It’s hard to overestimate the influence that “Flatland” has had over people in the last century-and-a-quarter, harder still when one considers its renown is entirely a sort of coincidental fluke. It’s remarkable that it was noticed at all, and nothing short of miraculous that it hit a point of ubiquity around 1900. Since then, it’s receded into the background like other trends of the day, much like hoop skirts and saddle sales. It’s of interest mostly only to crackpots, advanced geeks (Like me), antiquarians, mathematicians (Though not even very many of them), and Spiritualists, though more on this last group in a bit.
(Just as an aside, I first learned of this book through various comments by Rudy Rucker.)
Written by Edwin Abbot in 1884, this very slim book - maybe 2/3rds the length of a Hardy Boys novel - is basically an attempt to explain by allegory how the Fourth Dimension works and appears. The actual device for this is very clever: Rather than having a person from our world be greeted by a hyperdimentional entity bringing tales of glory and wonder, Abbot bumps the whole thing down a notch: Our protagonist is mister “A. Square,” who is, in fact, a square. (There’s a solid theory that his first name is “Albert,” which would make his name a joke, but they never state that in the book. I do think it’s true, though.) He lives in a two-dimensional world called “Flatland,” which has width and length but no height.
One day, Mr. Square is visited by A. Sphere, a three dimensional being from “Spaceland.” Sphere attempts to indoctrinate Square in the mysteries of 3-D space, but Square keeps taking Sphere to be a magician or possibly “A very clever juggler,” and won’t have any of it. After several failed attempts, Sphere simply grabs Square and drags him out of the plane of Flatland entirely, allowing him to see the whole thing from above, as we look at drawings on a page. Changed by his experiences, his esoteric knowledge, Mr. Square goes back to Flatland and attempts to “Preach the gospel of three dimensions,” but the authorities lock him in solitary prison for the rest of his life.
While this is all going on, we make brief side trips to Pointland and Lineland, zero and one-dimensional worlds, respectively.
There is admittedly not a lot of story there. The basic concept behind it is very, very clever however.
In the latter part of the 19th century, the idea that there *were* higher dimensions was very new, and very mysterious, and very imperfectly understood. Abbot, a schoolteacher and theologian, was basically attempting to explain this concept in a way that could be instantaneously understood by your average Victorian in a clear, logical fashion, without resorting to mumbo jumbo or complex mathematical equations (Like this one: V=½ Pi^2R^4 for finding the volume of a hyper sphere of radius “R”). The concept he used was basically extrapolative: The fourth dimension would seem to us as the third dimension seems to a two-dimensional being, and as a two-dimensional being would seem to a one-dimensional one, and as a one-dimensional being would seem to a no-dimensional one.
While we lack the ability to access or perceive the fourth dimension, we can at least extrapolate from this progression what aspects of it might be like. Making our characters simple geometric forms rather than, you know, actual ‘people’ cleans matters up considerably, too. Trying to describe a complex shape like a human passing through a plane is vastly more difficult - and quite a bit more distracting - than when a simple ball passes through. Abbot strips this all down to platonic ideals (More or less) to keep the concepts simple and easy to understand, and once you’ve got those basics, everything else is just a detail.
There’s a lot of illiterate and innumerate talk of the fourth dimension in Science Fiction. It’s become a magic happy land where anything can happen, and everything makes sense. In essence, it’s a great big literary wizard that lets anything the story needs take place. This trend is ignorant and kind of a waste, too, as the higher dimensions are really neat, and they *do* play by a concrete set of rules, just as our own lower dimensions do. It’s fun to learn this stuff because, well, it’s fun to learn anything, really, but beyond that it’s kind of neat to be able to spot when some hack writer or three-degrees-below-Dumbass TV producer