Books on Special Topics  

The following sources give interesting background material not usually found in textbooks. We include grade-level suggestions for use of the material (or for student reading), where appropriate.

Ethnomathematics: A Multicultural View of Mathematical Ideas   Graph Theory and Applications
Marcia Ascher; Chapman and Hall, 1991; $42.

This is a fascinating book, which will give you a new perspective on mathematical ideas that one often takes for granted, such as the words used for counting. Of special interest is a chapter exploring Eulerian paths, a standard topic in discrete mathematics (e.g., see [17]), as an artistic aid to story-telling in the South Pacific islands. This is the source of some of the material in the module Drawing Pictures with One Line, described on page . The following is an excerpt from a review by Susan Picker LP `90.

But Ethnomathematics explores more than the topic of graph theory as it presents the mathematical ideas of number, kin relations, games of chance and strategy, and symmetric strip decorations. ...It provides a comprehensive look at the meaning and use of similar mathematical ideas in different cultures, illuminating both the mathematics and the culture in which it appears, and through this showing the value of the study of mathematics in a multicultural setting.

Alan Turing: The Enigma   Codebreaking
Andrew Hodges, Simon and Schuster, NY, 1983. This is a popular book that tells the story of Alan Turing and his role in breaking the German ENIGMA code during WWII. It makes good background reading for teaching cryptography or computer science.

Visions of Symmetry   (7-up) Escher's Tessellations
Doris Schattschneider; W.H. Freeman, 1990; $25.

This excellent book on the work of M.C. Escher provides both historical and mathematical perspective on the evolution of his tessellation prints. Beginning with his sketches of mosaics in the Alhambra, the author guides the reader through Escher's notebooks, explaining how he developed an understanding of the mathematics of tessellation (as well as his own system of classification) in order to create his work. The book is illustrated with many color reproductions from his notebooks.

An Introduction to Tessellations   (4-12)
Dale Seymour and Jill Britton; Dale Seymour, 1989; $21. This is a good source of ideas and activities for students. It illustrates basic tessellation concepts and gives a variety of ways to create interesting artwork via geometric transformations of simpler tessellations.

Teaching Tessellating Art   (K-12)
Jill Britton and Walter Britton; Dale Seymour, 1992; $21. This book contains transparency masters and activities that are very useful for teaching Escher-style tessellation techniques, as well as transformation geometry. I (Franzblau) have used these (as well as pictures from Visions of Symmetry) for hands-on workshops for K-8 teachers and high school students.

Orderly Tangles   (8-up) Knots
Alan Holden; Columbia U. Press, 1983; $32.

This is an excellent, accessible introduction to knots. The author begins with a discussion of highway interchanges, and goes on to discuss knots in general. It is illustrated with photographs of wonderful knot models made out of wooden dowels.

The Knot Book   (College-up)
Colin Adams; W. H. Freeman, 1994; $33. Adams has led many undergraduate research projects on knots; this book grew out of his notes. It develops the mathematics of knot theory, and can be read at many levels. A good source of problems about knots.

Chaos, Fractals, and Dynamics: Computer Experiments in Mathematics  
Robert Devaney, Addison-Wesley, Menlo Park, NJ, 1989.

Devaney is one of the pioneers in the applications of fractals, as well as the introduction of dynamical systems and chaos in the undergraduate and high school curriculum. See also Devaney's article in this volume [10], his video (section 6), and the Web site he developed (section 8).

Fractals, The Patterns of Chaos  
John Briggs; Simon and Schuster, 1992; $20 (paper).

The following is a recommendation from a Leadership Program participant:

When I try to explain what fractals are and how they relate to our world, I sometimes come up short. I was wandering through my favorite bookstore and discovered this book. It's terrific. In addition to the pretty fractal pictures, he shows fractals in the real world--decaying leaves, weather systems, lightning, cauliflower, the human body, etc. He even relates fractals to art and architecture. In the appendix, the author lists and evaluates fractal software both for the IBM and the Mac (both shareware and commercial software). He also lists several fractal publications. (Edward Polakowski LP `92, email posting.)

Game Theory and Strategy   (12-College)
P. Straffin; MAA, 1993; $34. This book was recommended by Joe Malkevitch [25]:

This is a wonderfully rich book about the theory of games. It covers most of the major ideas in a motivated and succinct way, and has many examples.

Fair Division: From Cake Cutting to Conflict Resolution  
Steven Brams and Alan Taylor, Cambridge Univ. Press, 1996. $18. If you're interested in estate settlement, voting, auctions, and similar topics, this book comes highly recommended by high-school teacher L. Charles Biehl LP `90 (email posting). It was written by the authors of a well-known recent work on ``envy-free'' cake-cutting.

Basic Geometry of Voting  
Donald G. Saari, Springer-Verlag, NY, 1995; $39.

This recent book provides additional mathematical background for teaching election theory and voting paradoxes, topics which are covered in the books For All Practical Purposes, Excursions in Modern Mathematics, and Discrete Mathematics through Applications, discussed in Section 2.

Unit Origami   (7-up) Geometry of Polyhedra; Graph Problems
Tomoko Fusé; Japan Publications, 1990; $19 (available from Cuisenaire or Dale Seymour; a classroom guide is also available).

This is an introduction to a style of origami that involves building complex shapes, including regular polyhedra, by fitting together simple (usually identical) units. It is often recommended by teachers. Many of the constructions are suitable for teaching to students, and can provide inspiration for exploring the geometry of polyhedra. Creating the shapes out of small units also leads one naturally to discrete problems: counting vertices, edges, and faces (as a prelude to Euler's formula), as well as to questions on coloring edges and faces.

Build Your Own Polyhedra   (5-up)
Peter Hilton and Jean Pederson; Addison-Wesley, 1988; $28
(available from Dale Seymour).

This is a beautiful book with illustrated directions for constructing polyhedra. It also introduces some of the mathematics of polyhedra, such as Euler's formula. This is a good source for classroom projects at almost all grade levels. See [18] for a discussion of the value of introducing students to polygons and polyhedra long before beginning the formal study of geometry.