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Publisher:

Princeton University Press

Publication Date:

1991

Number of Pages:

144

Format:

Paperback

Price:

10.95

ISBN:

0-691-02525-8

Category:

General

The Basic Library List Committee considers this book essential for undergraduate mathematics libraries.

[Reviewed by , on ]

Fernando Q. Gouvêa

02/25/2006

*Flatland* is such a great mathematical classic that it's hardly necessary to give it a detailed review. The story is told by A. Square, a denizen of Flatland, a two-dimensional space inhabited by sentient polygons. The skeleton of the plot is fairly simple. Square is visited by a three-dimensional being, a Sphere, which of course first appears to him as a circle of varying size (produced by the intersection of the sphere with the plane). Sphere reveals to him the truth about three-dimensional space, which ends up getting Square into trouble.

The story has things to teach us on two levels. First, by making us think through what it would be like to live in a two-dimensional world, it helps us understand our own dimensionality better. Seeing how A. Square deals with trying to understand the three-dimensional world helps us begin to see how we might understand spaces of dimensions higher than three. As Banchoff's introduction says, this is a great space to start if one wants to be able to imagine (if not quite visualize) higher-dimensional spaces.

There is also an interesting layer of social satire here. Flatland is a rigidly hierarchical system in which status is directly dependent on the number of sides a polygon has. There is upwards social mobility, since the number of sides tends to increase in one's children. Women, however, are always almost one-dimensional, and so have extremely low social status. A. Square himself sees this as just "the way things are," which may make some readers a little uncomfortable. But there are many indications throughout that the author wants us to question Square's assumptions about society, just as we should question his assumptions on dimensionality.

This is a book every mathematician should read, and a book we should encourage our students to read. Pass it on!

Fernando Q. Gouvêa is professor of mathematics at Colby College.

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