January 26, 2009

In 1884, a humble English schoolteacher named Edwin Abbott Abbott published a modest little novella called Flatland: A Romance of Many Dimensions and inadvertently brought the concept of extra geometric dimensions out of the Ivory Tower and into the mainstream. Isaac Asimov once described the book as "The best introduction one can find into the manner of perceiving dimensions," and many a physicist and mathematician will tell you the tome is among their favorites. I first read it just out of college, and was immediately charmed.

Part science fiction, part satire (and a pointed commentary on Victorian social hierarchy and the Establishment's animosity towards revolutionary new ideas), Flatland takes place in a two-dimensional world inhabited by 2D circles, squares, rectangles and a variety of polygons. Our narrator is a nameless Square who dreams about a one-dimensional world (Lineland) where nobody believes that anything lies beyond their simple linear existence -- certainly not an entire world in two dimensions.

But then the little Square meets a three-dimensional Sphere, who tells him about Spaceland, existing just beyond the ken of Flatland's inhabitants. Seeing is believing, so the Sphere takes the little Square on a tour of Spaceland, literally broadening his horizons. Once back home, the Square tries to tell others about this brave new world, and is denounced and eventually imprisoned for his trouble. (There's also a dream sequence involving Pointland, inhabited by a single point who thinks the Square's attempts to talk to him are just his own thoughts -- solipsism personified.)

There have been several attempts to adapt the book to film, most recently via the animated short, Flatland: The Movie, featuring the voices of Martin Sheen, Kristen Bell, Michael York, and Tony Hale:

Flatland is not a Utopian society (it's awfully repressive intellectually, for starters), and all men (and women) are not created equal. Men are polygons, and the number of sides they have determines their social class -- triangles are the lowest of the low, while a circle is considered a perfect shape. Women are relegated to being comprised solely of lines. And since a line moving towards an observer invariably appears to be merely a point, women are required by law to sway back and forth so that the men can see them coming. Apparently there were some "accidents" where men in Flatland were stabbed to death by oncoming women. (Right. An "accident." Ahem. I'm just saying that maybe one of the bastards had it coming.)

Anyway, extra dimensions became all the rage, well into the 20th century, where the notion of a fourth (or more) dimension influenced major artists like Pablo Picasso and Salvador Dali. In fact, Dali's famous painting, "Crucifixion," depicts Christ nailed onto a four-dimensional hypercube as a cross; Dali subtitled the work "Corpus Hydrocubus."

Post-Einstein and his concept of a unified spacetime, of course, time is considered the fourth dimension, so when scientists in the early 20th century began contemplating extra dimensions, they spoke of the "fifth dimension" -- and beyond. But just as the 2D shapes in Flatland couldn't see Spaceland, we can't perceive the fifth dimension. Mathematicians and physicists wanted a solution to the conundrum, and a Swedish mathematician named Oskar Klein obliged in the 1920s by arguing that the fifth dimension could be so tiny -- curled up, or "compactified," into a tiny ball smaller than the Planck length -- that noy even atoms could pass into it.

Such so-called "Kaluza-Klein models" languished for a bit after that, until the 1970s, when string theorists adapted this extra-dimensional approach to unify all four fundamental forces (gravity, electromagnetism, strong and weak nuclear force) into a giant Theory of Everything. To get all those pieces to fit together in string theory requires a whopping nine dimensions of space and one dimension of time (ten spatial dimensions if we're talking about "M" theory). The extra dimensions are supposed to be "compactified," which is why we can't experience them directly.

So how do we know they're there? Well, we don't, any more than Abbott's plucky little Square knew Spaceland existed until he visited it. We have some very pretty math buttressing the argument, and while it's a long shot, it's possible that experiments at the Large Hadron Collider could provide evidence of extra dimensions. Seeing is still believing.

String theorists, fortunately, do not inhabit a repressive, willfully ignorant world like Flatland. They might take some heat from their more skeptical colleagues now and then, and in the last couple of years there's been a bit of a backlash. But their story has a happier ending than Abbot's little Square. They are free to openly explore radical new ideas, inspiring art, literature, theater and such in turn. Whether string theory turns out to be right, or is eventually replaced by a better model, the world is a richer place for it.

Photo: "Crucifixion (Corpus Hydrocubus)," Salvador Dali, 1954.