Once upon a dimension

A sequel to the classic "Flatland" brings to life the mind-bending world of cutting-edge mathematics and alternate universes.

Published June 18, 2001 10:13PM (EDT)

It was dark in the pantry. I'd padded down the stairs to have a bowl of cereal before bed, navigating through the room by the dim glow of the kitchen clock. I'd found a bowl and a spoon, and had set the cereal box on the counter. I never made it to the fridge.

"I can see right through your pants!" the voice boomed.

It came as though from everywhere at once -- from the very air itself. The spoon clattered to the floor. I looked down at my pants. They were fastened. It was dark. Seconds passed. There seemed nothing to say.

"Do you know who I am?" the voice boomed.

A point of purplish light winked into existence in the center of the room, and grew into a small, rotating cube. Vertiginous, flashbulb purple -- retinal purple, spinning like mad. I swallowed dryly. I was suddenly glad I hadn't eaten anything. The cube grew to the size of a hatbox, and then to the size of a stack of LPs.

"No," I said. "I don't have the vaguest ... What do you mean, you can see through my pants?"

"I can see through everyone's pants," the cube said. "I can see inside every locked door, every box and safe ... I can see the fast-food wrappers stuffed under your car seat, and read the expiration date on the milk in the fridge. Your milk," the voice settled into an imperious cadence, "is nearly expired."

There once again seemed nothing to say.

"It's all easy, you see." The cube spun madly and grew to the size of an overstuffed ottoman, or a crate of pineapples, or something like that. "When you exist in four spatial dimensions, as I do. To me, your whole universe looks like someone left the lid off. You have 38 cents in your pocket, and an ATM receipt that looks like it's been through the wash."

I looked. It was so.

"And when you're four-dimensional," the cube said, "you can do stuff like this!"

The cube winked out of sight, and I felt a mild tugging sensation amidships. A pair of jockey shorts appeared, twirling in midair. A vertiginous purple light strobed through the leg and waist holes.

"Hey! Those are mine! Take those off!"

The shorts vanished and the cube winked back, about the size of an iMac, spinning zanily. A ball of baked beans glopped to the floor beneath it. "You'll never guess where your shorts are now!" the voice boomed.

"OK," I sighed, opening the drawer under the counter and reaching for the can opener. "How did you do that?" I lifted the cans in the cupboard, one by one, until I found one that was much lighter than the rest.

"Although I might look like a cube to you," the voice intoned, "that's because you can only perceive things in three dimensions. What you're really seeing is the infinitesimally thin slice of my true form -- my 4-D form -- which intersects with your three-dimensional world.

"Imagine Flatland," the cube said. "Do you know what Flatland is?"

I said, "It's an imaginary place with only two dimensions: a standard trope of mathematical brainteasers, which first appeared in a novel called 'Flatland: A Romance of Many Dimensions,' by a man named E.A. Abbott, in 1884. It was a novel about geometry, and the main character was a two-dimensional square who was visited, in his two-dimensional house in his two-dimensional world, by a three-dimensional being -- a sphere who ... Oh, now I see what you're up to."

"Yes, quite," said the cube. "But let's back up a moment. Abbott's 'Flatland' is often called a novel these days, but it's really an example of an extinct form -- a kind of light, slim, often self-illustrated book that Victorian authors would turn out between more serious projects, as bagatelles; or that gentleman dilettantes would sometimes write instead of the standard travel narrative or rote historical study. The form granted license for bad verse, broad allegory and things like that -- these were something like children's books for adults.

"'Flatland,'" it continued, "was as much a work of broad Victorian social satire as it was a geometrical fantasy. The Flatland of the title, for instance, with its literal depthlessness, its snobbery and rigid social stratification, replicates the British class and gender hierarchy of the day in perfect, Swiftian form."

I was, I remembered vaguely, listening to a glowing purple cube that had teleported my underwear into a bean can.

