In Lunda Geometry, Paulus Gerdes uses the drawings associated with the storytelling tradition of the Cokwe people as a loose starting point for an extensive study of mirror curves. A mirror curve is generated by setting mirrors in line with and between the grid points of a rectangular grid within a box whose internal walls are mirrored, then starting a beam of light at a point on the edge of the box and tracing its path through this mirrored landscape. Through a series of articles Gerdes shows how this process can be used to generate the Cokwe drawings as well as other interesting patterns including some Celtic patterns that are similar to the Cokwe drawings.
The chapters of the text do not work as a single unit building on or around a single unified theme for the text. Instead each chapter seems to represent an independent monograph on its particular subject. Consequently one can almost, though not quite, read them in any order. Also as a consequence of this format there is a not insignificant amount of overlap from one chapter to the next lending a repetitive feel to the book.
Each chapter covers some new material but in some the main purpose is simply to give a complete catalogue of particular types of patterns (See Chapter 6, Symmetrical, Closed Lunda-Polyominoes; Chapter 7, Examples of Lunda-Spirals; and Chapter 8, Lunda Strip and Plane Patterns). In the chapters whose main focus is not cataloguing Gerdes introduces some very interesting ideas and fun concepts. However, his exposition of these is frequently weighed down by thick notation with only brief introduction. For example in Chapter 2 he takes the time to explain how to add two matrices, surely a very familiar concept, but he classifies certain patterns as type d4’ or c4’ without giving a description of these classifications only a reference. In places the weight of the notation even trips up the author leading to some confusing errors. As a final general note, his “proofs” of some of his claims are loose and fast, really giving at best a nice picture of why one might suspect that something is true.
Some specific successes can be found in the text. His ability to find interesting mathematics in the cultural drawings that he discusses is inspiring and encouraging. And while it is true that the mathematics being found in these pieces of art are most likely not inherent in them (from the view point of their creators), it can be found and used to create interest in what could otherwise be difficult and tedious mathematics. This is well illustrated in Chapter 4, where he draws connections between his work and the Fibonacci sequence (a popular middle and high school topic), and in Chapter 9, where he gives suggestions for some exercises that would allow students to further investigate mirror curves on their own. It would have been nice if he had done a little less cataloging so that more of these sorts of exploratory opportunities would have been available for students.
Lunda Geometry is an interesting read, but the notation can make it a difficult read. It could be used as a reference for a teacher looking to find ways of connecting culture, art, and mathematics. In particular, it can be used as an object lesson in finding patterns and trying to carefully analyze possibilities. However, I would not recommend it as a text for a casual reader.
Chuck Rocca is an associate professor at Western Connecticut State University. His research in the past has been in the field of combinatorial group theory but recently he has been moving toward history of mathematics. In particular he is interested in the history of cryptology and the history of mathematics/science around the seventeenth century. He is also interested in the exploration of the responsible use of technology in the classroom. He can be reached at firstname.lastname@example.org.