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A square, and then alert, in his own world, that is the ground, as long as he is by a forthcoming visit from the third level: one in the field, there is a disturbing manner poke more or less threw himself into to the ground, the emergence of a mysterious ring Plaza expansion and contraction. Engage in a square in the field, in a conversation with the philosophy category, in a briefing on his pointland, space, size, is a protagonist to be "his own one and all, really nothing, but he signs improve self-satisfied…           Dimensions of Existence     Mathematical fiction is, to put it very mildly indeed, not a large or prominent category of literature. One of the very few such works to have shown lasting appeal has been Edwin A. Abbott’s Flatland, first published in 1884 and still in print today.   Flatland is narrated by a creature who calls himself “A Square.” He is, in fact, a square, living in a two-dimensional world—the Flatland of the book’s title. Flatland is populated by various other living creatures, all having two-dimensional shapes of various degrees of regularity: triangles both isosceles (only two sides equal) and equilateral (all three sides equal), squares, pentagons, hexagons, and so on. There is a system of social ranks, creatures with more sides ranking higher, and circles highest of all. Women are mere line segments and are subject to various social disabilities and prejudices.   The first half of Flatland describes Flatland and its social arrangements. Much space is given over to the vexing matter of determining a stranger’s social rank. Since a Flatlander’s retina is one-dimensional (just as yours is two-dimensional), the objects in his field of vision are just line segments, and the actual shape of a stranger—and therefore his rank—is best determined by touch. A common form of introduction is therefore: “Let me ask you to feel Mr. So-and-so.”   In the book’s second half, A Square explores other worlds. In a dream he visits Lineland, a one-dimensional place, of which the author gives an 11-page description. Since a Linelander can never get past his neighbors on either side, propagation of the species presents difficult problems, which Abbott resolves with great delicacy and ingenuity.   A Square then awakens to his own world—that is, Flatland—where he is soon visited by a being from the third dimension: a sphere, who has a disturbing way of poking more or less of himself into Flatland, appearing to A Square as a circle mysteriously expanding and contracting. The sphere engages A Square in conversations of a philosophical kind, at one point introducing him to Pointland, a space of zero dimensions, inhabited by a being who “is himself his One and All, being really Nothing. Yet mark his perfect self-contentment, and hence learn this lesson, that to be self-contented is to be vile and ignorant, and that to aspire is to be blindly and ignorantly happy.”   …Spaces of zero, one, two, and three dimensions—why stop there? Probably most nonmathematicians first heard about the fourth dimension from H. G. Wells’s 1895 novel The Time Machine, whose protagonist says this:   Space, as our mathematicians have it, is spoken of as having three dimensions, which one may call Length, Breadth, and Thickness, and is always definable by reference to three planes, each at right angles to the others. But some philosophical people have been asking why three dimensions particularly—why not another direction at right angles to the other three?—and have even tried to construct a Four-Dimensional geometry.   … [Indeed,] ideas about the dimensionality of space were occurring to mathematicians here and there all through the second quarter of the 19th century. In the third quarter these occasional raindrops became a shower, allowing the great German mathematician Felix Klein (1849–1925) to make the following observation in hindsight: “Around 1870 the concept of a space of n dimensions became the general property of the advancing young generation [of mathematicians].”       — John Derbyshire, Unknown Quantity: A Real and Imaginary History of Algebra, Chapter 8, The Leap into the Fourth Dimension                 Indexes/10