Review: Mathematica in Action

Stan Wagon and I have exchanged a number of emails about Mathematica. A few messages into the dialogue I realized that I needed to review his latest book: Mathematica® in Action: Problem Solving Through Visualization and Computation. Before I even immersed myself in the book I knew I would like it because I enjoy Stan’s playful relationship with Mathematica and I enjoy receiving the simple and elegant little programs that Stan would send me.

Here are the questions I ask myself when looking for Mathematica books to recommend.

1. Is the book current? Many Mathematica books are years old, which means these books have old Mathematica code, which may or may not run on recent releases. Of course, much of the code that focuses on the core underpinnings of Mathematica runs fine on newer releases. But, I’ve run into one too many pieces of sample code from a website that just doesn’t run on my copy of Mathematica 8.

2. Is the material accessible to beginners?

3. Does the book teach via interesting examples? I personally don’t like books that are heavy in discussions of a language’s grammar and syntax and light in engagement. Mathematica is certainly not a spectator sport. The framework demands full engagement to master its subtle and not-so-subtle concepts. What better way to engage than through interesting programming projects and case studies?

4. Do the case studies foster investigation? After I’ve copied and pasted the sample code in the book, can I modify and extend the code to discover something?

5. Is the writing style engaging? Mathematica can be very “heady” stuff. It takes a gifted writer to keep the reader interested.

In other words, does a particular book do as good a job as I hope to do via this blog to encourage the reader that playing with Mathematica can be fun?

As you might suspect, Stan Wagon has done an outstanding job meeting all five of my criteria.

1. “Mathematica in Action” is in its third edition. I wrote to Stan asking him what version of Mathematica he used to develop the code. He replied that he has made everything compatible with Mathematica 7, that there were no big architectural changes in Mathematica 8 and that no readers have reported any incompatibilities.

2. Beginners will benefit from the book as will more experienced Mathematica programmers as the explorations build from simple introductory concepts and increase in difficulty as each chapter progresses. Additionally, the book begins with a brief introduction to Mathematica. And, all of the code in the book is on a CD.

3. The best feature of the book is that it teaches via interesting examples. I’m biased in that I enjoy pure math (math for its own sake) much more than calculus, differential equations and such. For me the joy of math is in finding patterns and relationships and in exploring ideas in a right-brain way. Let me play with an idea and let Mathematica do the tedious computation and I’m happy.

   “Mathematica in Action” has twenty “primary” chapters, each with its own investigation, plus a twenty-first chapter, Miscellany. Chapter 17, Coloring Planar Maps, is but one example of an interesting and accessible investigation. I would not have imagined that an amateur mathematician such as myself would have an opportunity to investigate the four color map theorem. Wagon has made the subject accessible and interesting with the help of Mathematica’s built in Combinatorica package plus a second package, MapColoring, that is provided electronically. The chapter assumes an understanding of elementary concepts of graph theory. Like the other chapters, Chapter 17 starts from a basic idea and builds in complexity and sophistication. Here’s the description of this chapter’s exploration:

   The four-color theorem states that any planar map can be colored in four colors, where countries that share a one-dimensional border must get different colors. A “country” is the interior of a simple, closed curve. Of course, on the computer we will restrict ourselves to countries with polygonal boundaries. Inspired by the suggestion of Joan Hutchinson that Mathematica could be used to illustrate many phenomena related to the four-color theorem, I set out to write a comprehensive package to deal with planar maps, planar graphs, and coloring algorithms. As usual, careful algorithmic thinking leads to some new ideas. The main new point here is that Kempe’s false proof of 1879 can be modified by the addition of one word to yield a reasonably good algorithm for four-coloring actual maps. Full details of the implementation of Kempe’s method and its use in getting a four-coloring algorithm will be given. This chapter assume that the reader is familiar with the elementary notions of graph theory.

   Diving into Chapter 17 provides the opportunity to create and edit planar graphs, to apply the concepts learned to maps of real country data (built into Mathematica), to apply Euler’s formula (v-e+f=2) to the investigation, to study Kempe’s failed proof of the theorem, to modify Kempe’s approach, and much more.

   I could go on for a long time about how interesting the investigations are in each of the chapters. You might want to look at a detailed table of contents at the publisher’s site and get your own sense of the subjects. The preface and sample pages are available at the publisher’s site as well.

4. The case studies most definitely inspire investigation. Since each chapter combines simple ideas to build up to more sophisticated constructions, an excellent strategy for investigation is to read the description of the next idea and to play with that idea before reading further. Of course, one can also take one of the author’s ideas and explore it in a new direction. And, in the example of the four-color theorem, the curious investigator can read about the theorem in outside sources and bring those outside ideas to an exploration. Because the material in the book is about classical math problems, there will be plenty of outside material to spark continued investigation.

   Chapter 10, Penrose Tilings, is an excellent example of how one can explore concepts beyond the scope of what is in the book. A Google search of “penrose tilings” (without the quotes) yields 53,200 web results and 39,500 images. Adding the word “activity” to the search produces plenty of web-sites with their own investigations. A few of the page titles that caught my interest include “Miles of Tiles,” Playing Penrose’s Tile Game,” and “Multiple transformations of shapes.” While there is a list of references in the book, the references are, unfortunately, not cross-referenced with the chapters and the references are typically to journal articles so readers will need access to an academic library to read them.

5. The writing is remarkably good. The author clearly loves to play with Mathematica and mathematical ideas and that passion comes across in his writing. Stan Wagon starts each chapter with a very clear idea of where he’s going and he is careful to explain new ideas as we encounter them. The book contains a good number of color illustrations and is rich in Mathematica output.

I give this book two thumbs up. Books that combine interesting explorations with clear and engaging writing are hard to come by. I’m delighted to have this book in my collection.