The cube continued, "But it's the book's geometrical conceit that's kept it in print, continuously, for the past 117 years. If you read 'Flatland' straight, it's a whimsical romp of a mathematical book -- more Lewis Carroll than Jonathan Swift. It explains advanced geometrical concepts in a way that makes total, intuitive sense, so as you read the book, you don't even realize how it's blowing up your brain and twisting it into balloon animals. Mathematicians love allegory anyway. Lewis Carroll is as much a literary icon to them as Ernst Haeckel and M.C. Escher are their anointed poster-artists.

"Anyway, as the years passed, 'Flatland' became a classic, and began a tradition of allegorical, often whimsical popular-scientific literature -- not infrequently with a progressive socioreligious message. Direct sequels to 'Flatland' started appearing in 1965, with Dionys Burger's 'Sphereland.' 'Planiverse,' by A.K. Dewdney, appeared in 1984. But some of the stories in the 1954 Abbott-inspired story collection 'Fantasia Mathematica' (and its sequel, 'The Mathematical Magpie') date from as far back as the 1920s.

"For a time, in the '70s and '80s, it was quantum physics, not math, that generated the biggest popular-science titles, with John Gribbin's 'In Search of Schroedinger's Cat' and Gary Zukav's 'The Dancing Wu Li Masters' and all those books like that. But in the past few years there's been an outpouring of books in the 'Flatland' tradition, including Clifford Pickover's 'Surfing Through Hyperspace: Understanding Higher Universes in Six Easy Lessons'; 'Hyperspace: A Scientific Odyssey Through Parallel Universes, Time Warps and the Tenth Dimension' by Michio Kaku; and the Abbott-inspired children's book 'MathTrek: Adventures in the Math Zone' by Ivars Peterson and Nancy Henderson. And recently, we've seen the most ambitious Flatland sequel yet, in Ian Stewart's 'Flatterland.'"

"Ian Stewart?" I said. "Wasn't he killed in a -- ?"

"No, that's the other, evil Ian Stewart -- vocalist of the British Nazi skinhead band Skrewdriver. This one's a math professor at the University of Warwick. And just like in the original Flatland, the story in his book begins with a three-dimensional spheroid entity visiting a 2-D world and spiriting off one of its inhabitants into realms theretofore unimaginable. Only instead of a square -- the Abbott character was named 'A. Square' -- the main character is a line named Victoria. That's a London subway pun, by the way. And instead of a sphere, the visitor is a being that looks like an inflatable hoppity-hop toy that British kids bounce around on -- a 'Spacehopper.' Like that Julian Cope song of the same name."

"Didn't he just die from -- "

"No, that was Julian Cole, the applied mathematician. Terrible loss for the field. It's like this: In Flatland, there are only two dimensions. So if a sphere, or a round-bottomed Spacehopper, were to visit you there, you'd see it not as a sphere but as a circle -- as the infinitesimally thin slice of a sphere that touched the 2-D plane you inhabited. Plus, you'd see it edgewise: Since there'd be no up or down for you -- at least none to speak of -- you'd see it as a line that looked circular as you moved around it. That's what it would look like in two-space. And if that round 3-D shape were to move down through the 2-D plane, what you'd see would be the circle getting bigger." The cube doubled in size. "And then smaller." It shrank to a glowing point, and then slowly swelled back to form. "And then vanishing altogether."

"But now," the cube continued, "imagine you're the Spacehopper. You'd look down on Flatland, and nothing would have a roof or a top on it, or anything. It would all be made up of lines, like a huge architectural drawing. You could hang, invisibly, inches above the surface, and reach right into everything. And you could do things like ..."

The cube vanished. I braced myself.


I felt my damp, beany underwear reappear under my trousers.

"I guess that all makes sense intellectually," I muttered.

"Oh, it's a great allegory," it replied. "Especially if you have no experience with this stuff. It's much easier to grasp at first than the Mulder-and-Scully detective story in 'Surfing Through Hyperspace,' Pickover's book, where higher-dimensional beings swoop down and kidnap the president. Abbott's book used his allegory to suggest how four-dimensional space would seem to a person like yourself, who lives in a 3-D world. Stewart goes much further, using it as a jump-off point to describe various types of curved and funny-shaped space, quantum and relativistic phenomena, black holes and superstring theory. Victoria and the Spacehopper use a device called a virtual unreality engine -- shades of Douglas Adams, a terrible loss for his own field -- to insert themselves into alternate universes, where particles sing, and a group called the Space Girls dances, and Stephen Hawking is the overlord of a black-hole construction empire.

