ‘Flatland: A Romance of Many Dimensions‘ by Edwin Abbott Abbott