A Dimension Not Only of Sight and Sound, but of Mind

After finishing Catton, I sneaked Edwin A. Abbott's Flatland into my schedule last week. I heard about this book from somebody when I was a kid; likely it was my uncle Russell, who used to give me mathematical brain teasers. I knew that the story involved a creature from a two-dimensional word who watched a sphere pass through his universe, and that the story eventually went to our world and challenged the book's readers to imagine a world of four dimensions. But Abbott surprised me with some other worlds of the imagination as well.

The most surprising world had nothing to do with geometry and everything to do with politics. In Flatland, all men are polygons, and the size of their interior angles indicates their mental capacity. The most menial laborers and guards are isosceles triangles with apexes of only a couple of degrees. The most advanced men have so many sides, they are practically circles; each angle of such a creature is almost 180 degrees. But at least the men have brains. Women are all just straight lines -- no interior space at all, no interior angles, no brains. The men tell the women that they adore them and talk to them of love, duty, right, wrong, pity, and hope. But behind their backs, the men say what they really think: women are mindless, and these women's topics are "irrational" concepts with no existence. They keep up the fiction with the women, explains the square first-person narrator, only "to control feminine exuberances." Women and small-brained men are disposable; the society kills them without hesitation if economics or state security suggests the deed. I know some real-world people tend to think this way, but to envision an entire culture living out this creed requires almost as much imagination as picturing the world of four dimensions. Doesn't the circle who kills a useless line who believes in duty throw the game by having done his duty?

After a full half of the book covering these social conditions of Flatland, I found the geometrically challenging images I expected in the second half. One day in moving around Flatland, the square finds a community of one-dimensional creatures living out their lives in a single line. The square talks to the one-dimensional king and tries to explain the second dimension to him. "Imagine space stretching out to the right and left," he says, but the king has no way to imagine right and left. The square tries moving in and out of the line to demonstrate his extra dimension, but the line king (I realized how that sounds only after I wrote it -- this is one of the few times you'll find the words "no pun intended" spoken truthfully) only sees a creature appear and then disappear. When the sphere visits Flatland and tries to tell the square to imagine space going up and down as well as north to south and east to west, he finds his task just as hopeless. Then when the sphere passes through Flatland, the square just sees a circle appear, get bigger, get smaller, and then disappear. So the sphere takes the square to Spaceland (three-dimensional space), where he suddenly gets the big picture. Interestingly, it's the square who tries to explore the idea of four-dimensional space, and his speculations make no sense to the sphere.

But the square's real audience is the reader, and his arguments and analogies worked on me to an extent. The first comment that conjured the vision in my mind had to do with what we consider "inside." In Lineland, people see each other as points; every person considers the line between his endpoints as his inside. But the square sees the lines in their entirety. The square himself has four linear, visible sides; since he can't see his interior area, he considers that his inside. But the sphere sees the square from the vantage point of three-dimensional space and sees the whole square at once. A cube, he points out, has six, two-dimensional, visible sides, and a three-dimensional interior volume hidden from view. But, says the square, if I see the inside of the king of Lineland, and you see my inside, wouldn't the cube's inside be visible to a creature from a four-dimensional world?

Notice that people always see one dimension fewer than the number they exist in. All creatures must infer the nth dimension by other means. The lines of Lineland see each other as no-dimensional points. They infer length and distance by sound. The polygons of Flatland see everyone as luminous one-dimensional lines (put your eye even with your table top and see how a piece of paper looks). They infer distance and shape by the fact that closer objects appear brighter. We in Spaceland see two-dimensional pictures; otherwise we would not be able to mimic our visions on paper, canvas, or computer screen. We infer the third dimension from shading, focus, and the diminishing size of receding objects. So people from the fourth dimension would be able to see three dimensions at once. Again, every part of the cube, "inside" and out, would present itself to them at a glance.

The sphere and the cube discuss the progression of figures from one dimension to the next by means of imagining one shape creating another. A no-dimensional point, for instance, moving in a consistent direction would trace a straight, one-dimensional line. In turn, this line moving perpendicular to itself for a distance as long as itself would trace the area of a two-dimensional square. The square on a table if lifted up for a distance equal to the length of its sides would leave a wake in the shape of a solid cube. Finally, a cube moved perpendicularly to itself (that's the direction we Spacelanders can't quite imagine) would trace the four-dimensional figure. The side of each of these shapes has the form of the previous shape. The line's two ends, for instance, are points. The square's four sides are lines. The cube's six faces are all squares. The book didn't completely explain the implications of this pattern, but it stands to reason that the four-dimensional figure would have eight sides all in the form of complete cubes. The only way I could begin to imagine this was with an expanding cube. The original cube forms one of the eight sides. Each of the six square faces as it moves outward traces something like a cube (a cube fatter at one end than the other). And the final, expanded cube forms the eighth face of this figure.

My second-to-last flight of fancy concerned Einstein's theory that time is our fourth dimension. We say that we "move" through time; it's not too hard to imagine a point leaving a linear trail on this journey. And if space itself is truly expanding through time, maybe my vision of the expanding cube isn't so fanciful. Once I thought of this, I noticed several other comparisons between time and the spatial dimensions as described in Flatland. The most notable has to do with perceiving the extra dimension. The sphere points out that the second and third dimensions really are present in Flatland and Lineland, or else its inhabitants would not be able to see each other. The square sees everyone as one-dimensional lines. But if the lines really had no thickness, he wouldn't see them at all; they must be at least as thick as the spider's web I ran into this morning without having seen it. Similarly, residents of Flatland see the Z-dimension; it's just too narrow to be measured. In the same way, I've read that we don't perceive the present as a point in time, infinitesimally dividing past from future. Augustine thought of the present as a point, so he couldn't locate it even though he intuited it. Well, maybe we intuit the present because it has some appreciable length -- a length, though, that's immeasurable by human means.

My last flight of fancy concerned God existing in a fifth dimension and seeing all of time and space at once. The sphere saw all of Flatland at once, and the square first thought of him as a god. I see two differences between God and the sphere, though. First, as the sphere moves away from Flatland, he sees more of it but in less detail; God on the other hand can see all things and all details at a glance. Second, the sphere has its existence entirely in the two dimensions he sees plus a third one, while God's existence does not depend upon the four (or more) dimensions He sees. Like the sphere, though, He can move into our space and exist in it for at least thirty-three years.