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
