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 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 about a visit to a one-dimensional world (Lineland) inhabited by "lustrous points". These points 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 is unable to do so. In the end, the monarch of Lineland tries to kill A Square rather than tolerate his nonsense any further.Following this vision, he is himself visited by a three-dimensional sphere. Similar to the "points" in Lineland, the Square is unable to see the sphere as anything other than a circle. The Sphere then levitates up and down through the Flatland, allowing Square to see the circle expand and contract. The Square is not fully convinced 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 hope 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 on. After this proclamation is made, many witnesses are massacred or imprisoned (according to caste), including the Square's brother.
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