*The Math Life* is a 50-minute documentary on mathematics for a general audience. It was created by filmmakers Wendy Conquest and Bob Drake and mathematician Dan Rockmore, with financial support from the NSF, and consists primarily of excerpts from interviews with fourteen well-known mathematicians, together with a number of video sequences as illustrations.

Rather than focusing on a narrow theme, *The Math Life* covers as much ground as possible given its length. It's divided into nine sections: Doing the Math, Shape, Working, Number, Proof, Uncertainty, The Real World, Connections, and The Last Word. Each section features comments from the interviews. Some of the time it shows the mathematician talking, perhaps with an overlay of another image, and some of the time it displays relevant video with the comments as a voice over.

Overall, *The Math Life* is very well done. Unfortunately, the video sequences are mixed in quality. As many of them as possible are crammed in: if a mathematician mentions drawing on fogged up windshields, an image of someone doing that is immediately overlayed on the screen. Occasionally the result is impressive. For example, when one interviewee mentions that water coming out of a fountain forms a parabola, a very nice image is displayed. I would not have thought of including it, and I'm glad the producers did. However, I found some others, like drawing on the windshield, distracting.

For me, the computer graphics were the worst part of the video sequences. Many sequences were generated by computer for no apparent reason (how is a computer-generated lobster, or sunrise over the ocean, better than the real thing?). Furthermore, some of the computer animation seemed hokey. I fear that kids watching the video will think to themselves "My video games look a lot more impressive than that." On the other hand, there are times when computer animation is used to great effect. For example, when topology comes up, the video shows a number of topologically equivalent objects morphing into each other. That's a time when computer animation has a clear and dramatic advantage over any other medium.

I wasn't terribly fond of the video sequences, but the interviews are the heart of *The Math Life*, and they are very well done. I wish the full interviews were available, since I'm sure there were many interesting comments that could not be included in a 50-minute video.

There were several comments I found particularly striking, which I intend to keep in mind for use when talking with non-mathematicians. Dorothy Wallace explains that what mathematicians do is search for patterns nobody has noticed before, and that abstraction enters the picture when one realizes there are abstract patterns that describe and control concrete patterns. Mike Freedman points out that one can be successful in mathematics by being idiosyncratic, seeing what other people don't look for and understanding things in new and different ways, even if one isn't "smarter" than everyone else in the area. David Mumford observes that randomness should be viewed not as the absence of a pattern, but as a pattern itself. Each of these comments is a brief but effective way of showing people ideas that mathematicians often take for granted and don't put into words. *The Math Life* contains many illuminating comments of this sort.

Overall, I was impressed with the selection of interview excerpts. There were a couple of times I felt that my favorite point of view wasn't represented, and an occasional comment that might give people a misleading impression. However, that is unavoidable with a video of this sort, and unlikely to do any harm.

The one thing I found awkward about the interviews in *The Math Life* is the lack of context. Most people watching it will have no idea who the people being interviewed are, except for names and affiliations that flash briefly the first time someone appears on screen. I found it entertaining to see lots of people I know, and others I don't know but have heard of. However, I suspect that a typical audience member will see just a steady stream of talking heads. It could be worse: there's some benefit simply in exposing people to the idea that mathematicians are diverse in personality, gender, race, background, interests, etc. However, I couldn't help feeling that the audience might find it easier to connect with the mathematicians if the documentary focused on fewer of them.

Despite my quibbles with the lack of context and some of the video sequences, *The Math Life* is quite an achievement. I consider it especially likely to have a positive impact on potential mathematicians who have never thought of mathematics as a plausible career choice, but even those with no intention of doing mathematics themselves should end up with a more realistic picture of what mathematicians do, and a better impression of them. The mathematics community would be healthier if more of us devoted our time to projects of this sort.

Henry Cohn is a researcher in the theory group at Microsoft Research, and an affiliate assistant professor in the University of Washington Department of Mathematics. His primary mathematical interests are number theory, combinatorics, and the theory of computation.