Take a tour of the 'mathiverse'

Edwin Irvanus, Contributor, Jakarta

Flatterlands Like Flat Land, Only More So, By Ian Stewart, Macmillan Publishers Ltd, 2001, 301 + xi pages

During my school days, I never found mathematics an exciting and enjoyable subject. Probably the teacher or the teaching methods meant mathematics was a nightmare in class.

But my opinion changed radically after I finished reading Flatterland by Ian Stewart, which made me wish he'd been one of my schoolteachers.

This book taught me about dimensions higher than I'd ever thought existed. Stewart's story of a young girl stumbling upon a disgraceful family secret and then venturing out on a voyage of discovery gives him the perfect means to slip mathematics painlessly into the narrative. He even manages to add subtle social commentary upon our own lives and times.

Books like this are not usually done as well as this. Stewart was the recipient in 1955 of the Royal Society's Michael Faraday medal for his outstanding contribution to the public understanding of science. He is the author of numerous works, including the best-selling Does God Play Dice? and The Science of Discworld with Jack Cohen and Terry Pratchett.

It would be exceedingly difficult to dislike the book. It has a stylistic brilliance that makes somewhat easy going of what is one of the most abstruse parts of modern mathematics, the geometry of what could be an infinite number of dimensions. I must emphasis the "somewhat" in the last sentence because of the truly mind-boggling complexities of the ideas involved. The reader's mind, although well and truly boggled, is always pleasantly boggled and that is no mean feat in a work like this.

In 1884, Edwin A. Abbott combined mathematics and philosophy in the classic tale Flatland, in which he made fun of Victorian England's stiff society while introducing the concept of life in four dimensions. In 2001, Stewart (professor of mathematics at the University of Warwick in the UK) has gone a push beyond in Flatterland, an updated postmodern account of two-dimensional characters confronted with the unbelievable: multiple dimensions!

Flatland is a story about A. Square who resides in a two dimensional world and through the machinations of a Sphere is taken on a tour of three dimensions and the wonders therein. Flatterland is the long-needed update to Flatland. The heroine of the present work is Victoria Line (in Flatterland males are Squares and Females are Lines, with the males bearing the names of the great squares of London such as Grosvenor Square and the females bearing the names of subway lines, such as Victoria Line).

While Flatland was exclusively about two things: satirizing Victorian English society and explaining a fourth Euclidian dimension, Stewart's Flatterland is about all sorts of geometries with which mathematicians play infinite projective geometries, in which there are a finite number of points and lines, interacting in specific ways; discrete binary geometries, described through digital encoding and the error-correcting codes used in things such as CDs and DVDs; hyperbolic geometry, in which there are an infinite number of lines parallel to a particular line, all going through the same point (as opposed to the usual one parallel line); and so on.

Thomas F. Banchoff writes a fairly accurate evaluation for the back cover: "Flatterland challenges readers to go beyond Flatland and deal with phenomena, not just in dimensions higher than four, but in many exotic geometric realms that stretch our imagination and powers of visualization. Readers who have enjoyed other works by Stewart will delight in his play with words and concepts."

Sequels are always tricky. One of the great virtues of a sequel is that it is likely to encourage readers to go back and reread the original. Flatland is worth several readings, revealing several levels of structure -- sociological, linguistic, pedagogical, and religious.

Stewart's bold step is to have the story told by a female, a line! This allows him to present a fresh view of Flatland, from the viewpoint of a young woman seeking recognition and liberation. The chance to leave her two-dimensional confines and explore a whole range of new spaces is an irresistible opportunity, and this series of visitations comprises most of the book.

Having a one-dimensional narrator of course calls for an even greater suspension of judgment, following the adventures of sentient beings possessing two-dimensional brains. How is it possible for a creature, whose brain is contained in a line segment, to process the wealth of information in modern mathematics and science? That is what happens in the book, a grand tour of geometry, or, more precisely, of geometries, followed by almost as long a tour of the geometry of modern physics. The author has a clever answer to this difficulty, which he poses but does not develop very much, in the closing chapters of the book. That is where he explains his title.

In fact, I really have only three complaints about the book, one of which Stewart specifically addresses himself. My first problem is that it seems a bit too "gutsy" to attempt to follow up a classic book in this way, but as the author assures us in the preface, we can look at the original and see that it remains unchanged: He has not altered a word in it in attempting this sequel.

Second, it is a bit too cute. I found myself reading it while squinting my eyes to shield myself from the blinding cuteness. Finally, and most seriously, the problem is that while Abbott's Flatland was clearly well-suited for demonstrating to people the idea of higher dimensional spaces (since our own world seems "higher" dimensional to its flat inhabitants), there is no particular reason to use this setup to introduce most of the mathematical topics covered here. (Except, I suppose, to be cute!)

After all, there are many excellent conceptual materials in this book. I have read much about mathematics and string theory; even though I knew some of the stuff in this book there are some very interesting conceptual tools presented. Plus, it is fun to read. Tons of interesting puns too!