Flatland

Abstract

We study the mathematical allegory in Edwin Abbott’s novella Flatland: A Romance of Many Dimensions, whose ideas influenced mathematics, science fiction, education, philosophy, cosmology, modern art, and computer graphics, among other areas. Abbott, a schoolmaster and biblical, English literature, and Classics scholar, used n-dimensional geometry, up to the fourth dimension, to demonstrate how it is possible to understand an idea or object that we cannot construct or perceive. We provide an overview of nineteenth-century geometry, leading up to the discovery of different geometries and subsequently, the exploration of higher dimensions. We demonstrate the dimensional analogy, a mathematical exercise that introduces the fourth dimension—Abbott’s mathematical allegory. We explore Flatland’s various dimensions: as an introduction to higher-dimensional geometry, as a model of Plato’s cave, and as satire of Victorian hierarchical society and its treatment of women, and we provide a brief survey of novels and films Flatland has inspired.