Synopses & Reviews
This easy-to-read book demonstrates how a simple geometric idea reveals fascinating connections and results in number theory, the mathematics of polyhedra, combinatorial geometry, and group theory. Using a systematic paper-folding procedure it is possible to construct a regular polygon with any number of sides. This remarkable algorithm has led to interesting proofs of certain results in number theory, has been used to answer combinatorial questions involving partitions of space, and has enabled the authors to obtain the formula for the volume of a regular tetrahedron in around three steps, using nothing more complicated than basic arithmetic and the most elementary plane geometry. All of these ideas, and more, reveal the beauty of mathematics and the interconnectedness of its various branches. Detailed instructions, including clear illustrations, enable the reader to gain hands-on experience constructing these models and to discover for themselves the patterns and relationships they unearth.
Review
"There is something very pleasing about seeing paper figures that are visual displays about how math works in the world. The difference from seeing tall buildings or watching planes fly is that it is even possible for children to apply the mathematics to build something. In my opinion, origami and other constructions using paper are one of the best ways to train mathematicians that will be teaching mathematics and this is independent of the level of mathematics that will be taught. While the mathematics in this book is at the level of the college student, people that just want directions on how to make the figures can still use it."
Charles Ashbacher, Journal of Recreational Mathematics
Review
"... a triumph of embodied learning, which applies direct experience with the mathematics of objects. This book should be in every library where a chance meeting with a willing student will surely produce a new mathematician. Highly recommended."
J. McCleary, Vassar College for Choice Magazine
Synopsis
Build paper polygons and discover how systematic paper folding reveals exciting patterns and relationships between seemingly unconnected branches of mathematics.
Synopsis
Using the simple geometric idea of systematic paper folding, the authors demonstrate the beauty of mathematics and the interconnectedness of its various branches. Detailed instructions show how to build three-dimensional polygons that help the reader unearth some surprising and delightful results.
About the Author
Peter Hilton is Distinguished Professor Emeritus in the Department of Mathematical Sciences at the State University of New York (SUNY), Binghamton.Jean Pedersen is Professor of Mathematics and Computer Science at Santa Clara University, California.Sylvie Donmoyer is a professional artist and freelance illustrator.
Table of Contents
Preface; 1. Flexagons - a beginning thread; 2. Another thread - 1-period paper folding; 3. More paper folding threads - 2-period paper-folding; 4. A number-theory thread - folding numbers, a number trick, and some titbits; 5. The polyhedron thread - building some polyhedra and defining a regular polyhedron; 6. Constructing dipyramids and rotating rings from straight strips of triangles; 7. Continuing the paper-folding and number theory threads; 8. A geometry and algebra thread - constructing, and using, Jennifer's puzzle; 9. A polyhedral geometry thread - constructing braided platonic solids and other woven polyhedra; 10. Combinatorial and symmetry threads; 11. Some golden threads - constructing more dodecahedra; 12. More combinatorial threads - collapsoids; 13. Group theory - the faces of the tri-hexaflexagon; 14. Combinatorial and group theory threads - extended face planes of the platonic solids; 15. A historical thread - involving the Euler characteristic, Descartes' total angular defect, and Pólya's dream; 16. Tying some loose ends together - symmetry, group theory, homologues, and the Pólya enumeration theorem; 17. Returning to the number theory thread - generalized quasi-order and coach theorems; References; Index.