A Mathematical Tapestry: Demonstrating the Beautiful Unity of Mathematics

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This easy-to-read book demonstrates how a simple geometric idea reveals fascinating connections and results in number theory, polyhedral geometry, 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.

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Editorial Reviews

From the Publisher
"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

"... 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

"This culminating work of these two mathematicians is essential reading. A Mathematical Tapestry, which reveals how a seemingly simple paper-folding technique can lead to some of the most powerful ideas in mathematics. What I find most compelling about the book are its accessible tone and its mathematical complexity as well as the detailed connections made across so many different strands. Hilton and Pedersen present a rare opportunity to embark on a true mathematical exploration."
Kasi C. Allen, Lewis and Clark College for Mathematics Teacher

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Product Details

  • ISBN-13: 9780521128216
  • Publisher: Cambridge University Press
  • Publication date: 7/22/2010
  • Pages: 306
  • Product dimensions: 6.80 (w) x 9.60 (h) x 0.70 (d)

Meet 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.

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Table of Contents

Preface xi

Acknowledgments xv

1 Flexagons - A beginning thread 1

1.1 Four scientists at play 1

1.2 What are flexagons? 3

1.3 Hexaflexagons 4

1.4 Octaflexagons 11

2 Another thread - 1 -period paper-folding 17

2.1 Should you always follow instructions? 17

2.2 Some ancient threads 20

2.3 Folding triangles and hexagons 21

2.4 Does this idea generalize? 24

2.5 Some bonuses 37

3 More paper-folding threads - 2-period paper-folding 39

3.1 Some basic ideas about polygons 39

3.2 Why does the FAT algorithm work? 39

3.3 Constructing a 7-gon 43

3.4 Some general proofs of convergence 47

4 A number-theory thread - Folding numbers, a number trick, and some tidbits 52

4.1 Folding numbers 52

4.2 Recognizing rational numbers of the form 58

4.3 Numerical examples and why 3 × 7 = 21 is a very special number fact 63

4.4 A number trick and two mathematical tidbits 66

5 The polyhedron thread - Building some polyhedra and defining a regular polyhedron 71

5.1 An intuitive approach to polyhedra 71

5.2 Constructing polyhedra from nets 72

5.3 What is a regular polyhedron? 80

6 Constructing dipyramids and rotating rings from straight strips of triangles 86

6.1 Preparing the pattern piece for a pentagonal dipyramid 86

6.2 Assembling the pentagonal dipyramid 87

6.3 Refinements for dipyramids 88

6.4 Constructing braided rotating rings of tetrahedra 90

6.5 Variations for rotating rings 93

6.6 More fun with rotating rings 94

7 Continuing the paper-folding and number-theory threads 96

7.1 Constructing an 11-gon 96

7.2 The quasi-order theorem 100

7.3 The quasi-order theorem when t = 3 104

7.4 Paper-folding connections with various famous number sequences 105

7.5 Finding the complementary factor and reconstructing the symbol 106

8 A geometry and algebra thread - Constructing, and using, Jennifer's puzzle 110

8.1 Facts of life 110

8.2 Description of the puzzle 111

8.3 How to make the puzzle pieces 112

8.4 Assembling the braided tetrahedron 115

8.5 Assembling the braided octahedron 116

8.6 Assembling the braided cube 117

8.7 Some mathematical applications of Jennifer's puzzle 118

9 A polyhedral geometry thread - Constructing braided Platonic solids and other woven polyhedra 123

9.1 A curious fact 123

9.2 Preparing the strips 126

9.3 Braiding the diagonal cube 129

9.4 Braiding the golden dodecahedron 129

9.5 Braiding the dodecahedron 131

9.6 Braiding the icosahedron 134

9.7 Constructing more symmetric tetrahedra, octahedra, and icosahedra 137

9.8 Weaving straight strips on other polyhedral surfaces 139

10 Combinatorial and symmetry threads 145

10.1 Symmetries of the cube 145

10.2 Symmetries of the regular octahedron and regular tetrahedron 149

10.3 Euler's formula and Descartes' angular deficiency 154

10.4 Some combinatorial properties of polyhedra 158

11 Some golden threads - Constructing more dodecahedra 163

11.1 How can there be more dodecahedra? 163

11.2 The small stellated dodecahedron 165

11.3 The great stellated dodecahedron 168

11.4 The great dodecahedron 171

11.5 Magical relationships between special dodecahedra 173

12 More combinatorial threads - Collapsoids 175

12.1 What is a collapsoid? 175

12.2 Preparing the cells, tabs, and flaps 176

12.3 Constructing a 12-celled polar collapsoid 179

12.4 Constructing a 20-celled polar collapsoid 182

12.5 Constructing a 30-celled polar collapsoid 183

12.6 Constructing a 12-celled equatorial collapsoid 184

12.7 Other collapsoids (for the experts) 186

12.8 How do we find other collapsoids? 186

13 Group theory - The faces of the trihexaflexagon 195

13.1 Group theory and hexaflexagons 195

13.2 How to build the special trihexaflexagon 195

13.3 The happy group 197

13.4 The entire group 200

13.5 A normal subgroup 203

13.6 What next? 203

14 Combinatorial and group-theoretical threads - Extended face planes of the Platonic solids 206

14.1 The question 206

14.2 Divisions of the plane 206

14.3 Some facts about the Platonic solids 210

14.4 Answering the main question 212

14.5 More general questions 222

15 A historical thread - Involving the Euler characteristic, Descartes' total angular defect, and Pólya's dream 223

15.1 Pólya's speculation 223

15.2 Pólya's dream 224

15.3 … The dream comes true 229

15.4 Further generalizations 232

16 Tying some loose ends together - Symmetry, group theory, homologues, and the Pólya enumeration theorem 236

16.1 Symmetry: A really big idea 236

16.2 Symmetry in geometry 239

16.3 Homologues 247

16 4 The Pólya enumeration theorem 248

16.5 Even and odd permutations 253

16.6 Epilogue: Pólya and ourselves - Mathematics, tea, and cakes 256

17 Returning to the number-theory thread - Generalized quasi-order and coach theorems 260

17.1 Setting the stage 260

17.2 The coach theorem 260

17.3 The generalized quasi-order theorem 264

17.4 The generalized coach theorem 267

17.5 Parlor tricks 271

17.6 A little linear algebra 275

17.7 Some open questions 281

References 282

Index 286

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