Though =Flatland=, by Edwin Abbott^2 100 years ago, was exclusively about 2 things: satirizing Victorian English society and explaining a 4th Euclidean dimension, Ian Stewart's =Flatterland= is about all sorts of geometries that mathematicians play in: finite projective geometries, in which there are a finite number of points and lines, interacting in specific ways; discrete binary geometries, which described digital encoding and the error-correcting codes used in things such as CDs and DVDs; hyperbolic geometry, in which there are and infinite number of line parallel to a particular line, all going through the same point (as opposed to the usual one parallel line); and on and on and on.

However, as a math grad student, I found the treatment too shallow, the puns too egregious (especially when I saw them coming) and too unrelenting (though now I've got plenty of math jokes to add to my arsenal), and some of the descriptions are somewhat confusing -- the only reason I knew what was going on is that =I knew= all these subjects from mathematics before. And, being a physics major in my undergrad life, I wasn't thrown off by the veering into modern physics topics.

This book is more like a travel brochure - letting you know what exciting sights are to be found in the strange lands of Geometry - but not giving you much of an experience of what's there. I think this book would be a great gift for a child who's interested in math - sure, they won't understand alot of it (and they'll miss many of the puns), but then my favorite math book, =Godel, Escher, Bach=, was given to me when I was 12, and I grew into it over the years through rereading it and learning more stuff in school. I can see this book as inspiring kids to learn more about strange concepts in math, but it would be nice to have a list of followup books for doing some =real= exploring as opposed to this travel guide. (I recommend Rudy Rucker's book =The 4th Dimension= for those who want to do more thinking about the 4th dimension).

If you're a math teacher, this book can come in handy in providing was to visualize some very odd concepts in math. And, again, there's the puns that you can try out on your class.

For those interested in getting a feel for what math is about, there's actually a great secret revealed inside the book - just what makes something a geometry. The answer doesn't seem evident when one compares the odd spaces and places visited by Victoria Line and the Spacehopper, but it does become clear. I will not give that secret away, but I will give another secret away that is also shown by the book - yes, mathematicians love to play with their math.

Mary Pat Campbell, last updated 11 June 2001