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?