Flatland: A Romance of Many Dimensions, and: Flatterland: like Flatland only More So (review)

Flatland: A Romance of Many Dimensions. By Edwin A. Abbott. New York: Dover, 1992. $1.50 (paper).

Flatterland: Like Flatland, only More So. By Ian Stewart. Cambridge: Perseus Books, 2001. $25.

Flatterland and its predecessor Flatland (first published in 1884) present popularized and generally accessible views of the modern geometry of their respective times. They each follow an inhabitant of a two-dimensional space whose mathematical and social viewpoint is expanded through the miraculous appearance of a visitor from a higher dimensional world. The earlier work, a product of the late Victorian era, brings us to see how worlds of one or two or four dimensions might appear to their respective inhabitants. Ian Stewart's Flatterland, an update of Flatland, gives us a perspective on the social concerns and the richer geometry of our own era. The more recent book also touches upon physics, giving brief descriptions of modern concerns about quantum theory, particle physics, and cosmology.

In Flatland, Edwin Abbott starts with an extended description of a dour and forbidding two-dimensional world. Each inhabitant has a social role fixed by birth and determined by its number of sides. Women, being line-like, have the fewest sides of all and stand at the very bottom of the rigid hierarchy. A three-dimensional visitor moves into this world and brings one of the inhabitants out to see a richer geometry. During his travels he—and the readers—get an artful geometry lesson. Upon return to Flatland, this Marco Polo-like inhabitant is disbelieved and imprisoned for life. The high-status members of the society see him as a danger to the social hierarchy and repressive political structure. The Dover edition states that:

Flatland has maintained a unique place in imaginative scientific literature for over a century. The product of Edwin A. Abbott (1838-1926), an English clergyman and Shakespearean scholar whose avocation was mathematics, this charming narrative of two-dimensional world has achieved renown both as an unequaled presentation of geometrical concepts and as a barbed satire of the hierarchical world of the Victorians.

I can endorse Dover's view of their product except that I cannot quite see the charm stressed in this quote.

On the other hand, I did find Flatterland rather charming. Its premise is that 100 years after the events described in Flatland, another traveler arrives and whisks away another Flatlander, this time a young woman, to see and learn about a wide variety of concepts in geometry and then physics. It is jokey. For example, the name of the heroine is Victoria Line and her mother is Jubilee Line. (On the London Tube—get it?) The guide on this tour is a childish toy, the "Space Hopper." Despite the jokes, the math is pretty well done. The Hopper explains the basic concepts of this mathematics and even describes some of its applications. We see lots of different kinds of spaces, many more than would have been mentioned in high school or basic college courses. We learn how geometry works in each of the spaces. It is all tied together by a modern description that defines the different spaces by their symmetry properties. It is good solid math. In addition, the treatment is light-hearted and mostly good fun.

About half-way through, the book switches from math to modern physics. Some of the physics discussions were excellent, and I found Stewart's description of Schrödinger's cat both correct and illuminating. But the book also includes some subjects, like string theory, that are not yet fully understood. Because string theory is often discussed by using spaces of two and 10 and 11 dimensions, this subject is a natural for the book, but its discussion is necessarily sketchy and not as illuminating as the math in the earlier part of the book.

Like Abbott, Stewart reaches beyond mathematics to political...