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This novel is a "firstperson" account of a twodimensional square who comes to appreciate a threedimensional world. The square describes his world as a plane populated by lines, circles, squares, triangles, and pentagons. Being twodimensional, the inhabitants of Flatland appear as lines to one another. They discern one another's shape both by touching and by seeing how the lines appear to change in length as the inhabitants move around one another. One day, a sphere appears before the square. To the square, which can see only a slice of the sphere, the shape before him is that of a twodimensional circle. The sphere has visited the square intent on making the square understand the threedimensional world that he, the sphere, belongs to. He explains the notions of "above" and "below," which the square confuses with "forward" and "back." When the sphere passes through the plane of Flatland to show how he can move in three dimensions, the square sees only that the line he'd been observing gets shorter and shorter and then disappears. No matter what the sphere says or does, the square cannot comprehend a space other than the twodimensional world that he knows. Only after the sphere pulls the square out of his twodimensional world and into the world of Spaceland does he finally understand the concept of three dimensions. From this new perspective, the square has a bird'seye view of Flatland and is able to see the shapes of his fellow inhabitants (including, for the first time, their insides). Armed with his new understanding, the square conceives the possibility of a fourth dimension. He even goes so far as to suggest that there may be no limit to the number of spatial dimensions. In trying to convince the sphere of this possibility, the square uses the same logic that the sphere used to argue the existence of three dimensions. The sphere, now the shortsighted one of the two, cannot comprehend this and does not accept the square's arguments—just as most of us "spheres" today do not accept the idea of extra dimensions. From 3D to 4D It's difficult for us to accept the idea because when we try to imagine even a single additional spatial dimension—much less six or seven—we hit a brick wall. There's no going beyond it, not with our brains apparently. Imagine, for instance, that you're at the center of a hollow sphere. The distance between you and every point on the sphere's surface is equal. Now, try moving in a direction that allows you to move away from all points on the sphere's surface while maintaining that equidistance. You can't do it. There's nowhere to go—nowhere that we know anyway. The square in Flatland would have the same trouble if he were in the middle of a circle. He can't be at the center of a circle and move in a direction that allows him to remain equidistant to every point of the circle's circumference—unless he moves into the third dimension. Alas, we don't have the fourdimensionsal equivalent of Abbott's threedimensional sphere to show us the way to 4D. (In mathematics, moving into ever higher dimensions is a walk in the park. See Multidimensional Math.) How about 10D? In 1919, Polish mathematician Theodor Kaluza proposed that the existence of a fourth spatial dimension might allow the linking of general relativity and electromagnetic theory. The idea, later refined by the Swedish mathematician Oskar Klein, was that space consisted of both extended and curledup dimensions. The extended dimensions are the three spatial dimensions that we're familiar with, and the curledup dimension is found deep within the extended dimensions and can be thought of as a circle. Experiments later showed that Kaluza and Klein's curledup dimension did not unite general relativity and electromagnetic theory as originally hoped, but decades later, string theorists found the idea useful, even necessary. The mathematics used in superstring theory requires at least 10 dimensions. That is, for the equations that describe superstring theory to begin to work out—for the equations to connect general relativity to quantum mechanics, to explain the nature of particles, to unify forces, and so on—they need to make use of additional dimensions. These dimensions, string theorists believe, are wrapped up in the curledup space first described by Kaluza and Klein. To extend the curledup space to include these added dimensions, imagine that spheres replace the KaluzaKlein circles. Instead of one added dimension we have two if we consider only the spheres' surfaces and three if we take into account the space within the sphere. That's a total of six dimensions so far. So where are the others that superstring theory requires? It turns out that, before superstring theory existed, two mathematicians, Eugenio Calabi of the University of Pennsylvania and ShingTung Yau of Harvard University, described sixdimensional geometrical shapes that superstring theorists say fit the bill for the kind of structures their equations call for. If we replace the spheres in curledup space with these CalabiYau shapes, we end up with 10 dimensions: three spatial, plus the six of the CalabiYau shapes, plus one of time. If superstring theory turns out to be correct, the idea of a world consisting of 10 or more dimensions is one that we'll need to become comfortable with. But will there ever be an explanation or a visual representation of higher dimensions that will truly satisfy the human mind? The answer to this question may forever be no. Not unless some fourdimensional lifeform pulls us from our threedimensional Spaceland and gives us a view of the world from its perspective. 




