One of the genres used in this Website is is “didactic”. I classify works of fiction as “didactic” if the intention of the author is to use the fiction to teach mathematics. For example, Enzensberger's “Der Zahlenteufel” is didactic because the story does not really matter at all; the purpose of that novel is to interest the reader in the real mathematics that it discusses. The idea is that many readers who have trouble with abstract mathematical thinking will understand it better if it is included in a story and given some sort of fictional “reality”.

Many people do have trouble conceiving of higher dimensional geometry, and a reference to Flatland is now commonly used by people who are trying to help others understand this difficult concept. It does seem to help people to imagine creatures living and thinking in a two-dimensional universe and to imagine how they would perceive the three-dimensional objects that are familiar to us. So, people certainly use Flatland as a didactic work of mathematical fiction.

However, I do not think Edwin Abbott Abbott was using math that way. It was not his goal to make the math more understandable and believable by including it in a story. Quite the opposite, in fact. I think Abbott thought of math as something that people would already understand and wanted to use that math to discuss certain non-mathematical ideas that were important to him. Moreover, he hoped that by using mathematics (a topic most people agree upon), he would be able to generate some agreement in discussing something more controversial. In particular, I think that what he really wanted to write about was not mathematics but the relationships between people and the relationship between people and the supernatural.

Consider this: the main character, “a square”, of Flatland has had an experience with something from beyond his own universe, something he cannot see entirely but can only glimpse in pieces. This has changed his view of his reality, changed his view of his relationship with the other creatures of Flatland, and he wants to share that information.

Now, notice that because of his strangely repetitive name (“Abbott Abbott”), the author of Flatland could also describe himself as “A Squared”=“A^2". Abbott was a theologian. He presumably also believed that he could perceive God's existence, but not entirely, only in pieces. And although some of his ideas seem to reflect an old fashioned bias to modern readers (e.g. that the females in Flatland are line segments while the males are polygons) he was actually somewhat progressive for his day. His view of the relationships between people was also rather introspective for Victorian England.

Consequently, I believe that the role of mathematics in Flatland was to provide Abbott with a language (the language of geometry) through which he could discuss non-mathematical ideas with the readers that he otherwise could not quite put into words.

One difference between my professional area of expertise (mathematics) and my hobby (literature) is that there is much more agreement in the former than the latter. This is one reason that people like to use mathematical language to discuss controversial non-mathematical topics. (They hope that the listener will be swayed by the supposed “objectivity” of mathematics.) But, for this very same reason, I think it is likely that people will entirely disagree with my analysis of Flatland and the author's intentions. Do you think I'm right or am I completely missing the point? Please let me know your opinion on this question.

Note: The entire text of Flatland is currently available for free in an electronic (ASCII) format from the Gutenberg Project at ibiblio.org. Note for 2002: The book has been rereleased with annotations by Ian Stewart. The new version is reviewed in the AMS Notices by AK Dewdney.

Note that this book has inspired many direct sequels. See Flatterland, An Episode of Flatland, A Message Found in a Copy of Flatland, Sphereland and Spaceland.

Perhaps you're thinking of the 1965 animated version made at Harvard University, featuring the voice of Dudley Moore and directed by Eric Martin?!? At least, that's the only candidate I can think of. Believe it or not, that version is still for sale from the DER. (Unfortunately, it's not cheap. But nobody ever said that romance would be, eh?)

[Added October 2006] It appears that "Flatland the Movie" (starring Martin Sheen) is in the works. Some have suggested that it is just a joke (oh, come on...it's not THAT bad), but the trailer available at flatlandthemovie.com is rather entertaining.

Yes, it's true. You can read about two different projects at FlatlandTheFilm.com and FlatlandTheMovie.com. I'm sure someone will get a wonderful term paper out of comparing and contrasting the two movies.

Thanks, Mark! (NB When visiting the IMDB link above, take note that there is also an unrelated TV series with the same title.)

Vijay and I agreed that "Geometric Regional Novel" by Gert Jonke is not sufficiently mathematical to justify having its own entry in this database. However, since it is similar to Flatland in some ways, I will post Vijay's summary of that book here for those who might be interested in it: