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A Mathematical Tapestry: Demonstrating the Beautiful Unity of Mathematicsby Peter Hilton
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.
Book News Annotation:
Each of the 17 chapters provides some arithmetic, some geometry, or some algebra, say Hilton (mathematical sciences, State U. of New York-Binghamton) and Pedersen (mathematics and computer science, Santa Clara U., California), and are laced together with at least nine threads, such as paper-folding, number-theory, polyhedral, geometry, algebra, combinatorial, symmetry, group-theory, and historical. It is possible to follow a single thread through the entire book, though they do not recommend it, noting that the chapters demonstrate how a single idea in mathematics can lead in many different directions. Anyone with a knowledge of high school algebra and plane geometry should be able to follow the argument, even if they do not want to fold up the paper models. Annotation ©2010 Book News, Inc., Portland, OR (booknews.com)
Build paper polygons and discover how systematic paper folding reveals exciting patterns and relationships between seemingly unconnected branches of mathematics.
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.
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Science and Mathematics » Mathematics » General
Science and Mathematics » Mathematics » Geometry » General
Science and Mathematics » Mathematics » Geometry » Geometry and Trigonometry
Science and Mathematics » Mathematics » Group Theory
Science and Mathematics » Mathematics » Number Theory