'Flatland: The Movie Edition' by Edwin A. Abbott, with Thomas Banchoff and the filmmakers of 'Flatland'

The Movie Edition

Edwin A. Abbott, with Thomas Banchoff and the filmmakers of "Flatland"

Princeton University Press: 168 pp., $15

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.