Flatland: A Romance of Many Dimensions is a satirical novella by the English schoolmaster Edwin A. 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.

The story describes a two-dimensional world occupied by geometric figures. The narrator is a square named “A Square”, who guides the readers through some of the implications of life in two dimensions. 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.

Following this vision, he is himself visited by a three-dimensional sphere named A 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 retract. The Square is not fully convinced until he sees Spaceland (a tridimensional world) for himself.

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.

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. Eventually the Square himself is imprisoned, and seven years later, A Square writes out the book Flatland in the form of a memoir, hoping to keep it as posterity for a future generation that can see beyond their two-dimensional existence.

Flatland features in The Big Bang Theory episode “The Psychic Vortex”, when Sheldon Cooper declares it one of his favourite imaginary places to visit.