Flatland

Mostly moved into my new place, and with the day off from work to watch the cable guy drill holes in our walls for hours on end, I decided to try out this 1884 novella. I'd heard great things about it: notably, that it offered as good a conceptual image of multi-dimensional space as anything that has ever been written. Oddly enough, the author, Edwin A. Abbott, was actually not even a mathematician or scientist, but a literature enthusiast who enjoyed the Greek classics. The titular world of "Flatland" exists on an infinite two-dimensional plane. All people, things, and objects in Flatland are two-dimensional shapes that can "see" the world around them only as a series of lines. Since the entire world is coplanar, no being's field of vision can consist of more than one dimension, much as our own vision presents us with two-dimensional images; just as we (in three-dimensional "Spaceland") can infer the third dimension using depth perception, the shapes of Flatland can infer "depth" as well, allowing them to directly view one dimension while perceiving another. The protagonist and narrator of the novella is a simple middle-class square. One day the square visits "Lineland," a land in which all beings and objects are nothing more than colinear line segments. Each line segment can only "see" the one in front of it and the one behind it, and each is thus perceived as being nothing more than a point. The square finds it difficult to explain a second dimension to the linear residents of Lineland who have only ever known - and can only infer the existence of - one dimension. Even as the square passes in and out of the solitary line, thus projecting himself as a line that grows and shrinks depending on the square's angle, the lines in Lineland see it as nothing more than a line segment growing and shrinking. Frustrated, the square departs. Later, the square is visited by a sphere from "Spaceland," a three-dimensional world much like our own (except apparently with talking floating spheres). The sphere projects onto Flatland as nothing more than a circle, and the square has just as much difficulty understanding and believing in the concept of three dimensions as the Lineland beings did with two dimensions. But at last the sphere is able to somehow take the square with him into three dimensional space, and the square can see his home and world from a bird's eye view for the first time, giving him more clarity than he ever had pictured his own home with. As the story concludes, the square asks the sphere to take him one level higher - to a land of four dimensions, or even five, six, or seven. The sphere, of course is just as incredulous about the idea of higher dimensions as were the lines and the squares. But by now, we the readers are, like the square, wondering why there shouldn't be four dimensions. Two points become a line. Four lines become a square. Six squares become a cube. Why shouldn't eight cubes become some basic four-dimensional shape? And if such a four-dimensional shape existed, wouldn't it project onto our own world as nothing more than a simple cube, albeit one that seems to grow and shrink at will? I have to give Abbott huge props for illustrating very effectively the folly of believing in the finiteness of dimensions as well as the difficulty (and impossibility, even) of being able to infer or detect any and all dimensions beyond the three our world consists of. There was even an amusing and brief trip to "Pointland," a zero-dimensional "universe" consisting of a single point who is the ruler, sole resident, and entirety of the universe itself. There was also a lot of strange social satire in Flatland; the more sides a polygon had in Flatland, the higher his social rank was. Also, women were nothing more than straight lines. (In Lineland, women were mere points. Weird, but it was 1884, so whatever.) But the big takeaway for me was definitely the exploration of extra-dimensional concepts and perceptions. At under 100 pages, it made for a simple afternoon read and I'd definitely recommend it to anyone who has ever wondered about how there could possibly be a fourth spatial dimension.