A Guide to Flatland: What It’s Like to Live in Two Dimensions

Valero Doval

Mathematician Ian Stewart wants us to see what he sees. Which is kind of a problem, because he’s accustomed to envisioning some pretty impossible shapes: snowflakes in fractional dimensions, hypercubes in 4-D, 11-dimensional superstrings. So when the University of Warwick professor and author writes about these freaky geometries, he relies on an analogy—a way for us blockheads to understand realities above, below, and in between our three dimensions. His guide is a little mathematical fantasy he read over half a century ago: the cult classic Flatland .

Written in the early 1880s by Edwin A. Abbott, the story follows A. Square, a regular quadrilateral who lives on a 2-D plane. He can’t conceive of depth, but his perspective expands when a sphere visits him from 3-D Spaceland. Though Square can’t experience all three of the sphere’s dimensions, he can see it in cross-section as a circle of various sizes. That’s the trick Stewart uses: By describing what it’s like for a flat object to imagine a solid one, he can help us imagine an object in four dimensions. For instance, a crazy 4-D sphere (called a glome) might appear to us as expanding and contracting spherical cross-sections. “Starting with Flatland‘s point of view,” he says, “you find a way in.”

Stewart considers Flatland one of the earliest works of popular science, a genre he’s been writing in for decades (his latest book came out in October). In addition to an annotated Flatland and a trippy follow-up (Flatterland , natch), he has collaborated on four educational books about the science of Terry Pratchett’s Discworld, in which wizards ponder the strange technology of Earth from a separate (magical) dimension.

In Flatland and Discworld, the act of removing yourself from your own reality allows you to understand it. And Stewart wants his readers to push that concept as far as they can. “How does the fourth dimension relate to the 11th?” Stewart asks. “We can crank up the analogy.” Whatever you say, Professor—we’ll keep an eye out for Stringland.