"You know, according to superstring theory," the cube said meaningfully, "there's a general, albeit fitful, consensus that there are actually nine spatial dimensions in this apparently 3-D universe of yours. Most of them are just really small -- almost flat -- so you can't see them."

I was puzzled. "Well, OK. That would definitely explain why subatomic particles seem to behave so weirdly. But what would a 10-dimensional universe look like?"

"Well, one of the strengths of Stewart's book, and of the whole allegorical approach, is that it's much easier to fathom it the other way around -- starting from your own 3-D perspective, and imagining how you'd interact with oddball lower-D geometries. But I'll show you."

The cube vanished. I braced myself.

The president of the United States winked into existence on my pantry floor. He was ashen, his limbs flailing, his face a mask of horror. He vanished again in midscream. His voice seemed to Doppler into the distance.

"Don't worry." The cube had returned. "We'll put him back again. Just decided to get a bit Lovecraft on his ass after all that business with the Kyoto accord, and the tax plan and everything. He seemed to lack ..." The cube might've been chuckling. "Perspective."

"I voted for Nader."

The cube continued as though it hadn't heard. "Yes, so superstring theory is pretty counterintuitive and hard to visualize -- as 'Hyperspace,' Kaku's book, says in no uncertain terms. Stewart might've bitten off more than he could chew in that regard. After some great chapters on alternate-shaped universes and curved space-time and such, he covers the contemporary stuff pretty broadly -- superstring theory, supersymmetry. It's as though he wanted to make 'Flatterland' a comprehensive book about all the strange implications of modern mathematics and physics -- while at the same time needed to step up his writing pace as he went on. Like Abbott, Stewart is the author of about 60 books, and sometimes 'Flatterland' reads as though he were already thinking about the next one: It gradually starts to seem a bit willy-nilly and half-cooked."

"I mean, if I didn't live in a state that Gore was supposed to win, I probably wouldn't have."

"And while Stewart understands that the original 'Flatland' was a work of satire -- an engaged work -- as well as a geometry book, he doesn't continue that project. He reimagines Abbott's Draconian 2-D world as a place where '60s-style lifestyle reforms are solving all of society's problems. It's a bit, you know, flat in view of the visionary current that's enlivened pop-science books throughout the century: Carl Sagan's wide-tie futurism, the neo-Taoist quantum physics of the '70s and '80s, the whole cyber-prophet thing of the last decade. Pickover, in his book, wonders whether he can find mankind's gods in four-dimensional geometry -- whether a "god" might just be a four-dimensional entity, able to act outside of the physical laws that people take for granted. It's wacky, maybe, but it gets you thinking about something more than math and physics. If there's one thing 'Flatland' did for popular-science writing, it's that it granted authors a license to be playful and visionary -- both at once."

I suddenly remembered the can opener I was holding. "So OK," I said, "how did you do that thing before, with my shorts?"

I felt a pull, from everywhere and nowhere at once. I saw my pantry as a maze of disconnected forms: cabinets askew, with gaping holes in every joint; the floor tilted, yet straight, with the basement showing beyond. The basement didn't look too good either. I shifted position and everything rearranged slightly, moving in and out of everything else. I could see my arm in front of me. It came from the wrong direction.

"Gaah!" I cried. And then everything was back to normal again. Except the pantry was facing the wrong way. The cereal box had all its letters backward, and the numbers on the kitchen clock were reversed.

"Sorry, hold on," the voice boomed.

The world righted itself.

"I flipped you by mistake," the cube said, still spinning zanily. "Listen, I've got to go; the president has an appointment coming up with that new drug czar of his. Take these."

A burlap sack fell from midair and thumped heavily to the floor. I looked inside. It was full of ballpoint pens, guitar picks, coins and socks.

"And try to keep better track of your stuff from now on, eh?" the voice warned. The cube vanished.

By Gavin McNett

Gavin McNett is a frequent contributor to Salon.

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