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 is spouting gibberish.
So: a very clever concept, very cleverly done, and I, personally liked it so much I’ve read it twice.
The central concept (As above, so below, dimensionally speaking) is a pretty easy one. I’ve set it down in less than two pages here, and while there’s a lot of finer points and implications, really you can cover the whole thing in fairly exhaustive detail in twenty pages or so, tops. Any more detail than that would bring math into it, and no one wants that. Well, a twenty-page novel isn’t a novel, now is it? In fact, it’s not even really a short book, more of a tract.
Abbot pads out his story with an *enormous* amount of social satire. The entire first half of the book is an elaborate description of how Flatlander civilization works. Basically they’re very class-conscious victorianesque folk, who’s status is dictated by the number and regularity of their angles, and who’s intelligence is dictated by the degree of their largest angle. Women are simple lines, and having no angles at all, are regarded as stupid chattel. The lower classes are all pointy isosceles triangles, the priestly class are circles, the nobility are manysided polygons. We get a lot of detail as to how they interact, how their schools work, how they feel about each other, and so forth. I’m told by people who claim to know such things that all this is a hilarious send up of the mores of the day. Well, maybe so, but I get the feeling these sections were every bit as flaccid then as they are now. There is, however, one really funny bit about how schools are very instrumental in keeping down the numbers of the lowest classes by chaining them up in classes for the students to gawk at, and thereby learn to recognize the poor. In some of the more liberal schools, they feed the poor subject, but in most of them they just leave the guy chained up until he starves, then chuck the corpse and grab another. After all, there’s always more poor, right?
All of this goes on way too long and in way too much detail, but it is needed to get across the idea of how Lineland works as a society, and not just a mathematical abstraction. For the allegory to be effective, we have to accept Square as a person (Stodgy prig though he may be). It also drives home the idea of what the place looks like from ‘inside’ - remember, the Flatlanders can only see a line. Their entire universe is just an infinite plane, so basically they can only see lines. A triangle is a line seen edge on, as is a circle, as is a square. They identify each other by touch, or by shading, since light and fog pervade the place (Don’t ask how).
This is tedious and time consuming, and it’s not helped any by Abbot’s hyper-florid style of writing. Having read a lot of Victorian SF, I feel reasonably certain that his peers would have read this and said, “Eddie, dude, you’re cramming sixteen prepositions into one line, what’s with all this 'ye olde timey' crap?” (Said with a British accent, of course)
The second half of the book works much better, as we get to the actual meat of the story, and such padding as there is really is quite compelling: We deal with society and government’s utter refusal to accept any new teachings that question the nature of reality, as their morals and position depend on a perpetual status quo, we get fear and loathing, and imprisonment as befitting all prophets, and we get a surprising bit where Mr. Square suddenly has a glimpse of inspiration that’s been denied to the Spheres themselves, which makes him - momentarily - their unlikely intellectual superior. This pisses them off, his heavenly ascent is ended, and he spends the rest of his days, like Paul in prison, waiting for the axe.
Which brings us to the ‘spiritual’ and ‘popular’ aspects of this book I mentioned above:
This book was championed by the “Spiritualist” Movement around the turn of the century. Spiritualists were folks who insisted séances were real, the spirits were real, and that all this stuff makde some kind of preternatural scientific sense. In fact, most of them didn’t really care about the scientific stuff so much - ninety-nine percent of what they were on about was pernicious nonsense - but science gave their hoo-hah the touch of validity they craved. The fourth dimension would seem to be ‘the dimension of spirit,’ or at least it would appear as such to us, and so they embraced it. There’s a zillion spiritualist tomes from the period (1890-1930 or so) that make the connection between hypermath and spookie-ookies. A number of them work in Atlantis and Madam Blavatsky to a greater or lesser extent as well. Abbott got there first, however, and he did it better, and without overtly tying it to the occult, even if he implies such a thing might be possible.
His critique of organized religion in the end - that it’s sent running higgledy-piggeldy by any new revelation - is certainly true. We see it in the bible itself: New prophets being opposed by the followers of old prophecy; we see it in politics and art as well. Anything that casts new light on old things is always feared, and Abbott correctly sums that up without getting bogged down in particulars. (My own personal shibboleth for this has always been: If St. Paul was sitting in your church on Sunday, would your preacher be happy about it, or would he try to cover it up for fear that Paul would damage his power and reputation?)
So: an obscure little mathematical allegory became required reading in a bunch of weirdo religious cults, became ubiquitous, and then had its popularity decline as the cults it’d been tied to withered and died. The book itself isn’t cultic at all, however, despite the guilt by association. Even so, it’s an extremely important benchmark in both Science Fiction and actual Math itself.
WILL CONSERVATIVES LIKE THIS BOOK?
I honestly don’t see why not. There’s no sex, no drugs, nothing bad, just a stodgy Victorian parallelogram debating the nature of the universe, and casting aspersions on the very government we fought a war to get away from. What’s not to like? It’s a challenging read, but ultimately a rewarding one, well worth the time if you’ve got the fortitude to wade through excess verbiage.