This book is unlike any previously considered for the Euler Prize. Indeed, it is unlike any book on hyperbolic geometry previously written, and it is in a different universe from any book on crochet previously written. But, when you look at it, the idea makes such perfect sense that it seems inevitable.
Eugenio Beltrami, who in 1868 first modeled the non-Euclidean geometry of Bolyai and Lobachevsky by surfaces of negative curvature, actually toyed with the idea of building such surfaces. He made a small fragment of such a surface out of paper, and the idea was taken up again by William Thurston in the 1970s. But the idea did not take off, let alone reach a wide audience, until Daina Taimiņa wrote this book. By bringing crochet technology to the subject, she makes it easy and fun to construct hyperbolic surfaces that vividly illustrate essential features of non-Euclidean geometry. The book is elegant, from both a visual and mathematical point of view.
Thus, Crocheting Adventures with Hyperbolic Planes is a novel approach to geometry that has brought a whole new audience to mathematics. In this respect it has greater outreach potential than any book we have previously considered. But it is much more than that; it is perfectly capable of standing on its mathematical feet as a clear, rigorous, and beautifully illustrated introduction to hyperbolic geometry. It is truly a book where art, craft, science, and mathematics come together in perfect harmony.
Forward, by William Thurston
1. What Is the Hyperbolic Plane? Can We Crochet It?
2. What Can You Learn from Your Model?
3. Four Strands in the History of Geometry
4. Tidbits from the History of Crochet
5. What Is Non-Euclidean Geometry?
6. How to Crochet a Pseudosphere and a Symmetric Hyperbolic Plane
7. Metamorphoses of the Hyperbolic Plane
8. Other Surfaces with Negative Curvature: Catenoid and Helicoid
9. Who Is Interested in Hyperbolic Geometry Now and How Can It Be Used?
Appendix: Paper Models