IN 1884, the English mathematician Edwin Abbott Abbott published an enchanting fable set in a two-dimensional world he called Flatland. Within this planar universe live triangles, squares, hexagons and other polygonal beings, who go about their business within a mere two degrees of freedom, working, playing and carrying on the processes of government without the luxury of depth. The hero and narrator, one A. Square, is a modest fellow, rather low down in the social hierarchy of Flatland but intellectually curious and a bit of a mathematician at heart. He likes to think about numbers and shapes, and at times he wonders whether there isn't, somehow, more to reality than meets the eye.
In A. Square's dexterously naive voice, addressed to "the Reader," we learn about the physics, physiology, educational system, history, governance and social hierarchy that pertain in his two-dimensional, Euclidean domain. Here, a rigid pecking order reigns: The more sides a citizen has, the higher is his class. Thus Triangles are the lowest class, with Isosceles even lower than Equilaterals; next come Squares, who serve as clerks, scribes and other literate functionaries; then Pentagons and Hexagons, who make up the professionals (physicians, lawyers); and so on up to the "infinitely-sided" Circles, the priestly and noble classes.
All this is delivered with the earnestness of a convert; toward the end of the tale, we learn that A. Square has been inducted into the mysteries of the Third Dimension by a magnificent stranger in the form of a Sphere. Under Lord Sphere's guidance, he has been vouchsafed a glimpse of the vast, expanded cosmos of three-dimensional space, herein known as Spaceland, in which reside the transcendently excellent figures of Cubes, higher-dimensional versions of his own lowly form.
Like so many other heroes who have seen the light of a higher order, from Jesus to Galileo, A. Square will suffer greatly for the illumination he offers his fellow citizens. In Flatland, any discussion of a third dimension is heresy, punishable by imprisonment or death. Indeed, A. Square narrates from prison, where he has been confined for the past seven years, having failed to stifle his enthusiasm over what he witnessed during his brief time in Spaceland.
To many students of mathematics, "Flatland" stands alongside "Alice's Adventures in Wonderland" as one of the most beloved stories of the modern age. Like Charles Dodgson (Lewis Carroll), Abbott was able to transform complex mathematical ideas -- in this case, an emerging understanding of multidimensional space -- into a fantastical story at once whimsical and serious, which, by confounding genre, remains as fresh and appealing as when it was first published.
Just as the Alice tales have done, "Flatland" has inspired many imitators and renditions in other media. Among its "sequels" are "Sphereland" (1965) by Dionys Burger; "The Planiverse" (1984) by A.K. Dewdney, mathematical games columnist for Scientific American; "Flatterland" (2001) by English mathematician Ian Stewart; and "Spaceland" (2002) by U.S. mathematician Rudy Rucker. In an episode of "Cosmos," Carl Sagan used "Flatland" to explain higher-dimensional spaces, and in a 1960s episode of "The Outer Limits," a character named Eck visited humans from his own two-dimensional world. In 1965, Dudley Moore narrated the first animated film version, which was followed by a second film in 1982, directed by mathematician Michele Emmer. There has even been an opera -- "VAS: An Opera in Flatland" (2002) by Steve Tomasula. My favorite theatrical rendition is a delightful puppet opera by Randall Wong, performed here last year at the Museum of Jurassic Technology.
Last year also saw the production of not one but two new film versions, the first a full-length animated feature directed by Ladd Ehlinger Jr., updating the story from Victorian England to contemporary America, and the second a 30-minute animation starring the voices of Martin Sheen, Kristen Bell and Michael York, which has just been released on DVD. Accompanying the latter is "Flatland: The Movie Edition," published by Princeton University Press, in which we get Abbott's novella plus the script of the movie, along with an introduction by Brown University mathematics professor Thomas Banchoff and commentary by the film's writer and producer (Seth Caplan), director (Jeffrey Travis) and chief animator (Dano Johnson).
All movies of beloved stories must struggle against the preconceptions of their fans, and I will be honest in confessing that as soon as I laid eyes on the lush color graphics in the Princeton book my heart began to sink. I first read "Flatland" when I was studying mathematics at Sydney University 25 years ago and fell in love with its subtle blend of fantasy, pedagogy and satire. Reading it again a quarter-century later, I was particularly struck by Abbott's incisive skewering of class-bound Victorian society, and particularly by his parodic rendition of Victorian attitudes to women. In "Flatland," women are the lowest class of all, being merely straight lines with no area at all, and hence literally no space for brains. When Flatland was first published, some readers misunderstood Abbott's point and accused him of misogyny, whereas in fact he was a brilliant teacher who supported the cause of women's education.
Beneath its fairy-tale trappings, "Flatland" was a subversive piece of social commentary, and it wasn't written for little children. The makers of the new film have stripped this multifaceted story of much of its depth and reduced it to a rather saccharine lesson. The basic educational message -- about one-, two-, three- and possibly higher-dimensional spaces -- is intact, but the social satire is gone. Instead, we get a simplified story in which there is little ambiguity about who is "good" and who is "bad." A. Square has been given a mathematically inclined granddaughter named Hex, who is a hexagon, and it is she who first plants in his mind the idea of a third dimension. (This has a parallel in the original, where A. Square has a smart hexagonal grandson.) Although it's nice to see a female character taking a mathematical lead, the very power of satire is in revealing what it is apparently obscuring. Hex's intellectual perkiness comes off as a politically correct gesture, as do several other simplifications in the film, which seems aimed at a primary-school audience.
The filmmakers have admirably attempted to inform the look of "Flatland: The Movie" with mathematical motifs, such as fractals, but I wish they'd paid less attention to visual effects and more to the texture of the story. Sadly, this is a very flattened version of Flatland. If you want to experience the full depth of this miniature masterpiece, pick up "The Annotated Flatland," with notes by Stewart. Its multidimensional treasures will leave you as starry-eyed as A. Square himself.
On the Web
For more photographs from Flatland, go to latimes.com/flatland.