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

- - - - - - - - - - - -
By Gavin McNett

June 18, 2001 | 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.



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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.

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