Title: Flatterland
author Ian Stewart
publisher Perseus Publishing
ISBN 0-7382-0442-0
pages 293
rating 6/10 (see below)
summary Ian Stewart tackles a wide array of mathematical topics, all related to spatial theories, following the scheme of Edward Abbot's 1884 classic, Flatland.
reviewer Stern

The original Flatland was written in 1884 by Edwin Abbott Abbott with three aims in mind: to teach about higher dimensional math, to reconcile religion and reason, and to satirize the social mores of the day. It's sad to think that best selling authors of today are primarily concerned with weight loss and "Chicken Soup for Dummies." Abbott Abbott (that's not a typo) teaches about math, argues about the nature of miracles, and advocates revolution in the role of women. Nowadays, nobody much cares for satirizing Victorian social mores, and there isn't much demand for reconciling reason and religion. However, the book has remained continuously in print for almost 120 years.

Why? People have always found the mathematical heart of the book to be worthwhile. Ian Stewart, in writing a sequel to Flatland, recognizes his audience is math teachers and students, so he more or less dispenses with the didactic intent of the original. Instead he uses its conceit — a high dimensional being in communication with its lower-dimension bretheren — to teach about a variety of 'fun math' not covered by the original (or indeed, not yet discovered at the time it was written). [An aside: I got my copy of Flatland from Professor Frank Morgan of Williams College for successfully tackling a version of the "four fours" problem posed in his Math Chat column years ago)

The content of Flatterland is therefore similar to the content of other mathematical surveys for the lay reader. We have some fractals, some chaos, non-Euclidian geometries, and more strangely, a bit of quantum mechanics (the superimposition of states and Schroedinger's cat), relativity, cosmology and string theory.

The question is whether or not this mathematical survey, with its higher dimension lectures to the learning Flatland line, is any better at teaching these topics than the thousands of more conventional surveys with the same goal. (Stewart himself has written several excellent ones).

In the opening chapters of the book, Stewart delivers many cutsie mathematical puns. The Flatland explorer "Marco Polygo", "Moobius the Cow", and so forth. It was a bit like the introduction to the 1994 "Flintstones" movie, in which every name was rendered with the names of rocks worked in. "A Steven SpielROCK production" har har har. These were irritating, and I found myself wishing that Stewart had written his book with fewer gimicks. One or two of the jokes were funny ("The Space Girls", for example), but the plot seemed to get in Stewart's way more often than it helped. The organization of this book plays to ease of marketing at the expense of its value in teaching math.

That said, the book has value. Its ambitious scope includes not only math but also quantum physics and relativity and other math-heavy topics normally taught in physics classes. These are all linked by the concept of space, defined in its broadest mathematical sense. Most books that attempt to cover so much ground end up worthless, yet this one keeps an edge, because Stewart retains a sense of wonder and majesty, and communicates this to the reader. Also because the book mixes topics that most readers will probably have encountered in the past (Heisenberg's Uncertainty Principle, transformations in the plane) with other topics which are less commonly discussed, or of more recent origin (the topology of the physical universe, and how we can test a hypothesis on this subject through observation). There are also amusing ruminations on subjects like the origins of the term "Klein Bottle" (and no, it's not as simple as you think).

The idea, I think, is not just that the book will appeal to audiences with a range of sophistication, but also that it bears multiple readings. Especially if you are studying math via other methods, you will appreciate more of this book each time you return to it.

Is the book worth reading?

Frankly, some of the chapters, including his discussion of the Mandlebrot Set, were manifestly more complex and less memorable than if he had just written about them straight. Others, such as his discussion of hyperbolic geometry, worked as well as any other lay introduction to the topic. Each chapter ends with a statement of relevancy which seems like a poor afterthought, seeming like something suggested by the editor late in the production process.

Do you like pop-math books? If so, by all means add this to the collection. However, if you are to buy only one book of this genre, as a gift for somebody for example, there are others that might be better appreciated. Stewart took risks here, and he shouldn't be punished for that, so I'm going to recommend others of his own books: go ahead and buy From Here to Infinity or Does God Play Dice. They are both very good.

Overall Ranking?

6/10 (with a 9/10 for effort).

note: the edition reviewed is an pre-publication copy from March. The book may have changed somewhat on its way to printing in May.

Stern is the president of Information Markets Corp.

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