"Like Flatland only more so" is the cover claim of the new book by my friend Ian Stewart, and, like most Flatland purists, I was very skeptical when I first encountered that description. But I ended up by writing a recommendation for the back cover that fairly accurately but slightly incompletely registers my evaluation: "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 Ian Stewart will delight in his play with words and concepts." In the spirit of Tom Lehrer's record jacket notes, I had added that if anyone did not appreciate the author's earlier puns and wordplays, he or she would likely not enjoy this book. Let that be a warning to potential readers. Let me illustrate my point.
A careful reader of Flatland will note that the author was originally indicated by a pseudonym "A Square", not "A. Square" as is commonly misstated. The "A" is his name, not his initial, so the author's search for the long form of his name seems misguided. On the other hand, it admirably suits the author's purpose, which is to set up a sequel that avoids some of the difficulties of other writers. For example, Dutch secondary school physics teacher Dionys Burger wrote Sphereland in the 1960's, from the viewpoint of A Hexagon, the grandson of A Square who plays a part in Flatland. After two further generations, the story would have to be told in a household of octagons or higher. Instead of facing this challenge, the author modifies the speed of a fundamental genetic principle in Flatland and claims that evolution proceeds much more cautiously. Instead of stating that the son of a figure automatically has more sides than the father, he posits that the sons of squares usually stay square, with only an occasional upward mutation. This allows the author to indulge in shameless wordplay by deciding that the "A" stands for Albert and that his square descendants should be named "Grosvenor", "Leicester", and "Berkeley". Even cleverer, also true to London place names, the female lead characters are lines called "Victoria" and "Lee" (for "Jubilee"). If you find such cleverness amusing, you will like the book. If not, stop here.
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. Sphereland was a natural sequel to the pedagogical purpose of its predecessor, leading the reader to appreciate concepts like the expanding universe and non-Euclidean geometry. Unfortunately it lacked the depth of the original book in its analysis of sociology, language, and theology. Other writers have included short pieces based on Flatland without claiming to be sequels. Jeff Weeks has one in The Shape of Space and Rudy Rucker includes several "updated Flatland" segments in his The Fourth Dimension. There are a number of other two-dimensional allegories, following the tradition begun by C. Howard Hinton in the early 1880s, and brought to a fascinating general treatment by A. K. Dewdney in 1984 with his The Planiverse, recently reprinted in a Copernicus edition from Springer-Verlag.
Ian 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 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 irresistibe opportunity, and this series of visitations comprises most of the book.
Having a one-dimenional narrator of course calls for an even greater suspension of judgement that 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 last chapters of the book. That is where he explains his title.
I would suggest that a reader should think of this volume as two books, one on math and one on physics. For the mathematics, read the first ten chapters, up to page 165, and then skip to Chapter 17. On another occasion, for a survey of modern geometrical physics, read chapters eleven through the end of the book. My personal taste leans toward geometry, and I think that I would have preferred getting to the message of the book right after the math tour, without having to read the physics, however interesting. It seems to me that the one-dimensionality of the narrator is more explicitly involved in the mathematics part, and only incidentally included in the physics sections. Nonetheless, I do appreciate the author's skill in portraying some of the contemporary physics topics that I had not seen so clearly expressed.
Thomas F. Banchoff (firstname.lastname@example.org) is Professor of Mathematics at Brown University and a former president of the MAA.