Flatland: A Romance of Many Dimensions is an 1884 book by Edwin A. Abbott (1838-1926) about an imaginary land which is a Euclidean plane, and whose citizens are geometrical figures. It is entertaining on many levels, not all intended by the author, and is particularly enjoyed by students learning geometry.
Flatland's civilization parallels that of Victorian England. The male inhabitants of Flatland are, for the most part, regular polygons, with their social status and intelligence increasing with the number of sides:
- Our Middle Class consists of Equilateral or Equal-Sided Triangles. Our Professional Men and Gentlemen are Squares (to which class I myself belong) and Five-Sided Figures or Pentagons. Next above these come the Nobility, of whom there
are several degrees, beginning at Six-Sided Figures, or Hexagons, and from thence rising in the number of their sides till they receive the honourable title of Polygonal, or many-Sided. Finally when the number of the sides becomes so numerous, and the sides themselves so small, that the figure cannot be distinguished from a circle, he is included in the Circular or Priestly order; and this is the highest class of all.
The lowest classes—"our Soldiers and Lowest Class of Workmen" are isosceles triangles. In Flatland, it is dangerous for an inhabitant to come into contact with a sharp, acute angle. Consequently, the upper classes live in fear of being attacked by the lower classes, whose acute angles make good weapons.
The women of Flatland are straight lines, and are thus lower in status and intelligence than even the isosceles triangles, and even more dangerous:
- If our highly pointed Triangles of the Soldier class are formidable, it may be readily inferred that far more formidable are our Women. For, if a Soldier is a wedge, a Woman is a needle; being, so to speak, all point, at least at the two extremities.
In the later chapters, the narrator is visited by a three-dimensional sphere who tries to explain the nature of three-dimensional space to him. He does not understand, until finally the sphere drags him into "spaceland." Having grasped the idea of three dimensions, the narrator in turn asks the sphere whether there might not be four dimensions, extending the arguments the sphere used on him.
- Flatland Online edition with illustrations