On the surface (pun intended), Flatland, written by Edwin Abbott Abbott, is the story of a two-dimensional world in which A Square is jolted out of his dimensional complacency into the realization that his worldview is not just flat, but downright ridiculous. A two-dimensional being’s inability to conceive of three dimensions is a powerful analogy for us as we try to imagine four-, five-, and more-dimensional space. As such, the book could be a nice tool for helping students of math think about higher dimensional spaces.
Looking more deeply, the book is more a social and political satire in the Swiftian tradition. Along with A Square, the reader comes to see class distinctions (the more sides you have the higher class and more intelligent you are, with the Circle as the epitome of both) and social abnormality (figures with irregular side lengths) as arbitrary divisions designed to systematically reinforce society’s current power structure. Heterodoxy in Flatland—e.g., failing your university exit examinations or, worse, talking about dimensions higher than two—is punished by life in prison or death. Women in Flatland are barely rational one-dimensional lines; they are good for breeding and emotions only and can’t follow the higher reasoning of the males; educating women is therefore a waste of time. Though published in 1884, Abbott’s pointed criticism of society is no less salient today.
In his dedication, Abbott admits that he aspires to contribute “to the Enlargement of The Imagination and the possible Development of that most rare and excellent Gift of Modesty.” I would put it somewhat differently: Abbott wants to jar the reader out of his or her arrogance about the very nature of the world and of society; he wants to challenge us to see beyond ourselves, to imagine other perspectives, to have compassion for other possible realities based on a humble awareness of our own myopia.
Perhaps the clearest example of this myopia is the inhabitant of Pointland. This singularity, a point in space, can perceive no dimensions and therefore sees itself as the whole universe. The resulting onanistic soliloquy would be laughable if I didn’t see in it the reflections of myself and other people I have known:
Infinite beatitude of existence! It is; and there is none else beside It . . . . It fills all Space . . . and what It fills, It is. What It thinks, that It utters; and what It utters, that it hears; and It itself is Thinker, Utterer, Hearer, Thought, Word, Audition; it is the One, and yet the All in All. Ah, the happiness ah, the happiness of Being!
When challenged by A Square to see outside itself, the Point, hears only itself:
Ah, the joy, ah, the joy of Thought! What can IT not achieve by thinking! Its own Thought coming to Itself, suggestive of Its disparagement, thereby to enhance Its happiness! Sweet rebellion stirred up to result in triumph! Ah, the divine creative power of the All in One! Ah, the joy, the joy of Being!
Realizing that the there is no chance of enlarging the Point’s perspective, A Square leaves it to itself—perhaps a lesson in knowing when a battle is not worth fighting.
Abbott was an educational and societal reformer. He wrote many over 40 books on education and other topics; this classic work endures because it works on many levels and because its implications continue to be worth pondering.