First published in 1884, Abbott's Flatland is a whimsical story about a square (or Mr. A. Square) living in a two dimensional world who is visited by a sphere from a three dimensional world. Eventually, Mr. A. Square is transported by the sphere into the three dimensional world so that he might understand the limitations of his own experience. Flatland is at once a lesson in geometry and an allegory on two levels. The geometry lesson is quite simple. Abbott provides the reader with a reasonably plausible account of how a two dimensional world might appear to a two dimensional being. At one point he also provides us with an account of how a one dimensional world might appear to a one dimensional being. However, it is the allegories that are the most interesting aspects of the book. One the most obvious level, Mr. A. Square's world is rife with the social and political hierarchies of the late 19th century. Class is represented by the number of sides the resident polygons have. Near circles are the highest, nearly priestly class, while on the other end of the hierarchy, women are nearly simple straight lines. On a less obvious level, the allegory is about how we are limited by the metaphysical contours of our world and our faculties of mind.
Abbott does not explicitly write of this, but his work clearly suggests that our ordinary manner of thinking about the world as being laid out in four dimensions might be a product of our faculties of mind. Indeed, theories in contemporary physics suggest that we in fact live in a world best described by numerous more dimensions than the four we commonly accept. For someone not acquainted with these theories, one feels like Mr. A. Square wrestling with the arguments of the sphere that assert unimaginable ideas.
Flatland is utterly charming.