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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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