Flatland and Dimensions

Dimensions: you keep running into them while reading your books and attending your lectures, and in most computations they are not very difficult to handle. But have you ever tried to imagine what all those more-dimensional spaces and objects look like? For example, the four-dimensional analogon of a cube? There are lots of people who will put this aside as nonsense, not worth spending your time on, but there have been others who found this a very intriguing question. One of those people was Edwin A. Abbott, a nineteenth-century schoolmaster and clergyman who was fond of mathematics and literature.

In 1884 he wrote Flatland, a small but very amusing book which is not only about spatial dimensions, but also houses an entire Victorian society of two-dimensional creatures. Flatland is divided in two parts. In the first part a Square, inhabitant of Flatland, gives a very amusing overview of Flatland society in all its aspects. Amusing, because Flatland society reveals itself to the careful reader as a subtle satire of the Victorian society in which Abbott lived: it is, for example, clearly hierarchically organized. All inhabitants of Flatland are geometrical figures, regular or irregular.

A Flatlander with a regular shape (i. e. a polygon) automatically belongs to the upper social class; the more sides he has, the higher his position. At the top of this structure stand the priests, who are circles, and whose judgement cannot be fought. The lower class consists of triangles with two equal sides (the so called isosceles), who form the plebs. Being a woman means that you are no more than a single line, and you continuously have to beware of severely wounding a Flatlander with your sharp, needle-like end.

Polygons, by having a good marriage, can have offspring with one additional side (thus automatically of higher class); women, however, can never be more than lines. In the second part of the book the Square tells the story of his own life. On the forenight of a new millennium, the peaceful life he lived with his wife and children is disturbed by the arrival of a Sphere. The Sphere tries to convince the Square that there are THREE dimensions by drawing analogies between the different dimensions.

The Square, failing to imagine the existence of such a thing, makes an effort to chase the Sphere away, but the Sphere lifts him out of his two-dimensional world into the third dimension! At first horribly frightened, the Square becomes more and more enthusiastic about the beautiful things he sees (and could never have imagined possible). When, however, he concludes that there should be even more dimensions than these, he runs into an argument with the Sphere, who appears to be very short-sighted in these matters.

The Square is then placed back into his two dimensions, and decides to spread the word about the existence of multiple dimensions among the people of Flatland. Naturally, in Victorian Flatland these unholy theories give him eventually more trouble than he wished himself. What makes Flatland fun to read, is that it is a popular scientific work and a social satire at the same time. Abbott succeeded in wrapping these themes in an entertaining story, which seems incapable of aging, even after more than a hundred years!

Naturally, there have been many who tried to follow Abbott, however, with only a mathematical goal (indeed, some kind of sequel to Flatland exists; it is called Sphereland, but I have never read it myself). In these much more recent books, higher dimensions are again explored in a popular way; also, some attention is given to visualizing these higher dimensions by drawing analogies. This is particularly interesting because truly imagining higher spatial dimensions seems to be an almost impossible business… A challenge awaits?


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