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Cambridge University Press 978-0-521-12821-6 - A Mathematical Tapestry: Demonstrating the Beautiful Unity of Mathematics Peter Hilton and Jean Pedersen Table of Contents More information

Contents

Preface page 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 ∗ ta −1 4.2 Recognizing rational numbers of the form tb−1 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

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© in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-0-521-12821-6 - A Mathematical Tapestry: Demonstrating the Beautiful Unity of Mathematics Peter Hilton and Jean Pedersen Table of Contents More information

viii Contents 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

Contents ix 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 Polya’s´ dream 223 15.1 Polya’s´ speculation 223 15.2 Polya’s´ dream 224 ∗15.3 ...Thedreamcomestrue 229 ∗15.4 Further generalizations 232 16 Tying some loose ends together – Symmetry, group theory, homologues, and the Polya´ enumeration theorem 236 16.1 Symmetry: A really big idea 236

x Contents ∗16.2 Symmetry in geometry 239 ∗16.3 Homologues 247 ∗16.4 The Polya´ enumeration theorem 248 ∗16.5 Even and odd permutations 253 16.6 Epilogue: Polya´ 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