Superfractals: Patterns of Nature Michael F. Barnsley Cambridge University Press (2006), 464 pp., Price: US $35.00 (hardcover) ISBN 978-0521844932 (hardcover).
Reviewer: Claire Postlethwaite, Engineering Science and Applied Mathematics, Northwestern University, Evanston, Illinois, U.S.A.

One of the first things I noticed about this book was its abundance of wonderful colourful pictures. The topic of fractals and superfractals is clearly amenable to being well-illustrated, and it is done so very well here, with clear pictures and useful captions throughout the book.

Much of the first couple of chapters of this book are taken up with a review of set theory, topology, measure theory and vector spaces. These sections are very well written and would be suitable for a first or second year undergraduate with some knowledge of calculus, even if they didn't read any of the rest of the book. The book continues on to talk about transformations of pictures and iterated function systems in chapter 3, and then moves onto details of fractals and superfractals in chapters 4 and 5.

Throughout the book there are many helpful and useful exercises and examples, and the author continually brings the reader back to earth when they could so easily be lost in higher mathematics. One of my favourite examples is in the first chapter, in which he discusses the concept of the distance between two sets which are bigger than a single point. He points out that if one is travelling by car, and sees a roadsign giving the distance to a town, this distance would normally be to the centre of the town. However, if one is driving in say, France, and sees a signpost to Italy, this distance will most likely be to the border. This was an idea I personally had never considered before, and I found it amusing and enlightening.

Overall, this is a lovely book to read, and I would recommend it to anyone intrigued by the mysteries of fractals, as well as anyone looking for a good introductory text to a wide variety of topics in pure mathematics.