## Geometrical Creatures

#### The Annotated Flatland: A Romance of Many Dimensions

#### Flatterland: Like Flatland, Only More So

### 1.

One feature of the world that few people stop to puzzle over is how many dimensions it has. Although it is a little tricky to say just what a dimension is, it does seem fairly obvious that we, the objects that surround us, and the space we move about in are structured by three dimensions, conventionally referred to as length, width, and depth. Even philosophers have tended to take this for granted. Aristotle, at the beginning of *On the Heavens*, declared that “the three dimensions are all there are.” Why? Because, he argued in a somewhat mystical vein, the number three comprises beginning, middle, and end. Therefore, it is perfect and complete.

The first philosophers to talk about a “fourth dimension” were the seventeenth-century Cambridge Platonists, but they seemed to have something more spiritual than spatial in mind. One of them, Henry More, suggested in 1671 that the fourth dimension was the abode of Platonic ideas and, quite possibly, ghosts. Around the same time, Descartes took the seemingly innocuous step of adding an extra variable to his coordinate geometry, which enabled him to define four-dimensional entities he called *sursolides*. Timid contemporaries found this intolerable; in 1685, the mathematician John Wallis denounced the *sursolide* as a “Monster in Nature, less possible than a Chimera or Centaur!” Kant, in his early writings at least, flirted with the idea that three-dimensional space might be contingent; perhaps, he conjectured, God created other worlds with different numbers of dimensions. By the time he wrote the *Critique of Pure Reason*, however, Kant had decided that space was not an objective feature of reality, but something imposed on it by the mind to give order to experience. Moreover, he held, its character was irrevocably Euclidean and three-dimensional—this we know with “apodictic certainty.”

Meanwhile, in mathematics, a revolution was getting underway. Some mathematicians began to investigate curved geometries, where the shortest distance between two points was no longer a straight line; others extended the Euclidean system to spaces of more than three dimensions. These developments were brought together in a magnificent synthesis by Georg Friedrich Bernhard Riemann (1826– 1866). In an 1854 lecture before the faculty of the University of Göttingen, titled “On the Hypotheses Which Lie at the Foundation of Geometry,” Riemann toppled the Euclidean orthodoxy that had dominated mathematics—and, indeed, Western thought—for two millennia. According to Euclid, a point has zero dimensions; a line, one; a plane, two; and a solid, three. Nothing could have four dimensions. Moreover, Euclidean space is “flat”: parallel lines never meet. Riemann transcended both these assumptions, rewriting the equations of geometry so that they could describe spaces with any number of dimensions, and with any kind of curvature. (In a spherical space, for example, there is no way that parallel lines could not meet.)

The Riemannian revolution destroyed the old notion of geometry as the science of physical space. Clearly, there was nothing metaphysically necessary about three dimensions. An endless variety of other spatial worlds was possible; they could…

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