Are you a fan of *The X-Files*? Do you love to pull a science fiction book off the shelf, looking for some ideas rather than a well-written story? Have you been meaning to re-read *Flatland*? If your answer to *all* of these questions is ``yes,'' then go ahead and pick up Cliff Pickover's book *Surfing Through Hyperspace* and you won't be disappointed.

About half of this book is written as a story of two FBI agents: an unnamed Chief Investigator of unexplained phenomena, and his beautiful colleague, Sally. As we begin, Sally does not believe in beings from the fourth dimension. This changes abruptly when she is abducted by them. She becomes the pupil of the Chief and, along with the reader, is off to explore higher dimensions.

To be fair, if one includes the pages of appendices, notes, and references, only one third of the book is this attempt at a science fiction story. And it is in these notes and appendices that much of the interesting mathematics and physics is to be found. Here Pickover provides the reader with some of the current ideas and theories in physics regarding, for example, string theory, wormholes, M theory. He also includes some nice computer generated pictures of Klein bottles and a bit of history about the quaternions. Unfortunately, though, there are not nearly as many spots of this sort in the book as I had hoped. To be sure, there are numerous references for the ambitious reader to pursue which range from *Scientific American* articles to articles in *Physical Review*.

The book is divided into six lessons and nine appendices. Lesson One, "Degrees of Freedom," begins exactly where one would expect, outlining the difference between intrinsic and extrinsic geometry, and setting up the universal mysterious appeal of hyperspace. Lesson Two provides the usual and necessary analogy to *Flatland*. This chapter includes the standard examples of what a difference an extra dimension can make - examples such as the inability of a 3-D prison to keep a 4-D being locked up. Lesson Three continues the exploration of the intersection of n-dimensional space with (n+1)-dimensional space. The author is quite concerned with the question of vision: how would the eyes of a 4-D being function? How does a 3-D being appear to a 4-D being? And, of course, once again we are led to contemplate how a 4-D being appears to a 3-D being. In Lesson Four, entitled "Hyperspheres and Tesseracts," Pickover does a nice job of unraveling a hypercube, and this chapter contains plenty of graphics to aid the exploration. This is the most mathematical chapter in the book, and ends with some interesting questions such as "can you stuff a whale into an 8-D unit sphere?". Lesson Five contains explorations of reflections and orientability. The Möbius strip makes its appearance here, along with the Klein bottle, and Pickover makes all the comments one would expect at this point. The last lesson, "Gods of Hyperspace," breaks with the previous convention of dividing the chapter into the story and the science behind the story. It's a disappointment that this chapter contains nothing more than the conclusion to the saga of Sally and the Chief. Then again, maybe there's no reason to be disappointed that it has ended after all.

*Surfing Through Hyperspace* is not a textbook. It is intended for those with no formal mathematics background who are interested in exploring the idea of higher dimensions. I was expecting this, of course, but I was still a mostly unhappy reader. I was often put off by the story line, and by musings that I found totally irrelevant such as ``how easy it would be for a 4-D being to impregnate a woman without being seen'' and how one ``might be able to knife a 4-D God and trap Him forever in our world''. And questions such as what the handwriting of a 4-D being might look like just don't capture my imagination. As I alluded to in the first paragraph, Pickover relies too heavily on his science fiction story and on *Flatland* to make his point. There are numerous references to Abbott's work and to the cross sections of a 4-D object passing through 3 space. This repetition of ideas frustrated me. I concede that these may be just the points that some readers find attractive about the book; I am not one of those readers.

Michele Intermont (

intermon@kzoo.edu) is Assistant Professor of Mathematics at Kalamazoo College. Her area of specialty is algebraic topology.